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+{"text":"\\section{Introduction} \\label{sec:1}\n\n\n\n\nThe exponential decay of unstable systems \nhas been a well-known\nlaw since the early days of quantum theory.\nThe quantum description of those systems, however, allows \ndeviation from exponential decay both at shorter and longer times \\cite{Khalfin(1957)}\nthan those times over which the exponential decay law dominates. \\cite{Fonda(1978),Nakazato(1996)}\nThe short time deviation was actually found in a quantum tunneling experiment,\n\\cite{Wilkinson(1997)}\nwhile the long time deviation seems still not to have been detected in any quantum system. \n\\cite{Greenland(1988)}\nThe main cause that hinders the detection \nis considered as the smallness of the deviation at such long times. \n\\cite{Delgado(2006)}\n\n\n\n\n\nIn a recent study, a method enhancing the long time deviation was proposed.\n\\cite{Jittoh(2005)}\nThe decay of the unstable systems is theoretically modeled in \nthe time evolution of the survival probability of unstable initial state. \nThe survival probability is just the probability of finding the initial state \nin the state at a later time $t$. \nSince it is rewritten in a Fourier integral of the spectral function, \nits behavior at long times \nis determined by that of the spectral function near the threshold of the energy continuum. \n\\cite{Fonda(1978),Nakazato(1996)} \nThe essential aspect of the method is then \ndistorting the spectral function from the Breit-Wigner form and dislocating its peak\ntoward the threshold energy. \nMathematically, this causes a divergence of the spectral function, i.e., \nthe resolvent at the threshold. \nThen, it is expected that the exponential decay period disappears and \nthe survival probability at long times is increased. \nA similar idea was also considered in a related context. \\cite{Rzazewski(1982),Greenland(1988)}\nIn addition, in the analysis of the Friedrichs model \\cite{Friedrichs(1948),Exner(1985)} \nthat is often used for the study on the decays of the unstable systems, \nthe survival probability at long times sometimes exhibits a power decay law \nslower than that in cases of no divergence. \\cite{Kofman(1994),Lewenstein(2000),Nakazato(2003)}\n\n\n\n\nThese facts remind the author \nof the zero energy resonance proposed by Jensen and Kato. \\cite{Jensen(1979)}\nAccording to them,\nsuch zero energy singularities are classified by \nthe zero energy eigenstates of the total Hamiltonian\nthat either belong to or do not belong to the Hilbert space.\nThe cases where such eigenstates exist are called the exceptional cases; \notherwise they are referred to as the regular case.\nThe result in Ref. \\makebox(9,1){\\Large \\cite{Jensen(1979)}}\\hspace{1mm}\nis concerned with the three-dimensional system of the one particle in short-range potentials, \nand they proved that the time evolution operator asymptotically decreases as \n$O(t^{-1\/2})$ for the exceptional cases, \nthat is slower than $O(t^{-3\/2})$ for the regular case. \nHowever, to the author's knowledge, \nthe zero energy resonance for the Friedrichs model seems not to have been examined \nin the previous studies \nincluding Refs. \\makebox(10,1){\\Large \n\\cite{Kofman(1994),Lewenstein(2000),Nakazato(2003),Jittoh(2005)}\n}\\hspace{10mm}, \nin spite of the wide applicability of the model to the various physical systems. \n\\cite{Facchi(1998),Antoniou(2001),Kofman(1994),Rzazewski(1982),Nakazato(2003)}\n\n\n\n\n\n\n\n\n\nIn the present paper, \nwe examine the zero energy singularities of the resolvent at the threshold energy \nfor the Friedrichs model from the viewpoint of the zero energy resonance, \\cite{Jensen(1979)} \nand clarify how the asymptotic behavior of the survival probability at long times is affected. \nThe Friedrichs model\\cite{Friedrichs(1948),Exner(1985)} \ndescribes the system of the finite discrete levels coupled with the continuous spectrum, \nin which the former can be interpreted as the unstable excited levels of atoms and the latter as \nthe environmental electromagnetic fields. \\cite{Kofman(1994),Facchi(1998),Antoniou(2001)}\nWe emphasize that the model is not restricted to the single level case\n\\cite{Friedrichs(1948),Exner(1985),Kofman(1994),Lewenstein(2000),Nakazato(2003),Rzazewski(1982),\nJittoh(2005),Facchi(1998),Antoniou(2001)}\nbut, rather, the $N$-level case, \n\\cite{Exner(1985),Davies(1974),Antoniou(2003),Antoniou(2004),Miyamoto(2004),Miyamoto(2005)}\nIn addition, we assume that the square modulus of the form factors \nvanishes at zero energy with an integer power, \\cite{Facchi(1998),Antoniou(2001),Seke(1994)}\nhowever it is treated without restriction to a specific form to some extent. \nFurthermore, since we only consider the initial state spanned by the discrete states, \nit is sufficient for us to see the reduced resolvent $\\tilde{R}(z)$ \nthat is just the restriction of the resolvent to the subspace spanned by the discrete states. \nThen, \nthe Fourier integral of $\\tilde{R}(z)$ that we call the reduced time evolution operator \n$\\tilde{U}(t)$ \nenables us to calculate the survival probability. \nIn fact \nit is expressed by the square modulus of the expectation value of $\\tilde{U}(t)$ \nin a given initial state. \nWe first study the zero energy eigenstates of the model \nwhich either belong to or do not belong to the Hilbert space.\nIt is then possible to estimate correctly \nthe asymptotic behavior of $\\tilde{R}(z)$ at small energies\nboth in the regular case and the exceptional cases. \nThe latter cases are examined in detail only for the first kind,\nwhere only the zero energy eigenstate not belonging to the Hilbert space exists. \nOn the basis of this analysis, \nwe can derive the long-time asymptotic formula for $\\tilde{U}(t)$ in those cases. \nIn particular,\nthe logarithmic decay proportional to $(\\log t)^{-1}$ of $\\tilde{U}(t)$\nis shown to occur in the exceptional case of the first kind for our form factors,\nwhich is extremely slower than the power decays in the regular case\nand in the exceptional case for another type of form factor. \n\\cite{Kofman(1994),Lewenstein(2000),Nakazato(2003)}\nThese results are shown in Theorems \\ref{thm:rffimLong} and \\ref{thm:rffLong1st}.\n\n\n\n\nThe organization of the paper is as follows.\nWe first explain in Sec. \\ref{sec:3} the $N$-level Friedrichs model \nwith an appropriate Hilbert space, and then \nin Sec. \\ref{sec:4} we introduce the reduced resolvent $\\tilde{R}(z)$. \nSection \\ref{sec:5} is devoted to the identification of\nzero energy eigenstates in this model.\nIt is then possible to obtain\nthe asymptotic expansion of $\\tilde{R}(z)$ at small energies\nin Sec. \\ref{sec:5.5}, where we examine the regular and the exceptional case of the first kind.\nBy making sure of the relation between $\\tilde{R}(z)$ and $\\tilde{U}(t)$ in Sec. \\ref{sec:6},\nthe asymptotic formula for $\\tilde{U}(t)$ in the regular and the exceptional case\nof the first kind\nare derived in Sec. \\ref{sec:7} . Concluding remarks are given in Sec. \\ref{sec:8}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Hilbert space and the $N$-level Friedrichs model}\n\\label{sec:3}\n\nWe shall use bracket notation; however it can be understood in a standard treatment\nbased on functional analysis as in Refs.\n\\makebox(6,1){\\Large \\cite{Exner(1985),Davies(1974)}}\\hspace{7.5mm}.\nThe Hilbert space describing the unstable multilevel systems is here defined by\n\\begin{equation}\n{\\cal H}:=\\mathbb{C}^N \\oplus L^2 ((0,\\infty )).\n\\label{eqn:5.10}\n\\end{equation}\nA vector $\\ket{c} \\in \\mathbb{C}^N$ is expressed by $\\ket{c}=\\sum_{n=1}^{N} c_n \\ket{n}$,\nwhere $\\ket{n}$'s are the orthonormal basis of $\\mathbb{C}^N$,\nso that $\\braket{n}{n'}=\\delta_{nn'}$, where $\\delta_{nn'}$ is Kronecker's delta.\n$L^2 ((0,\\infty ))$ is the Hilbert space of\nthe square-integrable complex function $\\ket{f}$ of the variable $\\omega$ defined on $(0,\\infty )$, i.e.,\n\\begin{equation}\n\\ket{f} \\in L^2 ((0,\\infty )) \\Leftrightarrow \\int_0^{\\infty} |f(\\omega)|^2 d\\omega <\\infty.\n\\label{eqn:5.10b}\n\\end{equation}\nIn a standard notation using\nthe (generalized) eigenstate $\\ket{\\omega}$ of the multiplication operator by $\\omega$,\n$\\ket{f}$ is nothing more than\n\\begin{equation}\n\\ket{f}=\\int_{0}^{\\infty} f(\\omega ) \\ket{\\omega} d\\omega,\n\\label{eqn:5.15}\n\\end{equation}\nwhere $\\braket{\\omega}{\\omega'}=\\delta (\\omega -\\omega')$ and\n$\\delta (\\omega -\\omega')$ is Dirac's delta.\nThen, an arbitrary vector $\\ket{\\Psi} \\in {\\cal H}$\ncomposed of $\\ket{c} \\in \\mathbb{C}^N$ and $\\ket{f} \\in\nL^2 ((0,\\infty ))$\nis denoted by\n\\begin{equation}\n\\ket{\\Psi}:=\n| c \\rangle +\\ket{f} ,\n\\label{eqn:5.20}\n\\end{equation}\nand the inner product between any two vectors $\\ket{\\Psi}$ and\n$\\ket{\\Phi} \\in {\\cal H}$ is defined by \\cite{innerproduct}\n\\begin{equation}\n\\braket{\\Phi}{\\Psi}\n:=\\braket{d}{c}+ \\braket{g}{f}\n=\\sum_{n=1}^{N} d_n^* c_n + \\int_0^{\\infty} g^* (\\omega ) f(\\omega) d\\omega ,\n\\label{eqn:5.30}\n\\end{equation}\nwhere ($^*$) denotes the complex conjugate and\n$\\ket{\\Phi}=\\ket{d}+\\ket{g}$\nwith $\\ket{d} \\in \\mathbb{C}^N$ and $\\ket{g} \\in L^2 ((0,\\infty ))$.\nIn particular, the associated norm of $\\ket{\\Psi}$ is\n$\\| \\Psi \\| := \\sqrt{\\braket{\\Psi}{\\Psi}}$, \nwhich is ensured to be finite for all $\\ket{\\Psi} \\in \\cal{H}$.\n\n\n\n\n\nLet us now introduce the $N$-level Friedrichs model\nfor a description of the decay of the unstable multilevel systems.\nThe Hamiltonian $H$ of this model is defined by\n\\begin{equation}\nH:=H_0 + \\lambda V,\n\\label{eqn:5.60}\n\\end{equation}\nwhere\n$H_0$ is the free part and $V$\nthe interaction part of $H$, respectively,\nand $\\lambda \\in \\mathbb{R}$ is the coupling constant.\n$H_0$ is defined by\n\\begin{equation}\nH_0 :=\\sum_{n=1}^N \\omega_n \\ketbra{n}{n}+\\int_{0}^{\\infty} \\omega \\ketbra{\\omega}{\\omega}d\\omega,\n\\label{eqn:5.40}\n\\end{equation}\nwhere $\\omega_n \\in \\mathbb{R}$ with $\\omega_1 \\leq \\omega_2 \\leq \\cdots \\leq \\omega_N$,\nand its action is prescribed by\n$H_0 \\ket{\\Psi}=\\sum_{n=1}^N \\omega_n c_n \\ket{n}+ \\omega \\ket{f}$\nfor any $\\ket{\\Psi}=\\ket{c}+\\ket{f} \\in D(H_0)$.\n$D(H_0)$ is the domain of $H_0$ defined by\n$\nD(H_0 ):=\\left\\{ \\ket{\\Psi} \\in {\\cal H} ~\\left|~\n\\int_{0}^{\\infty} |\\omega f(\\omega )|^2 d\\omega < \\infty \\right. \\right\\}\n$,\nand then the self-adjointness of $H_0$ is guaranteed.\nThe interaction part $V$ is defined by\n\\begin{equation}\nV:=\\sum_{n=1}^{N} \\int_{0}^{\\infty}\n\\bigl[ v_n^*(\\omega) \\ketbra{n}{\\omega} +v_n(\\omega) \\ketbra{\\omega}{n} \\bigr] d\\omega,\n\\label{eqn:5.70}\n\\end{equation}\nwhere we assumed that $\\ket{v_n} \\in L^2 ((0, \\infty ))$. \n\\cite{threshold}\nWe call the $L^2$-functions $v_n(\\omega)$ the form factors of the system under consideration. \nThe action of $V$ is then given by\n$V \\ket{\\Psi}=\\sum_{n=1}^N \\braket{v_n}{f} \\ket{n} + \\sum_{n=1}^{N} c_n \\ket{v_n}$\nfor any $\\ket{\\Psi}\\in {\\cal H}$.\nNote that since $D(V)=\\cal{H}$ and $V$ is a bounded self-adjoint operator,\n$H$ is self-adjoint with the domain $D(H)=D(H_0)\\cap D(V)=D(H_0)$.\n\n\n\n\n\n\nIn the whole of the paper,\nwe will restrict ourselves to the special kind of the form factor:\nSuppose that the product $v_m^* (\\omega) v_n (\\omega)$\nbetween an arbitrary pair of $v_m^* (\\omega)$ and $v_n (\\omega)$\nis written in a rational function, i.e.,\nit is expressed by\n\\begin{equation}\nv_m^* (\\omega) v_n (\\omega) =\\frac{\\pi_{mn} (\\omega) }{ \\rho_{mn} (\\omega)},\n\\label{eqn:formfactor1}\n\\end{equation}\nwhere $\\pi_{mn} (\\omega)$ and $\\rho_{mn} (\\omega)$ are the polynomials \nof the degree $M_{mn}$ and $N_{nm}$, respectively, \nand we assume that $\\rho_{mn} (\\omega)$ has no zeros in $[0, \\infty )$. \nIt is also assumed that $M_{mn}+2 \\leq N_{mn}$ and $\\pi_{mn} (0)=0$. \nThe former condition ensures that $v_m^* (\\omega) v_n (\\omega)$ is integrable in $[0, \\infty )$ \nand $\\lim_{\\omega\\to\\infty} v_m^* (\\omega) v_n (\\omega)=0$, \nwhile the latter condition implies that \nthe rational function $v_m^* (\\omega) v_n (\\omega)=O(\\omega)$ as $\\omega\\to +0$. \nThe form factors with such properties are often found in actual systems\ninvolving the process of the spontaneous emission of photons from the hydrogen atom,\n\\cite{Facchi(1998),Seke(1994)}\nand quantum dots. \\cite{Antoniou(2001)}\nWe do not treat the algebraic form factor \nthat behaves as $O(\\omega^{1\/2})$ as $\\omega\\to +0$ instead, \nassociated with \nthe photodetachment of electrons from the negative ion\n\\cite{Rzazewski(1982),Haan(1984),Lewenstein(2000),Nakazato(2003)}\nand the spontaneous emission from the atoms in the photonic crystals;\\cite{Kofman(1994)} \nhowever, the discussion developed in the following could be easily extended to such a case.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Reduced resolvent for the $N$-level Friedrichs model}\n\\label{sec:4}\n\n\n\nIn the following, we introduce the reduced resolvent that is simply the restriction\nof the resolvent of $H$ to the $N$ dimensional subspace $\\mathbb{C}^N \\oplus \\{ 0 \\}$.\nSince we only consider the initial state belonging to this subspace,\nthis restriction is sufficient for our study.\nIn a technical sense, this treatment corresponds to the appropriate choice\nof a weighted Sobolev space. \\cite{Jensen(1979),Murata(1982)}\nIn the later sections, we do not distinguish the vector in $\\mathbb{C}^N$ from\nthat in $\\mathbb{C}^N \\oplus \\{ 0 \\}$.\nAfter introducing the reduced resolvent,\nwe see the existence of the boundary values of\nthe reduced resolvent on the positive real line.\nThe large-energy behavior of the reduced resolvent is also examined,\nwhich is necessary for a rigorous estimation of the long time behavior of\nthe reduced time evolution operator.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Reduced resolvent}\n\n\n\nThe resolvent of $H_0$ and that of $H$ are defined by \n$R_0(z ) =(H_0 -z )^{-1}$ and $R(z ) =(H -z )^{-1}$, respectively, \nwhere we assume that $z \\in \\mathbb{C}\\backslash (\\sigma (H_0) \\cup \\sigma (H))$.\n$\\sigma (H_0)$ (or $\\sigma (H)$) is the spectrum of $H_0$ (or $H$),\ni.e., the set of the singular points of $R_0(z)$ (or R(z)).\nThen, we have\n\\begin{eqnarray}\nR(z ) - R_0(z )\n&=& -R_0(z )  V R(z )\n\\label{eqn:2.70} \\\\\n&=& -R_0(z ) V R_0(z ) +R_0(z ) V R_0(z )  V R(z ) .\n\\label{eqn:2.40}\n\\end{eqnarray}\nFrom Eq. (\\ref{eqn:2.70}), one obtains the equation $R(z)= (1+R_0(z ) V )^{-1} R_0(z )$,\nwhich is the starting point of the asymptotic expansion of $R(z)$\nfor the short-range potential systems. \\cite{Jensen(1979)}\nOn the other hand, we instead start from Eq. (\\ref{eqn:2.40}) to obtain\n\\begin{equation}\n[H_0 -z - V R_0(z )  V ] R(z ) = 1 -V R_0(z ) .\n\\label{eqn:2.110}\n\\end{equation}\nThis equation can be solved for our model if \nwe confine ourselves to the state subspace $\\mathbb{C}^N \\oplus \\{ 0 \\}$.\n\\cite{Exner(1985)}\nIn fact, from the fact that $\\bra{n} V R_0(z ) \\ket{n'} =0$\nfor any $\\ket{n}$ and $\\ket{n'} \\in \\mathbb{C}^N \\oplus \\{ 0 \\}$,\nEq. (\\ref{eqn:2.110}) reads\n\\begin{equation}\n\\sum_{m=1}^N\n[(\\omega_{n} -z ) \\delta_{n m} -\\lambda^2 S_{n m}(z) ]\n\\tilde{R}_{m n'}(z )\n=\\delta_{nn'} ,\n\\label{eqn:.20}\n\\end{equation}\nwhere $S (z ) $ and $\\tilde{R} (z ) $ are\nthe $N \\times N$ matrix defined\nwith the matrix components\n\\begin{equation}\nS_{mn}  (z )\n:=\n\\bra{m} V R_0(z ) V \\ket{n}\n=\n\\int_{0}^{\\infty}\n\\frac{v_m^* (\\omega) v_n (\\omega)}{\\omega -z}d \\omega,\n\\mbox{ and }\n\\tilde{R}_{mn}  (z ):=\\bra{m} R(z ) \\ket{n}.\n\\label{eqn:.40}\n\\end{equation}\nWe call $S(z) $ and $\\tilde{R}(z)$ the {\\it self energy} and the {\\it reduced resolvent}, \nrespectively.\nNote that $S(z)$ can be analytically defined for all $z \\in \\mathbb{C}\\backslash [0, \\infty)$.\nFor a later convenience, we also introduce the matrix ${K_0}$ and $K(z)$ by\n\\begin{equation}\n{K_0}_{mn}:=\\bra{m} H_0 \\ket{n}=\\omega_n  \\delta_{m n},\n\\quad \\mbox{and} \\quad\nK_{mn}(z):=[K_0 -\\lambda^2 S(z)]_{mn},\n\\label{eqn:.55}\n\\end{equation}\nrespectively. Then, Eq. (\\ref{eqn:.20}) is equivalent to\n\\begin{equation}\n[K(z)-z]\n\\tilde{R} (z ) =1, ~~~\n\\forall z \\in \\mathbb{C}\\backslash (\\sigma (H_0) \\cup \\sigma (H)),\n\\label{eqn:.60}\n\\end{equation}\nwhich implies that\n${\\rm det}[K(z)-z]\n{\\rm det} [\\tilde{R} (z ) ] =1 $,\nso that\n${\\rm det}[K(z)-z] \\neq 0$ and\n${\\rm det} [\\tilde{R} (z ) ] \\neq 0 $\nfor all $z \\in \\mathbb{C}\\backslash (\\sigma (H_0) \\cup \\sigma (H))$.\nThus, the inverse of $K(z)-z$\nexists, and we have\n\\begin{equation}\n\\tilde{R} (z )\n=\n[K(z)-z]^{-1}, ~~~\n\\forall z \\in \\mathbb{C}\\backslash (\\sigma (H_0) \\cup \\sigma (H)).\n\\label{eqn:.70}\n\\end{equation}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The boundary values of $\\tilde{R} (z )$ and its large energy behavior}\n\\label{subsec:4.2}\n\nFrom the assumption on the form factors, every $v_m^* (\\omega )v_n (\\omega )$ is \ncontinued to the whole complex plane as a meromorphic function which we merely denote \nas $v_m^* (z)v_n (z)$. \nIt may have a finite number of poles.\nThen, it follows from Lemma \\ref{lm:formfactor1} that\n${S} (z)$ can be reduced to the form\n\\begin{equation}\n{S} (z)\n=\nS(0) + A(z) -(\\log (-z)) {\\mit \\Gamma}(z) ,\n\\label{eqn:.76}\n\\end{equation}\nwhere we choose\n${\\rm arg} (-z) ={\\rm arg} (z)-\\pi$ and $0<{\\rm arg} (z)<2\\pi$.\nThe matrix ${\\mit \\Gamma}(z)$ is defined with the components\n\\begin{equation}\n{\\mit \\Gamma}_{mn} (z):=v_m^* (z)v_n (z),\n\\label{eqn:.100}\n\\end{equation}\nand satisfies ${\\mit \\Gamma}(z)\\to 0$ as $z\\to 0$ in $\\mathbb{C}$.\n$S(0)$ is the limit of $S(z)$ as $z\\to 0$ in $\\mathbb{C}\\backslash [0, \\infty)$,\nwhich turns out to be unique. Indeed, as we see \nfrom the Appendix in Ref. \\makebox(0,1){\\Large \\cite{Miyamoto(2005)}}\\hspace{4mm},\n$S_{mn}(0)=\\int_{0}^{\\infty}v_m^* (\\omega) v_n (\\omega)\/\\omega  \\ d \\omega$.\n$A(z)$ is then defined through Eq. (\\ref{eqn:.76}) and becomes\na Hermitian matrix for real $\\omega$,\nwhose components are the rational functions of $z$\nwithout any singularity on $[0, \\infty )$.\nBy definition, $A(z)$ satisfies $A(z)\\to 0$ as $z\\to 0$.\nOne sees that the boundary values of $S (z )$\nat the half line $(0,\\infty )$ exist and satisfy \\cite{Exner(1985)}\n\\begin{equation}\n\\lim_{\\epsilon \\to +0}\nS (\\omega \\pm i\\epsilon)\n=\n{D} (\\omega ) \\pm \\pi i {\\mit \\Gamma}(\\omega ),\n\\label{eqn:.80a}\n\\end{equation}\nwhere\n\\begin{equation}\n{D} (\\omega )\n:=\nS(0) + A(\\omega) -(\\log \\omega) {\\mit \\Gamma}(\\omega) .\n\\label{eqn:.80b}\n\\end{equation}\nThe matrix ${D} (\\omega )$ is just\nof the components\n\\begin{equation}\n{D}_{mn} (\\omega ):=P \\int_{0}^{\\infty}\n\\frac{v_m^* (\\omega' )v_n (\\omega' )}{\\omega'-\\omega} d\\omega' ,\n\\label{eqn:.90}\n\\end{equation}\nwhere $P$ denotes the principal value of the integral.\nNote that both ${D} (\\omega )$ and ${\\mit \\Gamma}(\\omega )$ are Hermitian\nmatrices and ${\\mit \\Gamma}(\\omega ) \\geq 0$.\n\n\n\n\nIn all the discussion developed in the following, we assume that\n\\begin{equation}\n{\\rm det}[K^\\pm (\\omega)-\\omega]\\neq 0, ~~~ \\forall \\omega >0,\n\\label{eqn:.110}\n\\end{equation}\nwhere we introduced\n\\begin{equation}\nK^\\pm (\\omega):= \\lim_{\\epsilon \\to +0} K(\\omega \\pm i\\epsilon )\n={K_0} -\\lambda^2 {D} (\\omega )\\mp \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega ),\n ~~~ \\forall \\omega >0.\n\\label{eqn:.112}\n\\end{equation}\nIt is worth noting that condition (\\ref{eqn:.110}) is\nequivalent to the requirement of no positive eigenvalues of $H_0$,\nwhose eigenstates are normalizable.\nIndeed, if\n${\\rm det}[K^\\pm (\\omega)-\\omega]=0\n$ for some $\\omega >0$, there is a non-zero vector\n$\\ket{\\eta}=\\sum_{n=1}^N \\eta_n \\ket{n} \\in \\mathbb{C}^N$ such that\n$[K^\\pm (\\omega)-\\omega] \\ket{\\eta}=0 $.\nSince both ${D} (\\omega )$ and ${\\mit \\Gamma}(\\omega )$ are Hermitian matrices,\nthe latter equation implies that\n\\begin{equation}\n\\bra{\\eta} [{K_0}-\\omega -\\lambda^2 {D} (\\omega )]\n\\ket{\\eta} =0 ~~\n\\mbox{and} ~~\n\\bra{\\eta} {\\mit \\Gamma}(\\omega ) \\ket{\\eta}\n=\\left|\\sum_{n=1}^{N} v_n(\\omega)\\eta_n \\right|^2=0 .\n\\label{eqn:.167}\n\\end{equation}\nNote that the latter relation means that\n${\\mit \\Gamma}(\\omega ) \\ket{\\eta}=0$ because ${\\mit \\Gamma}(\\omega ) \\geq 0$.\nThus, Eq. (\\ref{eqn:.167}) implies that\n${\\mit \\Gamma}(\\omega ) \\ket{\\eta} =0$ and\n$[{K_0} -\\lambda^2 {D} (\\omega )]\\ket{\\eta}=\\omega \\ket{\\eta}$, i.e.,\n\\begin{equation}\n\\sum_{n=1}^{N} v_n(\\omega)\\eta_n =0,\n~~ \\mbox{and} ~~\n\\sum_{n=1}^{N}\n[\\omega_{m}\\delta_{mn} -\\lambda^2 {D}_{mn} (\\omega )]\n\\eta_n =\\omega \\eta_m ,\n\\label{eqn:.169}\n\\end{equation}\nfor all $m=1, \\ldots, N$.\nThis is merely the condition for the existence of a positive\neigenvalue $\\omega$ of $H$. \\cite{Miyamoto(2005)}\n\n\n\n\n\n\n\\begin{lm\n\\label{lm:100}\nUnder the assumption (\\ref{eqn:.110}), it holds that\n$\\tilde{R}^{\\pm}(\\omega)\n:=\\lim_{\\epsilon \\to +0} \\tilde{R}(\\omega \\pm i \\epsilon )\n$\nexists for all $\\omega >0$ and\n$\\tilde{R}^{\\pm}(\\omega) =[K^\\pm (\\omega)-\\omega]^{-1}$.\n\\end{lm}\n\n\n{\\sl Proof} :\nUnder the assumption (\\ref{eqn:.110}),\n$[K^\\pm (\\omega)-\\omega]^{-1} $ exists. Then\n\\begin{eqnarray}\n&&\n\\|\n[K^\\pm (\\omega)-\\omega]^{-1} -\\tilde{R}(\\omega \\pm i \\epsilon )\n\\|\n\\nonumber \\\\\n&\\leq&\n\\|\n[K^\\pm (\\omega)-\\omega]^{-1}\n\\|\n\\|\n\\pm i\\epsilon +\\lambda^2 S (\\omega \\pm i \\epsilon)\n-\\lambda^2 {D} (\\omega )\n\\mp \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\|\n\\|\n\\tilde{R}(\\omega \\pm i \\epsilon )\n\\| .\n\\label{eqn:.120}\n\\end{eqnarray}\nNote that for any nonzero $\\ket{y} \\in \\mathbb{C}^N$ ($\\neq 0$)\nthere is a nonzero $\\ket{x} \\in \\mathbb{C}^N$ such that\n$\\ket{y}=[K(\\omega \\pm i \\epsilon )-\\omega \\mp i \\epsilon]\\ket{x}$.\nWe then obtain\n\\begin{eqnarray}\n\\frac{\n\\|\n\\tilde{R}(\\omega \\pm i \\epsilon )\\ket{y}\n\\|\n}\n{\\| y\\|}\n&\\leq&\n\\frac{\\| x \\|}\n{\n\\bigl|\n\\|\n[K^\\pm (\\omega)-\\omega] \\ket{x}\n\\|\n-\n\\|\n[\\pm i\\epsilon +\\lambda^2 S (\\omega \\pm i \\epsilon)\n-\\lambda^2 {D} (\\omega )\n\\mp \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )]\\ket{x}\n\\|\n\\bigr|\n}\n\\nonumber \\\\\n\\!&\\leq&\\!\n\\biggl[\n\\inf_{\\ket{x} \\neq 0, \\ket{x}\\in \\mathbb{C}^N}\\! \\!\n\\frac{\n\\|[K^\\pm (\\omega)-\\omega]\\|\n}{\\| x \\|}\n\\nonumber\\\\\n&&\n-\n\\|\n\\pm i\\epsilon +\\lambda^2 S (\\omega \\pm i \\epsilon)\n-\\lambda^2 {D} (\\omega )\n\\mp \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\|\n\\biggr]^{-1} \\hspace{-2mm},\n\\label{eqn:.130}\n\\end{eqnarray}\nwhich implies that\n\\begin{equation}\n\\mathop{\\overline{\\lim}}_{\\epsilon \\to +0}\n\\|\n\\tilde{R}(\\omega \\pm i \\epsilon )\n\\|\n\\leq\n\\left[\n\\inf_{\\ket{x} \\neq 0, \\ket{x}\\in \\mathbb{C}^N}\n\\frac{\n\\|K^\\pm (\\omega)-\\omega\\|\n}{\\| x \\|}\n\\right]^{-1} < \\infty ,\n\\label{eqn:.140}\n\\end{equation}\nwhere\nthe norm of an $N\\times N$ matrix $A$ is defined by\n$\\| A\\|=\n\\sup_{\\ket{x} \\neq 0, \\ket{x}\\in \\mathbb{C}^N} \\| A\\ket{x}\\| \/\\| x\\|$.\nIn Eq. (\\ref{eqn:.130}), we used the fact that\nthere is some $\\epsilon_0 >0$ such that\nfor any positive $\\epsilon < \\epsilon_0$\nand for any non zero $\\ket{x} \\in \\mathbb{C}^N$\n\\begin{eqnarray}\n\\frac{\n\\|\n[K^\\pm (\\omega)-\\omega] \\ket{x}\\|\n}{\\| x \\|}\n&\\geq&\n\\inf_{\\ket{x} \\neq 0, \\ket{x}\\in \\mathbb{C}^N}\n\\frac{\n\\|[K^\\pm (\\omega)-\\omega] \\ket{x}\\|\n}{\\| x \\|}\n\\nonumber \\\\\n&>&\n\\|\n\\pm i\\epsilon +\\lambda^2 S (\\omega \\pm i \\epsilon)\n-\\lambda^2 {D} (\\omega )\n\\mp \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\|\n\\nonumber \\\\\n&\\geq&\n\\frac{\n\\|\n[\\pm i\\epsilon +\\lambda^2 S (\\omega \\pm i \\epsilon)\n-\\lambda^2 {D} (\\omega )\n\\mp \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n]\\ket{x}\n\\|\n}{\\| x \\|},\n\\label{eqn:.150}\n\\end{eqnarray}\nwhere the assumption (\\ref{eqn:.110}) is taken into account.\nThus, by using Eq. (\\ref{eqn:.140}), Eq. (\\ref{eqn:.120}) leads us to\n\\begin{equation}\n\\lim_{\\epsilon \\to +0}\n\\|\n[K^\\pm (\\omega)-\\omega]^{-1}\n-\\tilde{R}(\\omega \\pm i \\epsilon )\n\\|\n=0,\n\\label{eqn:.160}\n\\end{equation}\nwhich completes the proof of the lemma.\n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{lm\n\\label{lm:Large-omega}\n:\nUnder the assumption (\\ref{eqn:.110}),\n$\\tilde{R}^\\pm (\\omega)$ is $r$-times differentiable in\n$\\omega \\in (0, \\infty)$, and it behaves as\n\\begin{equation}\n\\frac{d^r \\tilde{R}^\\pm (\\omega)}{d\\omega^r}\n=O(\\omega^{-r-1}) \\mbox{ as } \\omega \\to \\infty .\n\\label{eqn:Large10}\n\\end{equation}\n\\end{lm}\n\n{\\sl Proof} :\nWe first show the statement for $r=0$.\nFrom the assumption on the form factors and Lemma \\ref{lm:formfactor1}, one sees that\n\\begin{equation}\n\\lim_{\\omega \\to \\infty} {D} (\\omega) =0 \\mbox{ and }\n\\lim_{\\omega \\to \\infty} {\\mit \\Gamma} (\\omega) =0.\n\\label{eqn:Large40}\n\\end{equation}\nSince from the assumption (\\ref{eqn:.110})\n$K^\\pm(\\omega)-\\omega$ is invertible\nfor all $\\omega >0$,\nit holds that there is some positive $\\bar{\\omega}  > \\omega_N$\nsuch that for any $\\omega > \\bar{\\omega}$,\n\\begin{eqnarray}\n\\frac{\n\\|\n\\tilde{R}^\\pm (\\omega)\\ket{y}\n\\|\n}\n{\\| y\\|}\n&\\leq&\n\\frac{\\| x \\|}\n{\\displaystyle\n\\|\n({K_0}-\\omega)\\ket{x}\n\\|\n-\\lambda^2\n\\|\n[{D} (\\omega) \\pm \\pi i {\\mit \\Gamma}(\\omega)]\\ket{x}\n\\|\n}\n\\label{eqn:Large30a}\\\\\n&\\leq&\n\\frac{1}\n{\n\\omega -\\omega_N -\\lambda^2\n\\|\n{D} (\\omega) \\pm \\pi i {\\mit \\Gamma}(\\omega)\n\\|\n}\n=\nO(\\omega^{-1}),\n\\label{eqn:Large30}\n\\end{eqnarray}\nwhere\nthe last inequality is obtained as follows:\nwe can choose some positive $\\bar{\\omega}  > \\omega_N$\nsuch that for any $\\omega > \\bar{\\omega}\n$\\begin{equation}\n\\frac{\n\\|\n[{K_0}-\\omega]\\ket{x}\n\\|\n}{\\| x \\|}\n\\geq\n\\min_n \\{ \\omega-\\omega_n \\} =\\omega-\\omega_N\n>\n\\lambda^2 \\|{D} (\\omega) \\pm \\pi i {\\mit \\Gamma}(\\omega) \\|\n\\geq\n\\lambda^2\n\\frac{\n\\|\n[{D} (\\omega)\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega)\n]\\ket{x}\n\\|\n}{\\| x \\|} ,\n\\label{eqn:Large50}\n\\end{equation}\nwhere Eq. (\\ref{eqn:Large40}) was used.\nThus Eq. (\\ref{eqn:Large30}) reads just as Eq. (\\ref{eqn:Large10}) does for $r=0$.\nIn the case of $r\\geq 1$, we first note that from our assumptions on the form factors and\nLemma \\ref{lm:formfactor1} again, $A(\\omega ) $ and ${\\mit \\Gamma}(\\omega ) $,\nwhich are connected through ${D} (\\omega )=S_0 + A(\\omega ) -\\log \\omega {\\mit \\Gamma}(\\omega )$,\nalso satisfy\n\\begin{equation}\n\\frac{d^r A(\\omega )}{d\\omega^r }= O(\\omega^{-1-r}), ~~\n\\frac{d^r \\log \\omega {\\mit \\Gamma}(\\omega )}{d\\omega^r }= O(\\omega^{-1-r} \\log \\omega),\n\\label{eqn:Large80}\n\\end{equation}\nas $\\omega \\to \\infty$, where we used the estimation that\n$\\frac{d^r {\\mit \\Gamma}(\\omega )}{d\\omega^r }= O(\\omega^{-1-r})$.\nThus, for $r=1$, we have\n\\begin{equation}\n\\frac{d \\tilde{R}^{\\pm}(\\omega )}{d\\omega}\n=\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d}{d\\omega}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\tilde{R}^{\\pm}(\\omega )\n= O(\\omega^{-2}),\n\\label{eqn:Large90b}\n\\end{equation}\nas $\\omega \\to \\infty$, where Eq. (\\ref{eqn:Large10}) for $r=0$ was used.\nFor $r\\geq1$, we obtain\n\\begin{equation}\n\\frac{d^r \\tilde{R}^{\\pm}(\\omega )}{d\\omega^r}\n=\n\\sum_{j=1}^{r}\n{\\sum_{\\{s_i \\}_{i=1}^{j} }}'\na^{(r)} (\\{s_i \\}_{i=1}^{j} )\n\\left\\{\n\\prod_{i=1}^{j}\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^{s_i}}{d\\omega^{s_i}}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\right\\}\n\\tilde{R}^{\\pm}(\\omega ),\n\\label{eqn:Large100}\n\\end{equation}\nwhere $a^{(r)} (\\{s_i \\}_{i=1}^{j} ) $ is an appropriate positive integer.\nNote that the symbol ($\\ '$) means that\nthe summation over $\\{s_i \\}_{i=1}^{j}$ is taken under the condition that\n$s_i \\geq 1$ for all $i$\nand $\\sum_{i=1}^{j} s_i =r$.\nIf $r=1$, Eq. (\\ref{eqn:Large100}) reproduces Eq. (\\ref{eqn:Large90b})\nwith $a^{(1)} (\\{s_i \\}_{i=1}^{1} ) =1$.\nIn the general case, if Eq. (\\ref{eqn:Large100}) holds\nfor $r=k$, then its derivative is made up of a linear combination of\n\\begin{equation}\n\\left\\{\n\\prod_{i=1}^{j+1}\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^{s_i}}{d\\omega^{s_i}}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\right\\} \\tilde{R}^{\\pm}(\\omega ),\n\\label{eqn:Large110}\n\\end{equation}\nwhere $\\sum_{i=1}^{j+1} s_i =k+1$ for $1 \\leq j \\leq k$, and\n\\begin{equation}\n\\left\\{\n\\prod_{i=1}^{j}\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^{s_i}}{d\\omega^{s_i}}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\right\\} \\tilde{R}^{\\pm}(\\omega ),\n\\label{eqn:Large120}\n\\end{equation}\nwhere $\\sum_{i=1}^{j} s_i =k+1$ for $1 \\leq j \\leq k$.\nOn the other hand, they are actually included in the right-hand side (rhs) of\nEq. (\\ref{eqn:Large100}) for $r=k+1$. Thus Eq. (\\ref{eqn:Large100}) is\nvalid for all integer $r \\geq 1$.\nLet us now evaluate the asymptotic behavior of\n$d^r \\tilde{R}^{\\pm}(\\omega )\/d\\omega^r$ for large $\\omega$.\nOne can see that\nthe summand for $j=r$ in Eq. (\\ref{eqn:Large100}),\nwhere all $s_i =1$, contributes\n$O(\\omega^{-r-1})$ to $d^r \\tilde{R}^{\\pm}(\\omega )\/d\\omega^r$,\nwhile the other summands for $j<r$ specified by $\\{s_i \\}_{i=1}^{j}$\ncontribute $O(\\omega^{-r-1-2s_0 }(\\log \\omega )^{s_0}))$ at most,\nwhere $s_0$ is a number of $s_i$ satisfying $s_i \\geq 2$ and never vanishes for $j<r$.\nTherefore, the summand dominating for large $\\omega$ is\nthat for $j=r$. Since we recursively show\n$a^{(r)} (\\{s_1 \\}_{i=1}^{r} ) =r!$, which never vanishes, the statement is proved. \n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Classification of the zero-energy singularity of $\\tilde{R}^\\pm (\\omega ) $}\n\\label{sec:5}\n\n\n\nIn order to prescribe the zero energy resonance in the $N$-level Friedrichs model,\nwe should identify the zero energy eigenstates in this model\nwhich either belong to or do not belong to ${\\cal H}$.\nIn the case of the short-range potential systems, \\cite{Jensen(1979)}\nthis task needs some elaborate examination with an appropriately extended Hilbert space.\nOn the other hand, in our case,\nit is rather easily performed, as is seen in the following.\n\n\n\n\n\nLet us first\nsee whether the eigenvector $\\ket{\\psi} \\in \\mathbb{C}^N$ of ${K_0} -\\lambda^2 S(0)$,\nbelonging to the zero eigenvalue,\ncan be actually extended to the eigenvector of $H$ belonging to the zero eigenvalue of $H$.\nIf $\\ket{\\Psi}=\\ket{\\psi}+\\ket{f} \\in D(H) \\subset {\\cal H}$ is a zero eigenvector of $H$,\nit should satisfy $H\\ket{\\Psi}=0$, or equivalently \\cite{Miyamoto(2005)}\n\\begin{equation}\n\\omega_n \\psi_n +\\lambda \\braket{v_n}{f} =0 ~\\mbox{for}~ n=1, \\ldots, N,~~~\\mbox{and}~~~\n\\omega f(\\omega)+ \\lambda \\sum_{n=1}^{N}\\psi_n v_n (\\omega)=0.\n\\label{eqn:.333}\n\\end{equation}\nThe latter equation of Eq. (\\ref{eqn:.333}) is immediately solved as\n\\begin{equation}\nf(\\omega)=-\\lambda \\frac{\\sum_{n=1}^{N}\\psi_n v_n (\\omega)}{\\omega},\n\\label{eqn:.335}\n\\end{equation}\nwhich should be square integrable\nbecause we intend to find $\\ket{\\Psi}$ in ${\\cal H}$.\nIf this is the case, $\\omega f(\\omega) \\in L^2 ((0, \\infty))$, i.e.,\n$\\ket{\\Psi} \\in D(H)$ is ensured,\nand the substitution of Eq. (\\ref{eqn:.335}) into $\\braket{v_n}{f}$ is safely done.\nThen,\nwe find that the former equation of Eq. (\\ref{eqn:.333}) is nothing more than\n\\begin{equation}\n({K_0} -\\lambda^2 S(0) )\\ket{\\psi}=K(0)\\ket{\\psi}=0,\n\\label{eqn:.336}\n\\end{equation}\nwhere $K(0):=K^\\pm (0)={K_0} -\\lambda^2 S(0)$.\nHowever, it is noted that such an $f(\\omega)$ associated with $\\ket{\\psi}$ is not necessarily\nsquare integrable.\nHence, we shall decompose the zero eigenspace of $K(0)$, denoted by\n$M=\\{ \\ket{\\psi} \\in \\mathbb{C}^N |~ K(0)\\ket{\\psi}=0 \\}$,\ninto two kinds of subspaces:\n$M_1=(M_0 \\oplus M_2)^{\\perp}$ \nand $M_2 =\\{ \\ket{\\psi} \\in M | f(\\omega) \\in L^2 ((0, \\infty)) \\}$.\nHere $M_0=M^{\\perp}$, and\n$D^{\\perp}$ denotes the orthogonal complement of the subspace $D$.\nIn short we have $\\mathbb{C}^N = M_0 \\oplus M_1 \\oplus M_2$.\nThen, as is expected from the definition, we have\n\\begin{equation}\nM_1 \\subset \\{ \\ket{\\psi} \\in M | f(\\omega) \\notin L^2 ((0, \\infty)) \\} .\n\\label{eqn:.325}\n\\end{equation}\nNote that in general the subset on the rhs of the above is not a subspace.\nWe call $0$ the {\\it zero energy resonance} (or merely {\\it zero resonance}) \nof $H$ if $M_1$ is not empty.\nWe also introduce the projection operators $Q_0$, $Q_1$, and $Q_2$, associated with\n$M_0$, $M_1$, and $M_2$, respectively.\nWhat we next do is to introduce the terminology \nfollowing the study of Jensen and Kato. \\cite{Jensen(1979)}\n\n\n\n\\begin{df\n\\label{df:regular}\n{\\rm\nWe call the system a {\\it regular} case if it holds that\n$0 \\notin \\sigma (K(0))$, i.e.,\n\\begin{equation}\n{\\rm det}[K(0)]\n\\neq 0.\n\\label{eqn:.300}\n\\end{equation}\nIn this case, $0$ is said to be a {\\it regular} point for $H$.\n}\n\\end{df}\n\n\n\n\n\\begin{df\n\\label{df:exceptional}\n{\\rm \nWe call the system the {\\sl exceptional} case if, instead of Eq. (\\ref{eqn:.300}), it holds that\n$0 \\in \\sigma (K(0))$, i.e.,\n\\begin{equation}\n{\\rm det}[K(0)]\n= 0.\n\\label{eqn:.330}\n\\end{equation}\nIn particular, if $0$ is a resonance but not an eigenvalue\n($Q_1 \\neq 0$, $Q_2 =0$), $0$ is said to be an {\\it exceptional} point for $H$\nof the {\\sl first kind}.\nIf $0$ is not a resonance, but an eigenvalue ($Q_1 = 0$, $Q_2 \\neq 0$),\n$0$ is said to be an exceptional point\nof the {\\sl second kind}.\nIf $0$ is both a resonance and an eigenvalue ($Q_1 \\neq 0$, $Q_2 \\neq 0$),\n$0$ is said to be an exceptional point\nof the {\\it third kind}.\n}\n\\end{df}\n\n\n\n\n\n\nWe here remark that in general a non-trivial solution of Eq. (\\ref{eqn:.336}) does not exist, \nhowever we can find a special case where such a solution surely exists. \nSuppose that $N_+$ eigenvalues $\\omega_n$ of $H_0$ are positive, and \nall form factors $v_n (\\omega)$ satisfying the assumption (\\ref{eqn:formfactor1}) \nare linearly independent. \nThen increasing $\\lambda$ gradually form $0$ to $\\infty$, \nwe can find some critical values of $\\lambda$ \nfor which $K(0)$ has the zero eigenvalue. \nLet us denote the $n$-th eigenvalue of $K(0)$ by $\\kappa_n (0)$ where \n$\\kappa_1 (0)\\leq \\kappa_2 (0) \\leq \\cdots \\leq \\kappa_N (0)$. \nThen, $\\kappa_n (0)$ turns out to satisfy the inequality\n\\begin{equation}\n\\omega_n -\\lambda^2 \\sigma_N (0) \\leq \\kappa_n (0) \\leq \\omega_n -\\lambda^2 \\sigma_1 (0),\n\\label{eqn:zeroeigenvalue}\n\\end{equation}\nwhere both of $\\sigma_1 (0)$ and $\\sigma_N (0)$ \nare positive constants and ensured not to vanish.\\cite{Miyamoto(2005)} \nThus, for a sufficiently small $|\\lambda|$ \n$\\kappa_n (0)$ for each $n\\geq N-N_+ +1$ should be positive, \nwhile for a sufficiently large $|\\lambda|$ they should be negative. \nFurthermore, one easily sees that all $\\kappa_n (0)$ are continuous functions of $\\lambda^2$. \nTherefore, we conclude from the intermediate value theorem \nthat there is at least one critical value of $\\lambda$ to make \n$\\kappa_n (0)=0$ for each $n\\geq N-N_+ +1$. \nWe can actually find such special values of $\\lambda$ in Fig. 1 \ndepicted in Ref. \\makebox(8,1){\\Large \\cite{Miyamoto(2005)}}\\hspace{1mm}. \nThe example mentioned here could be treated in a more general way \nwith resort to the analytic Fredholm theorem \\cite{the analytic Fredholm theorem,Rauch(1978)}\nwhich tells us at most a finite number of the critical values exists. \n\n\n\n\n\nIt is also worth remarking that \nthe existence of the zero energy eigenstates that either belong to or not to \nthe Hilbert space necessarily prescribes the small energy behavior of \nthe form factors in the following way. \nRemember that under the assumption on the form factors, \n${\\mit \\Gamma}(\\omega )$ defined by Eq. (\\ref{eqn:.100}) has an asymptotic form like\n\\begin{equation}\n{\\mit \\Gamma}(\\omega ) = \\sum_{n=1}^{N} \\omega^n {\\mit \\Gamma}_n +O(\\omega^{N+1}),\n\\label{eqn:formfactor2}\n\\end{equation}\nas $\\omega\\to +0$. \nThen, if $\\ket{\\psi} \\in M_1$ exists, it should satisfy \n\\begin{equation}\n\\bra{\\psi} {\\mit \\Gamma}_1 \\ket{\\psi} \\neq 0. \n\\label{eqn:.340}\n\\end{equation}\nIn fact, if $\\bra{\\psi} {\\mit \\Gamma}_1 \\ket{\\psi} = 0$, we see that \n$f(\\omega)$ in Eq. (\\ref{eqn:.335}) has to satisfy \n$|f(\\omega)|^2=\\lambda^2\\bra{\\psi}{\\mit \\Gamma}(\\omega)\\ket{\\psi}\/\\omega^2=O(1)$ \nas $\\omega \\to +0$, however \nwhich concludes that $f(\\omega)$ is square integrable. \nThis contradicts the assumption that $\\ket{\\psi} \\in M_1$. \nIn order to make the condition (\\ref{eqn:.340}) be satisfied, \nat least ${\\mit \\Gamma}_1$ should not vanish identically. \nWe can find such form factors in the physical systems for \nthe spontaneous emission process of photons from the Hydrogen atom \n\\cite{Facchi(1998),Seke(1994)} and the quantum dot. \\cite{Antoniou(2001)} \nOn the other hand, the discussion mentioned here immediately implies the fact that \nif $\\ket{\\psi} \\in M_2$ exists, this time it should satisfy\n\\begin{equation}\n\\bra{\\psi} {\\mit \\Gamma}_1 \\ket{\\psi} = 0, \n\\label{eqn:.350}\n\\end{equation}\nwhich just ensures the requirement that $f(\\omega) \\in L^2 ((0, \\infty))$.\nHowever, note that Eq. (\\ref{eqn:.350}) does not imply ${\\mit \\Gamma}_1=0$ identically \nand only requires ${\\mit \\Gamma}_1=0$ on the subspace $M_2$.\n\n\n\n\n\n\n\n\n\\subsection{The small-energy behavior of $\\tilde{R}^\\pm (\\omega) $ in the regular case}\n\nIn this case, the same as in Eq. (\\ref{eqn:.160}), we can show that\n$\\tilde{R}^{\\pm}(0)\n=(K(0))^{-1}$.\nFurthermore, we can choose some positive $\\omega_0 >0$\nsuch that\n\\begin{equation}\n\\| (K(0))^{-1}\\|\n\\| [\\omega +\\lambda^2 {D} (\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n-\\lambda^2 S(0)\n] \\|\n< 1,\n\\label{eqn:.170}\n\\end{equation}\nfor all positive $\\omega < \\omega_0$. Then,\n$\\tilde{R}^{\\pm}(\\omega )$ is expanded as a Neumann series,\n\\begin{equation}\n\\tilde{R}^{\\pm}(\\omega )\n=\n\\{\nK(0)\n[\n1- (K(0))^{-1}\n[\\omega +\\lambda^2 {D} (\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n-\\lambda^2 S(0)\n]]\n\\}^{-1}\n=\n\\lim_{N\\to \\infty}S_N(\\omega),\n\\label{eqn:.180}\n\\end{equation}\nwhere\n\\begin{equation}\nS_N(\\omega)\n=\n\\sum_{j=0}^N\n\\{\n(K(0))^{-1}\n[\\omega +\\lambda^2 {D} (\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n-\\lambda^2 S(0)\n]\\}^j\n(K(0))^{-1}\n,\n\\label{eqn:.185}\n\\end{equation}\nfor all positive $\\omega < \\omega_0$ with\n$\\{\n(K(0))^{-1}\n[\\omega +\\lambda^2 {D} (\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n-\\lambda^2 S(0)\n]\\}^0 =1\n$.\nUnder our assumptions on the form factors,\n$A(\\omega ) $ defined in Eq. (\\ref{eqn:.80b}) is asymptotically expanded as\n\\begin{equation}\nA(\\omega) = \\sum_{n=1}^{N} \\omega^n A_n +O(\\omega^{N+1}),\n\\label{eqn:.220}\n\\end{equation}\nas $\\omega \\to 0$. By using Eqs. (\\ref{eqn:formfactor2}) and (\\ref{eqn:.220}), \nit also follows that \n\\begin{equation}\n{D} (\\omega )\n=\nS(0)\n-\\omega \\log \\omega {\\mit \\Gamma}_1\n+\\omega A_1 + O(\\omega^2 \\log \\omega ) ,\n\\label{eqn:.230}\n\\end{equation}\nas $\\omega \\to +0$. \nThen, Eq. (\\ref{eqn:.180}) tells us the dominant asymptotic behavior of $\\tilde{R}^{\\pm}(\\omega )$\nbecomes\n\\begin{equation}\n\\tilde{R}^{\\pm}(\\omega ) =(K(0))^{-1} +O(\\omega \\log \\omega),\n\\label{eqn:.240}\n\\end{equation}\nas $\\omega \\to +0$, where $(K(0))^{-1}$ never vanishes in the regular case.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The small-energy behavior of $\\tilde{R}(z) $\nin the exceptional case of the first kind}\n\\label{subsec:5.C}\n\n\n\nIn the exceptional case of the first kind,\nfrom the definition, $Q_1 \\neq 0$ while $Q_2 = 0$, so that $Q_0 +Q_1 =1$.\nThen, $\\tilde{R} (z)$ is divided into the following four terms,\n\\begin{equation}\n\\tilde{R}(z)\n=\nQ_0 \\tilde{R}(z) Q_1\n+Q_0 \\tilde{R}(z) Q_0\n+Q_1 \\tilde{R}(z) Q_0\n+Q_1 \\tilde{R}(z) Q_1 .\n\\label{eqn:5.C.1.30}\n\\end{equation}\nWe now introduce the four matrices,\n\\begin{equation}\nE_{kl}(z)\n=Q_k [K(z) -z]Q_l\n=\nQ_k [K_0 -z -\\lambda^2 [S(0)+A(z)-(\\log z) {\\mit \\Gamma}(z)+i\\pi {\\mit \\Gamma}(z)]]Q_l,\n\\label{eqn:5.C.1.40}\n\\end{equation}\nwhere $k, l =0, 1$, and $\\log(-z) -\\log z =-i \\pi$ is used.\nFrom the relation that\n$[K(z) -z]\\tilde{R}(z) =1$,\nthey satisfy\n\\begin{equation}\nE_{k0}Q_0 \\tilde{R}Q_l + E_{k1}Q_1 \\tilde{R}Q_l =Q_k \\delta_{kl},\n\\label{eqn:5.C.1.50}\n\\end{equation}\nfor $k,l=0,1$.\nTo solve the above equations we need to check whether\n$E_{11}$ and $E_{00}$ are invertible in the subspaces $M_1$ and $M_0$, respectively.\nBy using Eq. (\\ref{eqn:formfactor2}), $E_{11}(z)$ is rewritten as\n\\begin{equation}\nE_{11}(z)\n=\n\\lambda^2z  (\\log z) Q_1 {\\mit \\Gamma}_1 Q_1\n-z Q_1\n-\\lambda^2 Q_1 \\{ A(z) -(\\log z) [{\\mit \\Gamma}(z) -z {\\mit \\Gamma}_1 ]\n+i\\pi {\\mit \\Gamma}(z)\\} Q_1 ,\n\\label{eqn:5.C.1.60b}\n\\end{equation}\nwhere $Q_1K(0)Q_1 =0$ is used.\nNote that $A(z)=O(z)$ and ${\\mit \\Gamma}(z) -z {\\mit \\Gamma}_1 =O(z^2 )$\nfor our form factors,\nso that all terms excepting the first one of the rhs of Eq. (\\ref{eqn:5.C.1.60b}) are\nof the order of $O(z)$.\nFurthermore, the exceptional case of the first kind imposes the fact that\n$Q_1 {\\mit \\Gamma}_1Q_1 \\neq 0$ [see Eq. (\\ref{eqn:.340})], and\n$Q_1 {\\mit \\Gamma}_1Q_1$ is positive definite in $M_1$ and thus invertible in $M_1$.\nHence, $E_{11}(\\omega) $ is invertible for sufficiently small\n$|z|>0$, and the inverse can be expanded by the Neumann series as,\n\\begin{eqnarray}\n\\hspace*{-5mm}\nE_{11}^{-1}(z)\n&=&\n\\sum_{j=0}^{\\infty} (\\tilde{E}_{11}(z))^j\n\\frac{1}{\\lambda^2z  \\log z} (Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\label{eqn:5.C.1.90a}\\\\\n&=&\n\\frac{1}{\\lambda^2z  \\log z} (Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n +O(z^{-1} (\\log z)^{-2})=O(z^{-1} (\\log z)^{-1}) ,\n\\label{eqn:5.C.1.90b}\n\\end{eqnarray}\nfor small $|z|$, where we define\n\\begin{equation}\n\\tilde{E}_{11}(z)\n:=\n\\frac{1}{\\lambda^2z  \\log z} (Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\bigl\\{ z Q_1\n+\\lambda^2 Q_1 \\{ A(z) -(\\log z) [{\\mit \\Gamma}(z) -z {\\mit \\Gamma}_1 ]\n+ \\pi i {\\mit \\Gamma}(z) \\} Q_1\n\\bigr\\} ,\n\\label{eqn:5.C.1.90c}\n\\end{equation}\nwhich behaves as $1\/\\log z$ as $z\\to0$. For $E_{00}$ we have,\n\\begin{equation}\nE_{00}(z)\n=\nQ_0 K(0)Q_0\n-Q_0 [z +\\lambda^2 [A(z)-(\\log z) {\\mit \\Gamma}(z)\n+ i\\pi  {\\mit \\Gamma}(z)]]Q_0 ,\n\\label{eqn:5.C.1.100}\n\\end{equation}\nwhere the first term of the above is invertible in $M_0$,\nand the last term vanishes as $z \\to 0$.\nHence, $E_{00}(z) $ is invertible in $M_0$\nfor sufficiently small $|z|>0$ and is expanded as\n\\begin{equation}\nE_{00}^{-1}(z) \\!\n=\\!\n-\\sum_{j=0}^{\\infty}\n\\{[Q_0 K(0)Q_0 ]^{-1}\n[z +\\lambda^2 [A(z)-(\\log z) {\\mit \\Gamma}(z)\n+ i\\pi {\\mit \\Gamma}(z)]]Q_0 \\}^j\n[Q_0 K(0)Q_0 ]^{-1}\n=\nO(1) ,\n\\label{eqn:5.C.1.110b}\n\\end{equation}\nfor small $|z|$.\nFurthermore, we obtain\n\\begin{equation}\nE_{kl}(z) =\n-\\lambda^2 Q_k [A(z)-(\\log z) {\\mit \\Gamma}(z) +\\pi i {\\mit \\Gamma}(z) ]Q_l =O(z \\log z),\n\\label{eqn:5.C.1.130}\n\\end{equation}\nfor $k\\neq l$ as $z \\to 0$, because\n$Q_0 K(0)Q_1=Q_1 K(0)Q_0=0$.\nSolving Eq. (\\ref{eqn:5.C.1.50}), we obtain \\cite{MatrixAnalysis}\n\\begin{eqnarray}\nQ_0\\tilde{R}Q_0\n&=&\n(E_{00}-E_{01}E_{11}^{-1}E_{10})^{-1} =O(1),\n\\label{eqn:5.C.1.120a}\\\\\nQ_0\\tilde{R}Q_1\n&=&\n-E_{00}^{-1}E_{01} Q_1\\tilde{R}Q_1\n=\n-E_{00}^{-1}E_{01} (E_{11}-E_{10}E_{00}^{-1}E_{01})^{-1}\n=O(1),\n\\label{eqn:5.C.1.120b}\\\\\nQ_1\\tilde{R}Q_0\n&=&\n-Q_1\\tilde{R}Q_1 E_{10} E_{00}^{-1}\n=\n-(E_{11}-E_{10}E_{00}^{-1}E_{01})^{-1} E_{10} E_{00}^{-1}\n=O(1),\n\\label{eqn:5.C.1.120c}\\\\\nQ_1\\tilde{R}Q_1\n&=&\n(E_{11}-E_{10}E_{00}^{-1}E_{01})^{-1} =O(z^{-1} (\\log z)^{-1}).\n\\label{eqn:5.C.1.120d}\n\\end{eqnarray}\nIt is worth noting that \nsince the relation $[K(z) -z]\\tilde{R}(z) =1$\nis analytically continued to the second Riemann sheet\nthrough the cut $[0, \\infty)$, \nthe above-mentioned results are also valid for such a continued region and\nthe estimations obtained here can be applied without any corrections.\n\n\nWhen we consider the small energy behavior of $\\tilde{R}^- (\\omega)$,\nit is convenient to expand $E_{11}$, differently from Eq. (\\ref{eqn:5.C.1.60b}),\nas\n\\begin{eqnarray}\nE_{11}(z)\n&=&\n\\lambda^2z (\\log z -2\\pi i) Q_1 {\\mit \\Gamma}_1 Q_1\n\\nonumber\\\\\n&&\n-z Q_1\n-\\lambda^2 Q_1 \\{ A(z) -(\\log z -2\\pi i)[{\\mit \\Gamma}(z) -z {\\mit \\Gamma}_1 ]\n-i\\pi {\\mit \\Gamma}(z)\\} Q_1 .\n\\label{eqn:5.C.1.125b}\n\\end{eqnarray}\nAll the above-obtained results are only changed by replacing the term\n$\\log z$ with $\\log z -2\\pi i$.\nThen, we can obtain from Eq. (\\ref{eqn:5.C.1.60b})\n\\begin{equation}\nE_{11}^+ (\\omega)\n=\n\\lim_{\\epsilon \\to +0}E_{11}(\\omega+i\\epsilon)\n=\n\\lambda^2 (\\log \\omega) Q_1 {\\mit \\Gamma}(\\omega) Q_1\n-\\omega Q_1 -\\lambda^2 Q_1 [A(\\omega)-i\\pi {\\mit \\Gamma}(\\omega)]Q_1 ,\n\\label{eqn:5.C.1.127a}\n\\end{equation}\nwhile from Eq. (\\ref{eqn:5.C.1.125b})\n\\begin{equation}\nE_{11}^- (\\omega)\n=\n\\lim_{\\epsilon \\to +0}E_{11}(\\omega-i\\epsilon)\n=\n\\lambda^2 (\\log \\omega) Q_1 {\\mit \\Gamma}(\\omega) Q_1\n-\\omega Q_1 -\\lambda^2 Q_1 [A(\\omega)+i\\pi {\\mit \\Gamma}(\\omega)]Q_1 .\n\\label{eqn:5.C.1.127b}\n\\end{equation}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The small-energy behavior of $\\tilde{R}(z)$\nin the exceptional case of the second kind}\n\\label{subsec:5.D}\n\nIn the exceptional case of the second kind,\nit follows that $Q_1 = 0$, $Q_2 \\neq 0$, and $Q_0 +Q_2 =1$.\nLet us consider the asymptotic behavior of the reduced resolvent at small energies,\nwhich\nis written in the following form,\n$\\tilde{R} (z)=\\sum_{k,l=0,2}Q_k \\tilde{R} (z) Q_l $.\nWe now introduce the four matrices again,\n\\begin{equation}\nE_{kl}(z)\n=Q_k [K(z) -z]Q_l ,\n\\label{eqn:5.C.2.40}\n\\end{equation}\nwhere $k, l =0, 2$.\nFrom the relation that\n$[K(z) -z]\\tilde{R}(z) =1$,\nthey satisfy that\n\\begin{equation}\nE_{k0} Q_0 \\tilde{R} Q_l + E_{k2} Q_2 \\tilde{R} Q_l =Q_k \\delta_{kl},\n\\label{eqn:5.C.2.50}\n\\end{equation}\nfor $k, l=0, 2$.\nThis time,\n$E_{22}$ and $E_{00}$ are invertible in $M_2$ and $M_0$, respectively.\nIn fact, from Eqs. (\\ref{eqn:formfactor2}) and (\\ref{eqn:.220}) we have\n\\begin{equation}\nE_{22}(z) =-z Q_2(1+\\lambda^2 A_1) Q_2 -\\lambda^2 Q_2[A(z)-z A_1 -(\\log z) {\\mit \\Gamma}(z)\n+ i\\pi {\\mit \\Gamma}(z)]Q_2 ,\n\\label{eqn:5.C.2.60b}\n\\end{equation}\nwhere $Q_2K(0)Q_2 =0$ was used.\nNote that since $A(z)-z A_1=O(z^2)$ and $Q_2 {\\mit \\Gamma}_1 Q_2=0$\n[see Eq. (\\ref{eqn:.350})],\nthe second term of the rhs of Eq. (\\ref{eqn:5.C.2.60b}) is of the order of\n$O(z^2 \\log z)$.\nFurthermore, since $Q_2 A_1 Q_2 \\geq 0$ from $Q_2 {\\mit \\Gamma}_1 Q_2=0$\nand Lemma \\ref{lm:1st_order},\n$Q_2(1+\\lambda^2 A_1)Q_2 >0$ and invertible in $M_2$.\nThese facts bring us the fact that $E_{22}(z) $ is invertible in $M_2$ for sufficiently small\n$z>0$, that is\n\\begin{eqnarray}\nE_{22}^{-1}(z)\n&=&\n-\\sum_{j=0}^{\\infty}\n(-\\tilde{E}_{22}(z) )^j \\frac{1}{z}[Q_2 (1+\\lambda^2 A_1 )Q_2 ]^{-1}\n\\label{eqn:5.C.2.90a}\\\\\n&=&\n\\frac{1}{z}[Q_2 (1+\\lambda^2 A_1 )Q_2 ]^{-1}+O(\\log z)=O(z^{-1}),\n\\label{eqn:5.C.2.90b}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\tilde{E}_{22}(z)\n:=\n\\frac{1}{z}[Q_2 (1+\\lambda^2 A_1 )Q_2  ]^{-1}\n\\lambda^2 Q_2[A(z)-z A_1 -(\\log z) {\\mit \\Gamma}(z)+ i\\pi {\\mit \\Gamma}(z)]Q_2.\n\\label{eqn:5.C.2.90c}\n\\end{equation}\nFor $E_{00}$, we next have\n\\begin{equation}\nE_{00}(z)\n=\nQ_0 K(0)Q_0\n-Q_0 [z +\\lambda^2 [A(z)-(\\log z) {\\mit \\Gamma}(z)\n+ i\\pi {\\mit \\Gamma}(z)]]Q_0,\n\\label{eqn:5.C.2.100}\n\\end{equation}\nwhere the first term of the above is invertible in $M_0$,\nand the last term vanishes as $|z| \\to 0$.\nHence, $E_{00}(z) $ is invertible in $M_0$\nfor sufficiently small $|z|>0$, and the inverse is obtained as a Neumann series.\nOn the other hand, $E_{20}$ and $E_{02}$ behave as\n\\begin{equation}\nE_{kl}(z) =\nQ_k [-z(1+\\lambda^2 A_1 )\n-\\lambda^2 [A(z)-z A_1-(\\log z) {\\mit \\Gamma}(z)\n+ \\pi i {\\mit \\Gamma}(z)] ]Q_l =O(z),\n\\label{eqn:5.C.2.130}\n\\end{equation}\nfor small $|z|$ where $k\\neq l$.\nSolving Eqs. (\\ref{eqn:5.C.2.50}) as in Eqs. (\\ref{eqn:5.C.1.120a}) to (\\ref{eqn:5.C.1.120d}),\none sees that\n\nQ_0\\tilde{R}Q_0=O(1),\n$\nfor $k,l=0,2$, except\n\\begin{equation}\nQ_2\\tilde{R}Q_2\n=\n(E_{22}-E_{20}E_{00}^{-1}E_{02})^{-1} =O(z^{-1}),\n\\label{eqn:5.C.2.120d}\n\\end{equation}\nas $z\\to 0$.\nIn particular, the last equation is expanded as,\n\\begin{equation}\nQ_2 \\tilde{R} (z) Q_2\n=\n\\sum_{j=0}^{\\infty}\n\\Bigl[ E_{22}^{-1}E_{20}E_{00}^{-1}E_{02} \\Bigr]^j\nE_{22}^{-1}\n=\n-\\frac{1}{z}[Q_2 (1+\\lambda^2 A_1 )Q_2 ]^{-1} +O(\\log z),\n\\label{eqn:5.C.2.140d}\n\\end{equation}\nfor small $|z|$, where we used Eq. (\\ref{eqn:5.C.2.90b}).\n\n\n\n\n\nWe now remark that the zero energy eigenspace of $H$\ndenoted by ${\\cal N}_0$ is completely characterized by $M_2$.\nThat is, there is a bijection from $M_2\\oplus \\{0\\}$ to ${\\cal N}_0$.\nFrom the discussion concerning Eqs. (\\ref{eqn:.333}), (\\ref{eqn:.335}), and (\\ref{eqn:.336}),\nfor any $\\ket{\\Psi} \\in {\\cal M}_0$, there is a vector $\\ket{\\psi}\\in M_2\\oplus \\{0\\}$ such that\n\\begin{equation}\n\\ket{\\Psi}=\\ket{\\psi}-\\lambda \\int_0^\\infty \\frac{\\sum_{n=1}^N v_n (\\omega)\\psi_n}{\\omega}\n\\ket{\\omega}d\\omega\n=\n[1-\\lambda R_0 (0)V]\\ket{\\psi},\n\\label{eqn:5.C.2.150}\n\\end{equation}\nwhere\n$V$ is restricted to $\\mathbb{C}^N \\oplus \\{0\\}$ and\n$R_0 (0)$ is the (unbounded) multiplication operator of $1\/\\omega$ in $L^2((0,\\infty))$.\nThen we see $V\\ket{\\psi} \\in D(R_0 (0))$ because $\\ket{\\psi}\\in M_2\\oplus \\{0\\}$.\nThus $1-\\lambda R_0 (0)V$ is well defined as an operator from $M_2\\oplus \\{0\\}$ to ${\\cal H}$.\nNow, Eq. (\\ref{eqn:5.C.2.150}) tells us that\n$1-\\lambda R_0 (0)V$ is a surjection from $M_2\\oplus \\{0\\}$ to ${\\cal N}_0$.\nOn the other hand, for any $\\ket{\\Psi} \\in {\\cal N}_0$, if $\\ket{\\Psi}=0$, i.e.,\n$0=\\braket{\\Psi}{\\Psi}$, Eq. (\\ref{eqn:5.C.2.150}) implies that\n$0=\\braket{\\Psi}{\\Psi}\\geq \\braket{\\psi}{\\psi}$. Therefore,\n$1-\\lambda R_0 (0)V$ is also an injection from $M_2\\oplus \\{0\\}$ to ${\\cal N}_0$,\nand the proof is completed.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The small-energy behavior of $\\tilde{R}^\\pm (\\omega) $\nin the exceptional case of the third kind}\n\\label{subsec:5.E}\n\n\n\nIn the exceptional case of the third kind,\nfrom the definition, $Q_1 \\neq 0$, $Q_2 \\neq 0$, and $Q_0 +Q_1 +Q_2 =1$.\nThe reduced resolvent is written in the form,\n$\\tilde{R}^\\pm (\\omega)\n=\n\\sum_{k,l=0}^{2}\nQ_k \\tilde{R}^\\pm (\\omega) Q_l\n$.\nThis time, we need nine matrices,\n\\begin{equation}\nE_{kl}^{\\pm}(\\omega) =Q_k [K^\\pm (\\omega) -\\omega]Q_l ,\n\\label{eqn:5.C.3.40}\n\\end{equation}\nfor $k, l = 0, 1, 2$.\nFrom the relation that\n$[K^\\pm (\\omega) -\\omega]\\tilde{R}^\\pm (\\omega) =1$,\nthey satisfy that\n\\begin{equation}\nE_{k0}^{\\pm} Q_0 \\tilde{R}^\\pm Q_l + E_{k1}^{\\pm} Q_1 \\tilde{R}^\\pm Q_l\n+ E_{k2}^{\\pm} Q_2 \\tilde{R}^\\pm Q_l = Q_k \\delta_{kl},\n\\label{eqn:5.C.3.50}\n\\end{equation}\nfor $k, l = 0, 1, 2$.\nThe asymptotic behaviors of $E_{kl}^{\\pm}(\\omega) $\nare essentially examined\nin the preceding subsections, except for $E_{12}^{\\pm}(\\omega)$ and $E_{21}^{\\pm}(\\omega)$.\nThen, $E_{12}^{\\pm}(\\omega)$ becomes\n\\begin{eqnarray}\nE_{12}^{\\pm}(\\omega)\n&=&-\\omega \\lambda^2 Q_1 A_1 Q_2\n\\nonumber \\\\\n&&\n-\\lambda^2 Q_1 \\bigl[A(\\omega)-\\omega A_1 -(\\log \\omega) [{\\mit \\Gamma}(\\omega)-\\omega {\\mit \\Gamma}_1 ]\n\\pm \\pi i [{\\mit \\Gamma}(\\omega)-\\omega {\\mit \\Gamma}_1 ] \\bigr]Q_2\n\\label{eqn:5.C.3.60b}\\\\\n&=&-\\omega \\lambda^2 Q_1 A_1 Q_2+O(\\omega^2 \\log \\omega ),\n\\label{eqn:5.C.3.60c}\n\\end{eqnarray}\nwhere $Q_1K(0)Q_2 =0$, $Q_1 Q_2 =0$,\nand ${\\mit \\Gamma}_1 Q_2 =0$ are used. The last relation follows from the fact that $Q_2 {\\mit \\Gamma}_1 Q_2 =0$ and\n${\\mit \\Gamma}_1 \\geq 0$.\nIn addition, since $Q_1 {\\mit \\Gamma} (\\omega)Q_2=O(\\omega^2)$,\nwe see that $Q_1 A_1 Q_2=\\int_0^\\infty Q_1 {\\mit \\Gamma} (\\omega) Q_2 \\omega^{-2} d\\omega$.\nBy the same way, we also see that\n\\begin{equation}\nE_{21}^{\\pm}(\\omega)\n=-\\omega \\lambda^2 Q_1 A_1 Q_2+O(\\omega^2 \\log \\omega ).\n\\label{eqn:5.C.3.60d}\n\\end{equation}\nTo solve Eqs. (\\ref{eqn:5.C.3.50}), let us now put the $N\\times N$ matrix $\\bf E$ as\n\\begin{equation}\n{\\bf E}=\n\\left[\n\\begin{array}{ccc}\nE_{00}&E_{01}&E_{02} \\\\\nE_{10}&E_{11}&E_{12} \\\\\nE_{20}&E_{21}&E_{22}\n\\end{array}\n\\right] ,\n\\label{eqn:5.C.3.65a}\n\\end{equation}\nand partition it into\n${\\bf A}=\n\\left[\n\\begin{array}{cc}\nE_{00}&E_{01} \\\\\nE_{10}&E_{11}\n\\end{array}\n\\right]\n$,\n${\\bf B}=\n\\left[\n\\begin{array}{c}\nE_{02} \\\\\nE_{12}\n\\end{array}\n\\right]\n$,\n${\\bf C}=\n\\left[\n\\begin{array}{cc}\nE_{20}&E_{21}\n\\end{array}\n\\right]\n$,\n${\\bf D}=\n\\left[\n\\begin{array}{c}\nE_{22}\n\\end{array}\n\\right]$. %\nThen, from the inverse matrix formula again,\n${\\bf E}^{-1}$ ($=\\tilde{R}$) is expressed as \\cite{MatrixAnalysis}\n\\begin{equation}\n{\\bf E}^{-1}=\n\\left[\n\\begin{array}{cc}\n[{\\bf A}-{\\bf B}{\\bf D}^{-1}{\\bf C}]^{-1}\n& -{\\bf A}^{-1}{\\bf B}[{\\bf D}-{\\bf C}{\\bf A}^{-1}{\\bf B}]^{-1}\\\\\n-[{\\bf D}-{\\bf C}{\\bf A}^{-1}{\\bf B}]^{-1} {\\bf C} {\\bf A}^{-1}\n&[{\\bf D}-{\\bf C}{\\bf A}^{-1}{\\bf B}]^{-1}\n\\end{array}\n\\right] .\n\\label{eqn:5.C.3.67}\n\\end{equation}\nThe validities of ${\\bf A}^{-1}$ and ${\\bf D}^{-1}$ are already ensured in\nthe exceptional cases of the first and second kinds, respectively.\nThen, one sees that\nsince\n${\\bf A}^{-1}=O(\\omega^{-1} (\\log \\omega)^{-1})$,\n${\\bf B}=O(\\omega)$,\n${\\bf C}=O(\\omega)$, and\n${\\bf D}^{-1}=O(\\omega^{-1})$,\nit holds that ${\\bf A}^{-1}{\\bf B}{\\bf D}^{-1}{\\bf C}=O((\\log \\omega)^{-1})$.\nThus, $[{\\bf A}-{\\bf B}{\\bf D}^{-1}{\\bf C}]^{-1}$ exists for small $\\omega$\nand $[{\\bf A}-{\\bf B}{\\bf D}^{-1}{\\bf C}]^{-1}=O(\\omega^{-1}(\\log \\omega)^{-1})$.\nWe also show that $[{\\bf D}-{\\bf C}{\\bf A}^{-1}{\\bf B}]^{-1}$ exists for small $\\omega$\nand $[{\\bf D}-{\\bf C}{\\bf A}^{-1}{\\bf B}]^{-1}=O(\\omega^{-1})$.\nTo obtain the asymptotic forms of the matrix components of ${\\bf E}^{-1}$ explicitly,\nsome redundant calculation is required; however, it could be achieved by \na manner as similar to that used in the preceding subsections.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Asymptotic expansion of the reduced resolvent at small $z$ }\n\\label{sec:5.5}\n\n\n\nWe examine the small-energy behavior of the reduced resolvent\nonly for the regular case and the exceptional case of the first kind.\nThis analysis is crucial for determining the asymptotic behavior of\nthe reduced time evolution operator at long times.\n\n\n\n\\subsection{The regular case}\n\nHere, we introduce $\\tilde{A}(\\omega ) := \\omega\/\\lambda^2 +A(\\omega ) $\nand suppose that $\\tilde{A}(\\omega )$ and ${\\mit \\Gamma}(\\omega ) $ behave as\n\\begin{equation}\n\\tilde{A}(\\omega ) := \\frac{1}{\\lambda^2} \\omega +\nA(\\omega ) = \\sum_{n=n_a}^{n_a+N} \\omega^n \\tilde{A}_n +O(\\omega^{n_a+N+1}), ~~~\n{\\mit \\Gamma}(\\omega )\n=\\sum_{n=n_b}^{n_b+N} \\omega^n {\\mit \\Gamma}_n +O(\\omega^{n_b+N+1}),\n\\label{eqn:rff.220}\n\\end{equation}\nas $\\omega \\to 0$, respectively,\nthat is, $\\tilde{A}_{n} =0$ for all $n< n_a$ and ${\\mit \\Gamma}_{n_b}=0$ for all $n< n_b$, \nwhile $\\tilde{A}_{n_a}\\neq0$ and ${\\mit \\Gamma}_{n_b}\\neq0$.\nThen, we obtain\n\\begin{equation}\n\\frac{1}{\\lambda^2} \\omega +\n{D} (\\omega )\n=\nS(0) +\\tilde{A}(\\omega )-(\\log \\omega){\\mit \\Gamma}(\\omega)\n=\nS(0)\n-\\omega^{n_b} \\log \\omega {\\mit \\Gamma}_{n_b}\n+\\omega^{n_a} \\tilde{A}_{n_a} + O(\\omega h(\\omega) ) ,\n\\label{eqn:rff.230}\n\\end{equation}\nas $\\omega \\to +0$, where\n\\begin{equation}\nh(\\omega)\n=\n\\left\\{\\begin{array}{cc}\n\\omega^{n_b} \\log \\omega & (n_b \\leq n_a) \\\\\n\\omega^{n_a}  & (n_b > n_a)\n\\end{array}\\right. .\n\\label{eqn:rff.245}\n\\end{equation}\n\n\nIt is important to note that the values of two parameters $n_a$ and $n_b$ \nare not determined independently. We shall here consider $n_b$ as a controllable one. \nWe first note that if $n_b \\geq 2$ then $n_a =1$ should be concluded, \nbecause from Lemma \\ref{lm:1st_order} we have $A_1 >0$, so that \n$\\tilde{A}_1 =1\/\\lambda^2  +A_1 >0$ holds. \nTherefore, the conditions $n_b \\leq n_a$ and $n_b > n_a$ \ncan be realized only in the situations \n\\begin{equation}\nn_b =1 ~\\mbox{and}~ n_a \\geq 1,\\quad\\mbox{and}\\quad n_b \\geq 2~ \\mbox{and}~ n_a = 1,\n\\label{eqn:rff.246}\n\\end{equation}\nrespectively. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{lm} \\label{lm:rffremainder}\n: Assume that $0$ is a regular point for $H$.\nThen the $r$-th derivative of $\\tilde{R}^\\pm (\\omega)$ asymptotically\nbehaves as\n\\begin{equation}\n\\frac{d^r \\tilde{R}^\\pm (\\omega)}{d\\omega^r}\n=\n\\left\\{\n\\begin{array}{cc}\nO(1) & (r=0)\\\\\nO(\\omega^{1-r}(\\log \\omega)^{\\theta(1-r)}) & (r\\geq1)\n\\end{array}\n\\right. ,\n\\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(1) & (r=0)\\\\\nO(\\omega^{[1-r]^+}) &(1\\leq r < n_b) \\\\\nO(\\omega^{n_b-r} (\\log \\omega)^{\\theta(n_b-r)}) &(n_b\\leq r \\leq 2n_b)\n\\end{array}\n\\right. ,\n\\label{eqn:rffSmall3}\n\\end{equation}\nfor $n_b =1$, or $n_b \\geq 2$, \nrespectively, as $\\omega \\to 0$,\nwhere $[x]^+ =\\max \\{x, 0 \\}$\nand $\\theta(x)=1$ for $x\\geq 0$ or $0$ for $x<0$.\nIn addition, the $r$-th derivative of $\\tilde{R}^\\pm (\\omega)$\nis approximated by that of a finite series\n\\begin{equation}\n(K(0))^{-1}\n+\n(K(0))^{-1}\n\\left[\n-\\omega^{n_b} (\\log \\omega) \\lambda^2 {\\mit \\Gamma}_{n_b}\n+\\omega^{n_a} \\lambda^2 \\tilde{A}_{n_a} \\pm \\lambda^2  \\pi i \\omega^{n_b} {\\mit \\Gamma}_{n_b}\n\\right]\n(K(0))^{-1} ,\n\\label{eqn:rffSmall5}\n\\end{equation}\nthat is, it is shown that\n\\begin{eqnarray}\n&&\n\\Biggl\\|\n\\frac{d^r}{d\\omega^r}\n\\biggl\\{\n\\tilde{R}^\\pm (\\omega)\n\\nonumber \\\\\n&&\n-\n(K(0))^{-1}\n-\n(K(0))^{-1}\n\\left[\n-\\omega^{n_b} (\\log \\omega) \\lambda^2 {\\mit \\Gamma}_{n_b}\n+\\omega^{n_a} \\lambda^2 \\tilde{A}_{n_a} \\pm \\lambda^2  \\pi i \\omega^{n_b} {\\mit \\Gamma}_{n_b}\n\\right]\n(K(0))^{-1}\n\\biggr\\}\n\\Biggr\\|\n\\nonumber \\\\\n&=&\n\\begin{array}{cc}\nO(\\omega^{2-r} (\\log \\omega)^{1+\\theta(2-r)})\n& (r \\geq 0)\n\\end{array}\n\\nonumber \\\\\n&&\n\\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[2-r]^+}) & (0\\leq r\\leq n_b) \\\\\nO(\\omega^{n_b+1-r} (\\log \\omega)^{\\theta(n_b+1-r)}) & (n_b+1 \\leq r \\leq 2n_b)\n\\end{array}\n\\right. ,\n\\label{eqn:rffSmall7}\n\\end{eqnarray}\nfor $n_b =1$, or $n_b \\geq 2$, \nrespectively, as $\\omega \\to 0$. Here, $n_a$ is restricted to the condition (\\ref{eqn:rff.246}). \n\\end{lm}\n\n\n\n\n\n\n\n\n\n\n\n{\\sl Proof} :\nThe left-hand side (lhs) of Eq. (\\ref{eqn:rffSmall7}) is written as follows:\n\\begin{eqnarray}\n&&\\hspace*{-10mm}\n\\Biggl\\|\n\\frac{d^r}{d\\omega^r}\n\\biggl\\{\n\\tilde{R}^\\pm (\\omega)\n\\nonumber \\\\\n&&\\hspace*{-10mm}\n-\n(K(0))^{-1}\n-\n(K(0))^{-1}\n\\left[\n-\\omega^{n_b} (\\log \\omega) \\lambda^2 {\\mit \\Gamma}_{n_b}\n+\\omega^{n_a} \\lambda^2 \\tilde{A}_{n_a} \\pm \\lambda^2  \\pi i \\omega^{n_b} {\\mit \\Gamma}_{n_b}\n\\right]\n(K(0))^{-1}\n\\biggr\\}\n\\Biggr\\|\n\\nonumber \\\\\n&&\\hspace*{-10mm}\n\\leq\n\\left\\|\n\\frac{d^r}{d\\omega^r}\n\\bigl\\{\n\\tilde{R}^\\pm (\\omega)-S_1 (\\omega)\n\\bigr\\}\n\\right\\|\n\\nonumber \\\\\n&&\\hspace*{-10mm}\n~~~+\n\\Biggl\\|\n\\frac{d^r}{d\\omega^r}\n\\biggl\\{\nS_1 (\\omega)\n\\nonumber \\\\\n&&\\hspace*{-10mm}\n~~~-\n(K(0))^{-1}\n-\n(K(0))^{-1}\n\\left[\n-\\omega^{n_b} (\\log \\omega) \\lambda^2 {\\mit \\Gamma}_{n_b}\n+\\omega^{n_a} \\lambda^2 \\tilde{A}_{n_a} \\pm \\lambda^2  \\pi i \\omega^{n_b} {\\mit \\Gamma}_{n_b}\n\\right]\n(K(0))^{-1}\n\\bigg\\}\n\\Biggr\\| ,\n\\label{eqn:rffSmall50}\n\\end{eqnarray}\nwhere $S_N(\\omega)$ is defined by Eq. (\\ref{eqn:.185}).\nWhen $r=0$, the first term on the rhs of the above is estimated from the special case of the below\nfor $N=1$,\n\\begin{equation}\n\\left\\|\n\\tilde{R}^\\pm (\\omega)-S_N (\\omega)\n\\right\\|\n\\! \\leq \\!\n\\frac{\n\\|\n\\omega +\\lambda^2 {D} (\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n-\\lambda^2 S(0)\n\\|^{N+1}\n\\|\n(K(0))^{-1}\n\\|^{N+2}\n}\n{\n1-\\|(K(0))^{-1}\\|\n\\| [\\omega +\\lambda^2 {D} (\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n-\\lambda^2 S(0)\n]\\|\n}\n=O(h(\\omega)^{N+1} ),\n\\label{eqn:rffSmall60c}\n\\end{equation}\nas $\\omega \\to 0$. When $r\\geq 1$, instead we have\n\\begin{eqnarray}\n&&\n\\left\\|\n\\frac{d^r}{d\\omega^r}\n\\bigl\\{\n\\tilde{R}^\\pm (\\omega)\n-\nS_1 (\\omega)\n\\bigr\\}\n\\right\\|\n\\nonumber \\\\\n&\\leq&\n\\left\\|\n\\sum_{j=2}^{r}\n{\\sum_{\\{s_i \\}_{i=1}^{j} }}'\na^{(r)} (\\{s_i \\}_{i=1}^{j} )\n\\left\\{\n\\prod_{i=1}^{j}\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^{s_i}}{d\\omega^{s_i}}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\right\\}\n\\tilde{R}^{\\pm}(\\omega )\n\\right\\|\n\\nonumber \\\\\n&&\n+\n\\left\\|\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^r }{d\\omega^r}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\tilde{R}^{\\pm}(\\omega )\n-\n\\frac{d^r}{d\\omega^r}\nS_1 (\\omega)\n\\right\\| ,\n\\label{eqn:rffSmall70b}\n\\end{eqnarray}\nwhere Eq. (\\ref{eqn:Large100}) is used, and here $s_i \\geq 1$ and $\\sum_{i=1}^{j} s_i =r$\nshould be satisfied.\nNote that the first term on the rhs of Eq. (\\ref{eqn:rffSmall70b})\nappears only for $r\\geq 2$, which is estimated in the following.\nIn the following estimations, we temporarily forget the restriction (\\ref{eqn:rff.246}) \nand consider the two general cases: $n_b \\leq n_a$ and $n_b > n_a$.\nIn the case of $n_b \\leq n_a$, we can obtain for $r\\geq 2$\n\\begin{eqnarray}\n&&\n\\left\\|\n\\sum_{j=2}^{r}\n{\\sum_{\\{s_i \\}_{i=1}^{j} }}'\na^{(r)} (\\{s_i \\}_{i=1}^{j} )\n\\left\\{\n\\prod_{i=1}^{j}\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^{s_i}}{d\\omega^{s_i}}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\right\\}\n\\tilde{R}^{\\pm}(\\omega )\n\\right\\|\n\\nonumber \\\\\n&&\\leq\n\\sum_{j=2}^{r}\n{\\sum_{\\{s_i \\}_{i=1}^{j} }}'\na^{(r)} (\\{s_i \\}_{i=1}^{j} )\n\\left\\|\n\\tilde{R}^{\\pm}(\\omega )\n\\right\\|^{j+1}\nO(\\omega^{jn_b-r})\n\\prod_{i=1}^{j}\nO((\\log \\omega)^{\\theta(n_b-s_i)})\n\\nonumber \\\\\n&&=\nO(\\omega^{2n_b-r} (\\log \\omega)^{\\theta(n_b+1-r)+\\theta(2n_b+1-r)}),\n\\label{eqn:rffSmall80a}\n\\end{eqnarray}\nas $\\omega \\to 0$.\nFor $n_a < n_b$,\n\\begin{eqnarray}\n&\n\\left\\|\n\\sum_{j=2}^{r}\n{\\sum_{\\{s_i \\}_{i=1}^{j} }}'\na^{(r)} (\\{s_i \\}_{i=1}^{j} )\n\\tilde{R}^{\\pm}(\\omega )\n\\prod_{i=1}^{j}\n\\left\\{\n\\frac{d^{s_i}}{d\\omega^{s_i}}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\tilde{R}^{\\pm}(\\omega )\n\\right\\}\n\\right\\|\n\\nonumber \\\\\n&&=\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[2n_a-r]^+}) &(2\\leq r\\leq n_a+n_b-1) \\\\\nO(\\omega^{n_a+n_b-r} (\\log \\omega)^{\\theta(n_a+n_b-r)})\n&(n_a+n_b\\leq r \\leq 2n_b)\n\\end{array}\n\\right. ,\n\\label{eqn:rffSmall80b}\n\\end{eqnarray}\nas $\\omega \\to 0$.\nWe here used that\n\\begin{eqnarray}\n&&\n\\frac{d^r}{d\\omega^r }\n[\n\\omega+\\lambda^2 {D}(\\omega )\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n-\\lambda^2 S(0)\n]\n\\nonumber\\\\\n&&\n=\nO(\\omega^{n_b-r} (\\log \\omega)^{\\theta(n_b-r)})\n, \\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[n_a-r]^+}) & (0\\leq r< n_b) \\\\\nO(\\omega^{n_b-r} (\\log \\omega)^{\\theta(n_b-r)}) & (r\\geq n_b)\n\\end{array}\n\\right.\n,\n\\label{eqn:rffSmall98b}\n\\end{eqnarray}\nfor $n_b\\leq n_a$, or $n_a< n_b$, respectively, as $\\omega \\to 0$.\nEq. (\\ref{eqn:rffSmall98b}) follows from\n\\begin{equation}\n\\frac{d^r \\tilde{A}(\\omega )}{d\\omega^r }\n=O(\\omega^{[n_a-r]^+}) ,\n~~~\n\\frac{d^r {\\mit \\Gamma}(\\omega )}{d\\omega^r }\n=O(\\omega^{[n_b-r]^+}) ,~~~\n\\frac{d^r (\\log \\omega) {\\mit \\Gamma}(\\omega )}{d\\omega^r }\n=O(\\omega^{n_b-r} (\\log \\omega)^{\\theta(n_b-r)}) ,\n\\label{eqn:rffSmall90b}\n\\end{equation}\nas $\\omega \\to 0$.\nIncorporating Eqs. (\\ref{eqn:rffSmall80a}), (\\ref{eqn:rffSmall80b}),\nand Eq. (\\ref{eqn:rffSmall98b}), with\n\\begin{eqnarray}\n&&\n\\left\\|\n\\frac{d^r \\tilde{R}^\\pm (\\omega)}{d\\omega^r}\n\\right\\|\n\\nonumber\\\\\n&\\leq&\n\\left\\|\n\\sum_{j=2}^{r}\n{\\sum_{\\{s_i \\}_{i=1}^{j} }}'\na^{(r)} (\\{s_i \\}_{i=1}^{j} )\n\\left\\{\n\\tilde{R}^{\\pm}(\\omega )\n\\prod_{i=1}^{j}\n\\frac{d^{s_i}}{d\\omega^{s_i}}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\right\\}\n\\tilde{R}^{\\pm}(\\omega )\n\\right\\|\n\\nonumber \\\\\n&&\n+\n\\left\\|\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^r }{d\\omega^r}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\tilde{R}^{\\pm}(\\omega )\n\\right\\| ,\n\\label{eqn:rffSmall99b}\n\\end{eqnarray}\nwe have\n\\begin{eqnarray}\n&&\\hspace*{-10mm}\n\\left\\|\n\\frac{d^r \\tilde{R}^\\pm (\\omega)}{d\\omega^r}\n\\right\\|\n\\nonumber\\\\\n&&\\hspace*{-10mm}\n=\n\\left\\{\n\\begin{array}{cc}\nO(1) & (r=0) \\\\\nO(\\omega^{n_b-r}(\\log \\omega)^{\\theta(n_b-r)})) & (r\\geq 1 )\n\\end{array}\n\\right.\n\\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(1) & (r=0) \\\\\nO(\\omega^{[n_a-r]^+}) & (1\\leq r< n_b) \\\\\nO(\\omega^{n_b-r} (\\log \\omega)^{\\theta(n_b-r)}) & (n_b \\leq r \\leq 2n_b)\n\\end{array}\n\\right. ,\n\\label{eqn:rffSmall99c}\n\\end{eqnarray}\nfor $n_b\\leq n_a$, or $n_a< n_b$, respectively, as $\\omega \\to 0$.\nThen, the first part of the statement can be shown under the restriction (\\ref{eqn:rff.246}).\nLet us next examine the second term on the rhs of Eq. (\\ref{eqn:rffSmall70b}),\nwhich reads for $r\\geq 1$,\n\\begin{eqnarray}\n&&\n\\Biggl\\|\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^r }{d\\omega^r}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\tilde{R}^{\\pm}(\\omega )\n-\n\\frac{d^r }{d\\omega^r}\nS_1(\\omega)\n\\Biggr\\|\n\\nonumber \\\\\n&&\\leq\n2 \\left\\|\n\\tilde{R}^{\\pm}(\\omega )\n-(K(0))^{-1}\n\\right\\|\n\\left\\|\n\\frac{d^r }{d\\omega^r}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\right\\|\n\\left\\|\n\\tilde{R}^{\\pm}(\\omega )\n\\right\\|\n\\nonumber \\\\\n&&=\nO(\\omega^{2n_b-r} (\\log \\omega)^{1+\\theta(n_b-r)}),\n\\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{n_a+[n_a-r]^+}) & (r< n_b) \\\\\nO(\\omega^{n_a+n_b-r} (\\log \\omega)^{\\theta(n_b-r)}) & (r\\geq n_b)\n\\end{array}\n\\right. ,\n\\label{eqn:rffSmall100}\n\\end{eqnarray}\nfor $n_b \\leq n_a$ or $n_b > n_a$ respectively,\nas $\\omega \\to 0$.\nWe here used Eq. (\\ref{eqn:rffSmall60c}) with $N=0$.\nTherefore, substituting Eqs. (\\ref{eqn:rffSmall80a}),\n(\\ref{eqn:rffSmall80b}), and (\\ref{eqn:rffSmall100}) into\nEq. (\\ref{eqn:rffSmall70b}), one has  for $r\\geq 1$,\n\\begin{eqnarray}\n&&\\hspace*{-12mm}\n\\left\\|\n\\frac{d^r}{d\\omega^r}\n\\bigr\\{ \\tilde{R}^\\pm (\\omega)- S_1 (\\omega) \\bigr\\}\n\\right\\|\n\\nonumber\\\\\n&&\\hspace*{-12mm}\n=O(\\omega^{2n_b-r} (\\log \\omega)^{1+\\theta(n_b+1-r)}) ,\n\\mbox{or}\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[2n_a-r]^+}) &(r\\leq n_a+n_b-1) \\\\\nO(\\omega^{n_a+n_b-r} (\\log \\omega)^{\\theta(n_a+n_b-r)})\n&(n_a+n_b\\leq r \\leq 2n_b)\n\\end{array}\n\\right. \\label{eqn:rffSmall110}\n\\end{eqnarray}\nfor $n_b \\leq n_a$ or $n_b > n_a$, respectively, as $\\omega \\to 0$.\nNote that this estimation is also valid for $r=0$ because it reproduces\nEq. (\\ref{eqn:rffSmall60c}) for $N=1$.\n\n\n\n\n\nLet us now evaluate the last term in Eq. (\\ref{eqn:rffSmall50}).\nFor $r\\geq 0$, we have\n\\begin{eqnarray}\n&&\n\\Biggl\\|\n\\frac{d^r}{d\\omega^r}\n\\biggr\\{\nS_1 (\\omega)\n\\nonumber\\\\\n&&\n-\n(K(0))^{-1}\n-\n(K(0))^{-1}\n\\left[\n-\\omega^{n_b} (\\log \\omega) \\lambda^2 {\\mit \\Gamma}_{n_b}\n+\\omega^{n_a} \\lambda^2 \\tilde{A}_{n_a} \\pm \\lambda^2  \\pi i \\omega^{n_b} {\\mit \\Gamma}_{n_b}\n\\right]\n(K(0))^{-1}\n\\bigg\\}\n\\Biggr\\|\n\\nonumber \\\\\n&&\\leq\n\\left\\|\n(K(0))^{-1}\n\\right\\|^2\n\\nonumber \\\\\n&&\n~~~\\times\n\\left\\|\n\\frac{d^r }{d\\omega^r}\n\\left[\n-(\\log \\omega) \\lambda^2 ({\\mit \\Gamma}(\\omega)- \\omega^{n_b} {\\mit \\Gamma}_{n_b} )\n+\\lambda^2 (\\tilde{A}(\\omega)- \\omega^{n_a} \\tilde{A}_{n_a} ))\n\\pm \\lambda^2  \\pi i\n\\left(\n{\\mit \\Gamma}(\\omega)- \\omega^{n_b} {\\mit \\Gamma}_{n_b}\n\\right)\n\\right]\n\\right\\|\n\\nonumber \\\\\n&&=\nO(\\omega^{n_b+1-r} (\\log \\omega)^{\\theta(n_b+1-r)}),\n\\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[n_a+1-r]^+}) & (0\\leq r\\leq n_b) \\\\\nO(\\omega^{n_b+1-r} (\\log \\omega)^{\\theta(n_b+1-r)}) & (r\\geq n_b+1)\n\\end{array}\n\\right. ,\n\\label{eqn:rffSmall130b}\n\\end{eqnarray}\nfor $n_b \\leq n_a$ or $n_b > n_a$ respectively,\nas $\\omega \\to 0$ for any $r\\geq 0$.\nWe here used that for $r\\geq 0$,\n\\begin{equation}\n\\frac{d^r ({\\mit \\Gamma}(\\omega)- \\omega^{n_b} {\\mit \\Gamma}_{n_b})}{d\\omega^r}\n=O(\\omega^{[n_b+1-r]^+}) ,~~\n\\frac{d^r (\\log \\omega) ({\\mit \\Gamma}(\\omega)- \\omega^{n_b} {\\mit \\Gamma}_{n_b})}{d\\omega^r }\n=\nO(\\omega^{n_b+1-r} (\\log \\omega)^{\\theta(n_b+1-r)})\n\\label{eqn:rffSmall140}\n\\end{equation}\nas $\\omega \\to 0$, and so forth.\nThus,\nsetting Eqs. (\\ref{eqn:rffSmall110}) and (\\ref{eqn:rffSmall130b})\ninto Eq. (\\ref{eqn:rffSmall50}), we conclude that\n\\begin{eqnarray}\n&\n\\Biggl\\|\n\\frac{d^r}{d\\omega^r}\n\\biggl\\{\n\\tilde{R}^\\pm (\\omega)\n\\nonumber\\\\\n&&\n-\n(K(0))^{-1}\n-\n(K(0))^{-1}\n\\left[\n-\\omega^{n_b} (\\log \\omega) \\lambda^2 {\\mit \\Gamma}_{n_b}\n+\\omega^{n_a} \\lambda^2 \\tilde{A}_{n_a} \\pm \\lambda^2  \\pi i \\omega^{n_b} {\\mit \\Gamma}_{n_b}\n\\right]\n(K(0))^{-1}\n\\biggr\\}\n\\Biggr\\|\n\\nonumber \\\\\n&\n=\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{2n_b-r} (\\log \\omega)^{1+\\theta(n_b+1-r)})\n& (r \\geq 0, n_b=1) \\\\\nO(\\omega^{n_b+1-r} (\\log \\omega)^{\\theta(n_b+1-r)})\n& (r \\geq 0, n_b\\geq2)\n\\end{array}\n\\right. ,\n\\nonumber\\\\\n&&\n~~~\n\\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[n_a+1-r]^+}) & (0\\leq r\\leq n_b) \\\\\nO(\\omega^{n_b+1-r} (\\log \\omega)^{\\theta(n_b+1-r)}) & (n_b+1 \\leq r \\leq 2n_b)\n\\end{array}\n\\right. ,\n\\label{eqn:rffSmall60}\n\\end{eqnarray}\nfor $n_b \\leq n_a$ or $n_b > n_a$, respectively,\nas $\\omega \\to 0$.\nBy taking into account the restriction (\\ref{eqn:rff.246}), we can show the last part of the lemma.\n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\n\n\n\n\n\n\n\n\n\n\nTo estimate the long time behavior of the reduced time evolution operator,\nthe above-mentioned lemma seems not precisely appropriate \nbecause the reduced time evolution operator is obtained from\nthe Fourier transform of the imaginary part of $\\tilde{R}^+ (\\omega)$,\nnot from $\\tilde{R}^+ (\\omega)$ itself, which is explained in the next section.\nHence, the following lemma is more appropriate for our purpose. \n\n\n\\begin{lm} \\label{lm:rffimremainder}\n: Assume that $0$ is a regular point for $H$.\nThen the $r$-th derivative of\n${\\rm Im}\\tilde{R}^+ (\\omega):=(\\tilde{R}^+ (\\omega) -\\tilde{R}^- (\\omega))\/2i$\nis approximated by that of\n\\begin{eqnarray}\n(K(0))^{-1}\n\\lambda^2 \\pi\n\\omega^{n_b} ({\\mit \\Gamma}_{n_b} +\\omega {\\mit \\Gamma}_{n_b +1} +\\omega^2 {\\mit \\Gamma}_{n_b +2})\n(K(0))^{-1} ,\n\\label{eqn:rffimSmall5}\n\\end{eqnarray}\nin the sense that for $0\\leq r \\leq n_b+1$ the remainder is estimated as,\n\\begin{eqnarray}\n&&\n\\left\\|\n\\frac{d^r}{d\\omega^r}\n\\bigl\\{\n{\\rm Im}\\tilde{R}^+ (\\omega)-\n(K(0))^{-1}\n\\lambda^2 \\pi\n\\omega^{n_b} ({\\mit \\Gamma}_{n_b} +\\omega {\\mit \\Gamma}_{n_b +1} +\\omega^2 {\\mit \\Gamma}_{n_b +2})\n(K(0))^{-1}\n\\bigr\\}\n\\right\\|\n\\nonumber\\\\\n&&\n=\nO(\\omega^{2-r}\\log \\omega)\n, ~\\mbox{or} ~\nO(\\omega^{1+n_b-r})\n,\n\\label{eqn:rffimSmall60}\n\\end{eqnarray}\nfor $n_b=1$, or $n_b \\geq 2$, respectively, as $\\omega \\to 0$. \nFor $r=n_b +2$, the estimation is replaced by\n$O(\\omega^{-1})$ for $n_b=1$, $O(\\log \\omega)$ for $n_b = 2$, or $O(1)$ for $n_b \\geq 3$, \nrespectively, as $\\omega \\to 0$.\n\\end{lm}\n\n\n\n\n\n\n\n\n{\\sl Proof} :\nSince ${\\rm Im}\\tilde{R}^+ (\\omega)=\\lambda^2 \\pi \\tilde{R}^+ (\\omega) {\\mit \\Gamma}(\\omega)\n\\tilde{R}^- (\\omega)$, one has\n\\begin{eqnarray}\n&&\n\\left\\|\n\\frac{d^r}{d\\omega^r}\n\\bigl\\{\n{\\rm Im}\\tilde{R}^+ (\\omega)\n-\n(K(0))^{-1}\n\\lambda^2 \\pi\n\\omega^{n_b} ({\\mit \\Gamma}_{n_b} +\\omega {\\mit \\Gamma}_{n_b +1} +\\omega^2 {\\mit \\Gamma}_{n_b +2})\n(K(0))^{-1}\n\\bigr\\}\n\\right\\|\n\\label{eqn:rffimSmall40a} \\\\\n&\\leq&\n\\lambda^2 \\pi\n\\sum_{\\scriptsize\n\\begin{array}{c}\ns\\geq0, t\\geq0, u\\geq0, \\\\\n(s+t+u=r)\n\\end{array}}^{r}\n[F_1(s,t,u)+F_2(s,t,u)+F_3(s,t,u)] ,\n\\label{eqn:rffimSmall40c}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n&\\displaystyle F_1(s,t,u)=C_{stu}\n\\left\\|\n\\frac{d^s \\tilde{R}^+ (\\omega)}{d\\omega^s}\n\\right\\|\n\\left\\|\n\\frac{d^t {\\mit \\Gamma}(\\omega)}{d\\omega^t}\n\\right\\|\n\\left\\|\n\\frac{d^u}{d\\omega^u}\n\\bigl\\{ \\tilde{R}^- (\\omega)-(K(0))^{-1} \\bigr\\}\n\\right\\| ,&\n\\label{eqn:rffimSmall41a}\\\\\n&\\displaystyle F_2(s,t,u)=C_{stu}\n\\left\\|\n\\frac{d^s \\tilde{R}^+ (\\omega)}{d\\omega^s}\n\\right\\|\n\\left\\|\n\\frac{d^t}{d\\omega^t}\n\\bigl\\{\n{\\mit \\Gamma}(\\omega)\n-\n\\omega^{n_b}({\\mit \\Gamma}_{n_b} +\\omega {\\mit \\Gamma}_{n_b +1} +\\omega^2 {\\mit \\Gamma}_{n_b +2})\n\\bigr\\}\n\\right\\|\n&\n\\nonumber\\\\\n&\\hspace*{-57mm}\n\\displaystyle\n\\times\n\\left\\|\n\\frac{d^u(K(0))^{-1}}{d\\omega^u}\n\\right\\|,&\n\\label{eqn:rffimSmall41b}\\\\\n&\\displaystyle\nF_3(s,t,u)=C_{stu}\n\\left\\|\n\\frac{d^s}{d\\omega^s}\n\\bigl\\{\n\\tilde{R}^+ (\\omega) -(K(0))^{-1}\n\\bigr\\}\n\\right\\|\n\\left\\|\n\\frac{d^t}{d\\omega^t}\\omega^{n_b} ({\\mit \\Gamma}_{n_b} +\\omega {\\mit \\Gamma}_{n_b +1} +\\omega^2 {\\mit \\Gamma}_{n_b +2})\n\\right\\|\n&\n\\nonumber\\\\\n&\\hspace*{-67mm}\n\\displaystyle\n\\times\n\\left\\|\n\\frac{d^u(K(0))^{-1}}{d\\omega^u}\n\\right\\| ,\n&\n\\label{eqn:rffimSmall41c}\n\\end{eqnarray}\nwhere $C_{stu}$'s are appropriate constants.\nFor $1\\leq r\\leq n_b +1$,\nthe summation of the first summand in Eq. (\\ref{eqn:rffimSmall40c}) can be estimated as\n\\begin{eqnarray}\n&&\n\\sum_{\\scriptsize\n\\begin{array}{c}\ns\\geq0, t\\geq0, u\\geq0, \\\\\n(s+t+u=r)\n\\end{array}}^{r}\nF_1(s,t,u)\n\\nonumber\\\\\n&&\n=\nF_1(0,r,0)\n+\n\\sum_{t\\geq 0,u\\geq 1}^r\nF_1(0,t,u)\n+\n\\sum_{s\\geq 1,t\\geq 0}^r\nF_1(s,t,0)\n+\n\\sum_{\\min\\{ s, u\\}\\geq 1}^r F_1(s,t,u)\n\\nonumber \\\\\n&&\n=\nO(F_1(0, r-1, 1))\n=\nO(\\omega^{2n_b -r} \\log \\omega ) ,~~\n\\mbox{or}~~\nO(\\omega^{n_a+n_b -r}) ,\n\\label{eqn:rffimSmall47a}\n\\end{eqnarray}\nfor $n_b \\leq n_a$ or $n_b > n_a$, respectively, as $\\omega \\to 0$.\nNote that from Eq. (\\ref{eqn:rffSmall60c}) for $N=0$ this estimation is valid for $r=0$ too.\nFor $r=n_b+2$, it is estimated as\n\\begin{equation}\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{n_b-1}\\log \\omega) & (n_b\\geq 2)\\\\\nO((\\log \\omega)^2 ) & (n_b=1)\n\\end{array}\n\\right. ,~~\n\\mbox{or}~~\nO(\\omega^{n_a-1}) ,\n\\label{eqn:rffimSmall47b}\n\\end{equation}\nfor $n_b \\leq n_a$ or $n_b > n_a$, respectively, as $\\omega \\to 0$.\nThe summation of the second summand in Eq. (\\ref{eqn:rffimSmall40c}) for $0\\leq r\\leq n_b +3$\nis also estimated as\n\\begin{equation}\n\\lambda^2 \\pi\n\\sum_{\\scriptsize\n\\begin{array}{c}\ns\\geq0, t\\geq0, u\\geq0, \\\\\n(s+t+u=r)\n\\end{array}}^{r}\nF_2 (s,t,u)\n=\n\\sum_{(s+t=r)}^r\nF_2(s, t, 0)\n=\nO(F_2(0, r, 0))\n=\nO(\\omega^{n_b +3 -r}),\n\\label{eqn:rffimSmall47c}\n\\end{equation}\nboth for $n_b \\leq n_a$ and for $n_b > n_a$, as $\\omega \\to 0$.\nThe summation of the last summand in Eq. (\\ref{eqn:rffimSmall40c}) for $0\\leq r\\leq n_b +1$\nis estimated as\n\\begin{eqnarray}\n\\lambda^2 \\pi\n\\sum_{\\scriptsize\n\\begin{array}{c}\ns\\geq0, t\\geq0, u\\geq0, \\\\\n(s+t+u=r)\n\\end{array}}^{r}\nF_3 (s,t,0)\n&=&\n\\sum_{(s+t=r)}^r\nF_3(s, t, 0)\n=\nO(F_3(1, r-1, 0))\n\\nonumber\\\\\n&=&\nO(\\omega^{2n_b -r } \\log \\omega), ~\n\\mbox{or} ~\nO(\\omega^{n_a +n_b -r}),\n\\label{eqn:rffimSmall47d}\n\\end{eqnarray}\nfor $n_b \\leq n_a$ or $n_a < n_b $,\nrespectively, as $\\omega \\to 0$.\nFor $r=n_b +2$, the estimation is replaced by\n\\begin{equation}\nO(\\omega^{n_b -2}(\\log \\omega)^{\\theta(n_b-2)}) ,\n\\mbox{ or } ~\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[n_a -2]^+}) & (n_b \\geq 3)\\\\\nO(\\log \\omega) & (n_b = 2)\n\\end{array}\n\\right. ,\n\\label{eqn:rffimSmall50a}\n\\end{equation}\nfor $n_b \\leq n_a$ or $n_a < n_b $, respectively,\nas $\\omega \\to 0$.\nThen, by summarizing the above-noted estimations from Eqs.\n(\\ref{eqn:rffimSmall47a}) to (\\ref{eqn:rffimSmall50a}),\nand by taking into account the restriction (\\ref{eqn:rff.246}) again,\nthe proof of the lemma is completed.\n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The exceptional case of the first kind}\n\nIn this case, we first remember that from the discussion around Eq. (\\ref{eqn:.340}) \nit necessarily holds that ${\\mit \\Gamma}_1\\neq 0$, i.e., $n_b=1$ in Eqs. (\\ref{eqn:rff.220}).\n\n\n\n\\begin{lm} \\label{lm:1stremainder+}\n: Assume that $0$ is an exceptional point of the first kind for $H$.\nThen the $0$-th and the first derivative of $\\tilde{R} (z)$\nare approximated by those of a finite series\n\\begin{equation}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n+\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} ,\n\\label{eqn:1stSmall5}\n\\end{equation}\nthat is, it is shown that\n\\begin{eqnarray}\n&&\n\\Biggl\\|\n\\frac{d^r}{dz^r}\n\\Biggl[\n\\tilde{R} (z)\n-\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\Biggr]\n\\Biggr\\|\n\\nonumber \\\\\n&&=\nO(z^{-1} (\\log z)^{-3}) ~~\\mbox{for}~~r=0,~~~\\mbox{or}~~~\nO(z^{-2} (\\log z)^{-3}) ~~\\mbox{for}~~r=1,\n\\label{eqn:1stSmall7}\n\\end{eqnarray}\nas $z \\to 0$.\n\\end{lm}\n\n\n\n\n{\\sl Proof} :\nLet us first consider the quantity that\n\\begin{eqnarray}\n&&\n\\Biggl\\|\n\\frac{d^r}{dz^r}\n\\Biggl[\n\\tilde{R} (z)\n-\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\Biggr]\n\\Biggr\\|\n\\nonumber \\\\\n&&\\leq\n\\left\\|\n\\frac{d^r }{dz^r}\n\\left[\n\\tilde{R} (z) - Q_1 \\tilde{R} (z) Q_1\n\\right]\n\\right\\|\n+\n\\left\\|\n\\frac{d^r}{dz^r}\n\\Biggl[\nQ_1 \\tilde{R} (z) Q_1\n-\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\right.\n\\nonumber \\\\\n&&~~~\n\\left.\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\Biggr]\n\\right\\| .\n\\label{eqn:1stSmall50}\n\\end{eqnarray}\nFor $r= 0$, the first term on the rhs of the above is estimated as follows:\n\\begin{equation}\n\\left\\|\n\\tilde{R} (z)-Q_1 \\tilde{R} (z) Q_1\n\\right\\|\n\\leq\n\\left\\|\nQ_0 \\tilde{R} (z) Q_0\n\\right\\|\n+\n\\left\\|\nQ_0 \\tilde{R} (z) Q_1\n\\right\\|\n+\n\\left\\|\nQ_1 \\tilde{R} (z) Q_0\n\\right\\|\n=O(1),\n\\label{eqn:1stSmall60}\n\\end{equation}\nas $z \\to 0$,\nwhere Eqs. (\\ref{eqn:5.C.1.120a})--(\\ref{eqn:5.C.1.120c}) are used.\nFor $r=1$, one obtains\n\\begin{equation}\n\\left\\|\n\\frac{d\\tilde{R} (z)}{dz}\n-\n\\frac{dQ_1 \\tilde{R} (z) Q_1}{dz}\n\\right\\|\n\\leq\n\\left\\|\n\\frac{dQ_0 \\tilde{R} (z) Q_0}{dz}\n\\right\\|\n+\n\\left\\|\n\\frac{dQ_0 \\tilde{R} (z) Q_1}{dz}\n\\right\\|\n+\n\\left\\|\n\\frac{dQ_1 \\tilde{R} (z) Q_0}{dz}\n\\right\\| =O(z^{-1}).\n\\label{eqn:1stSmall70}\n\\end{equation}\nIn fact, by using expression (\\ref{eqn:5.C.1.120a})\nthe first term on the rhs of the above is estimated as follows:\n\\begin{eqnarray}\n\\frac{d Q_0 \\tilde{R} (z) Q_0}{dz}\n&=&\n-\nQ_0 \\tilde{R} (z) Q_0\n\\left( \\frac{d E_{00}}{dz} -\\frac{d E_{01}}{dz} E_{11}^{-1} E_{10}\n-E_{01} \\frac{dE_{11}^{-1} }{dz} E_{10}\n-E_{01} E_{11}^{-1} \\frac{dE_{10}}{dz}\n\\right)\n\\nonumber \\\\\n&&\n\\times\nQ_0 \\tilde{R} (z) Q_0\n\\label{eqn:1stSmall71b}\\\\\n&=&\nO(\\log z).\n\\label{eqn:1stSmall71c}\n\\end{eqnarray}\nFour derivatives in Eq. (\\ref{eqn:1stSmall71b})\nhave the same order, which can be shown from the use of\nEqs. (\\ref{eqn:5.C.1.60b}), (\\ref{eqn:5.C.1.90b}),\n(\\ref{eqn:5.C.1.100}), and (\\ref{eqn:5.C.1.130}): We here note that\n\\begin{equation}\n\\frac{dE_{00}}{dz}\n=O(\\log z),~~~\n\\frac{dE_{01}}{dz}\n=O(\\log z),~~~\n\\frac{dE_{10}}{dz}\n=O(\\log z),~~~\n\\label{eqn:1stSmall72}\n\\end{equation}\nand from Eqs. (\\ref{eqn:5.C.1.60b}) and (\\ref{eqn:5.C.1.90b})\n\\begin{eqnarray}\n\\frac{dE_{11}^{-1} }{dz}\n&=&\nE_{11}^{-1}\n\\left\\{\n\\frac{d}{dz}\nQ_1 \\left\\{\nz +\\lambda^2 [ A(z) -\\log z {\\mit \\Gamma}(z) + \\pi i {\\mit \\Gamma}(z) ]\n\\right\\} Q_1\n\\right\\}\nE_{11}^{-1}\n\\label{eqn:1stSmall73a}\\\\\n&=&\nE_{11}^{-1}\nQ_1 \\left\\{\n1 +\\lambda^2 \\left[ \\frac{dA(z)}{dz}-{\\mit \\Gamma}(z)\/z\n-\\log z \\frac{d{\\mit \\Gamma}(z)}{dz}\n+ \\pi i \\frac{d{\\mit \\Gamma}(z)}{dz} \\right]\n\\right\\} Q_1\nE_{11}^{-1}\n\\label{eqn:1stSmall73b}\\\\\n&=&\nO(z^{-2} (\\log z)^{-1} ).\n\\label{eqn:1stSmall73c}\n\\end{eqnarray}\nIn the same way, the second term on the rhs of Eq. (\\ref{eqn:1stSmall70})\nis also estimated as follows:\n\\begin{eqnarray}\n\\frac{d Q_0 \\tilde{R} (z) Q_1}{dz}\n&=&\n-\\left(\n\\frac{dE_{00}^{-1}}{dz} E_{01}\n+\nE_{00}^{-1} \\frac{dE_{01}}{dz}\n\\right)\nQ_1\\tilde{R}Q_1\n+\nE_{00}^{-1} E_{01}\nQ_1\\tilde{R}Q_1\n\\nonumber \\\\\n&&\n\\times\n\\left( \\frac{d E_{11}}{dz} -\\frac{d E_{10}}{dz} E_{00}^{-1} E_{01}\n-E_{10} \\frac{dE_{00}^{-1} }{dz} E_{01}\n-E_{10} E_{00}^{-1} \\frac{dE_{01}}{dz}\n\\right)\nQ_1\\tilde{R}Q_1\n\\label{eqn:1stSmall74b}\\\\\n&=&\nO(z^{-1}),\n\\label{eqn:1stSmall74c}\n\\end{eqnarray}\nwhere we used Eq. (\\ref{eqn:5.C.1.120b}) and the fact that\n\\begin{equation}\n\\frac{dE_{00}^{-1} }{dz}=O(\\log z),~~~\n\\frac{dE_{11}}{dz}=O(\\log z).\n\\label{eqn:1stSmall75}\n\\end{equation}\nIn a similar manner, we can also show\n\\begin{equation}\n\\frac{d Q_1 \\tilde{R} (z) Q_0}{dz}=O(z^{-1}).\n\\label{eqn:1stSmall76}\n\\end{equation}\n\n\n\nLet us next consider the last term in Eq. (\\ref{eqn:1stSmall50}). For $r=0$, it reads\n\\begin{eqnarray}\n&&\n\\left\\|\nQ_1 \\tilde{R} (z) Q_1\n-\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\right.\n\\nonumber \\\\\n&&\n\\left.\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\right\\|\n\\label{eqn:1stSmall77a}\\\\\n&\\leq&\n\\left\\|\nE_{11}^{-1}\n-\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\right.\n\\nonumber \\\\\n&&\n\\left.\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\right\\|\n\\nonumber \\\\\n&&\n+\\| (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}E_{11}^{-1} -E_{11}^{-1} \\|\n\\label{eqn:1stSmall77c}\\\\\n&\\leq&\n\\biggl\\|\n\\tilde{E}_{11}(z)\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\Biggl\\|\n\\sum_{j=2}^{\\infty}\n(\\tilde{E}_{11}(z))^j\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\Biggr\\|\n+\nO(1)\n\\label{eqn:1stSmall77d}\\\\\n&=&\nO( z^{-1}  (\\log z )^{-3} ),\n\\label{eqn:1stSmall77f}\n\\end{eqnarray}\nwhere in the second inequality we used Eq. (\\ref{eqn:5.C.1.90a}) and that\n\\begin{equation}\n\\| (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}E_{11}^{-1} -E_{11}^{-1} \\|\n\\leq\n\\frac{\\| E_{11}^{-1} E_{10} E_{00}^{-1} E_{01} \\|\n\\| E_{11}^{-1} \\|}{1-\\| E_{11}^{-1} E_{10} E_{00}^{-1} E_{01} \\| }\n=O(1),\n\\label{eqn:1stSmall78}\n\\end{equation}\nas $z \\to 0$.\nSubstituting Eqs. (\\ref{eqn:1stSmall60}) and (\\ref{eqn:1stSmall77f})\ninto Eq. (\\ref{eqn:1stSmall50}), we can obtain the estimation (\\ref{eqn:1stSmall7}) for $r=0$.\n\n\n\n\nFor $r=1$, we can obtain\n\\begin{eqnarray}\n&&\n\\Biggl\\|\n\\frac{d}{dz}\n\\biggl[\nQ_1 \\tilde{R} (z) Q_1\n-\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr]\n\\Biggr\\|\n\\label{eqn:1stSmall80a}\\\\\n&\\leq&\n\\Biggl\\|\n\\frac{d}{dz}\n\\biggl[\nE_{11}^{-1}\n-\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr]\n\\Biggr\\|\n\\nonumber \\\\\n&&\n+\\biggl\\| \\frac{d}{dz}\n\\bigl\\{ (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}E_{11}^{-1}\n-E_{11}^{-1}\n\\bigr\\}\n\\biggr\\|\n\\label{eqn:1stSmall80b}\\\\\n&\\leq&\n\\biggl\\|\nE_{11}^{-1}\n\\left\\{ \\frac{d}{dz}\nQ_1 \\left\\{\nz +\\lambda^2 [ A(z) + \\pi i {\\mit \\Gamma}(z) ]\n\\right\\} Q_1 \\right\\}\nE_{11}^{-1}\n\\nonumber \\\\\n&&\n-\n\\frac{1}{\\lambda^4 z^2 (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\biggl\\|\n-E_{11}^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\nE_{11}^{-1}\n-\\frac{d}{dz}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\nz\n\\biggl\\{\n\\frac{d}{dz}\n\\frac{1}{\\lambda^4 z^2 (\\log z)^2}\n\\biggr\\}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\\biggl\\|\n\\bigl\\{ (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1} -Q_{11} \\bigr\\}\n\\frac{dE_{11}^{-1}}{dz}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\biggl\\|\n\\left\\{\n\\frac{d}{dz}\n(Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}\n\\right\\}\nE_{11}^{-1}\n\\biggr\\|\n\\label{eqn:1stSmall80d}\\\\\n&=&O(z^{-2}(\\log z)^{-3}),\n\\label{eqn:1stSmall80e}\n\\end{eqnarray}\nwhere we used the expression for $dE_{11}^{-1}\/dz$\nin Eq. (\\ref{eqn:1stSmall73a}).\nActually, the first term in Eq. (\\ref{eqn:1stSmall80d}) is estimated as\n\\begin{eqnarray}\n&&\\hspace*{-13mm}\n\\biggl\\|\nE_{11}^{-1}\n\\left\\{ \\frac{d}{dz}\nQ_1 \\left\\{\nz +\\lambda^2 [ A(z) + \\pi i {\\mit \\Gamma}(z) ]\n\\right\\} Q_1 \\right\\}\nE_{11}^{-1}\n\\nonumber \\\\\n&&\\hspace*{-13mm}\n-\n\\frac{1}{\\lambda^4 z^2 (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\\hspace*{-13mm}\n\\leq\n\\biggl\\|\nE_{11}^{-1}-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\biggl\\|\n\\frac{d}{dz}\nQ_1 \\left\\{\nz +\\lambda^2 [ A(z) + \\pi i {\\mit \\Gamma}(z) ]\n\\right\\} Q_1\n\\biggr\\|\n\\| E_{11}^{-1} \\|\n\\nonumber \\\\\n&&\\hspace*{-13mm}\n~~~+\n\\frac{\\| ( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} \\| }{\\lambda^2 |z \\log z|}\n\\biggl\\|\n\\frac{d}{dz}\nQ_1 \\left\\{\nz +\\lambda^2 [ A(z) + \\pi i {\\mit \\Gamma}(z) ]\n\\right\\} Q_1\n\\biggr\\|\n\\biggl\\|\nE_{11}^{-1}-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\\hspace*{-13mm}\n~~~+\n\\frac{\\|( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} \\|^2}{\\lambda^4 z^2 (\\log z)^2 }\n\\biggl\\|\n\\frac{d}{dz}\nQ_1 \\left\\{\nz +\\lambda^2 [ A(z) + \\pi i {\\mit \\Gamma}(z) ]\n\\right\\} Q_1\n-\nQ_1 (1+ \\lambda^2 A_1 + \\lambda^2  \\pi i {\\mit \\Gamma}_1 )Q_1\n\\biggr\\|\n\\label{eqn:1stSmall90a}\\\\\n&&\\hspace*{-13mm}\n=\nO(z^{-2}(\\log z)^{-3}),\n\\label{eqn:1stSmall90b}\n\\end{eqnarray}\nas $z \\to 0$.\nOn the other hand, the second term in Eq. (\\ref{eqn:1stSmall80d}) is\nslightly complicated to evaluate:\n\\begin{eqnarray}\n&&\n\\biggl\\|\n-E_{11}^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\nE_{11}^{-1}\n-\\frac{d}{dz}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\nz \\biggl\\{\n\\frac{d}{dz}\n\\frac{1}{\\lambda^4 z^2 (\\log z)^2}\n\\biggr\\}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\label{eqn:1stSmall95a}\\\\\n&\\leq&\n\\biggl\\|\n-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\\frac{d}{dz}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\tilde{E}_{11}\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\\tilde{E}_{11}\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n+\nz\n\\frac{2}{(\\lambda^2 z \\log z )^3}\n\\biggl(\n\\frac{d}{dz}\n\\lambda^2 z \\log z\n\\biggr)\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\biggl\\|\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\sum_{j=2}^{\\infty}\\tilde{E}_{11}^j \\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\biggl\\|\n\\tilde{E}_{11} \\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\sum_{j=1}^{\\infty}\\tilde{E}_{11}^j \\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\biggl\\|\n\\sum_{j=2}^{\\infty}\\tilde{E}_{11}^j \\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\nE_{11}^{-1}\n\\biggr\\|\n\\label{eqn:1stSmall95b}\\\\\n&\\leq&\n\\biggl\\|\n-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 \\log z Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\\frac{d}{dz}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\biggl\\|\n-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\tilde{E}_{11}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n+\nz \\frac{1}{(\\lambda^2 z \\log z )^3}\n\\biggl(\n\\frac{d}{dz}\n\\lambda^2 z \\log z\n\\biggr)\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\nz (Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\biggl\\|\n-\\tilde{E}_{11}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n+\nz\n\\frac{1}{(\\lambda^2 z \\log z )^3}\n\\biggl(\n\\frac{d}{dz}\n\\lambda^2 z \\log z\n\\biggr) ( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+O(z^{-2}(\\log z)^{-3} )\n\\label{eqn:1stSmall95c} \\\\\n&=&O(z^{-2}(\\log z)^{-3} ).\n\\label{eqn:1stSmall95d}\n\\end{eqnarray}\nIn fact, the first term in Eq. (\\ref{eqn:1stSmall95c}) reads\n\\begin{eqnarray}\n&&\n\\biggl\\|\n-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 \\log z Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber\\\\\n&&\n-\\frac{d}{dz}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber\\\\\n&\\leq&\n\\biggl\\|\n-(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n+\\frac{d}{dz}\n\\lambda^2 z (\\log z) Q_1\n\\biggr\\|\n\\frac{\\| ( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} \\|}{(\\lambda^2z  \\log z)^{2}}\n\\label{eqn:1stSmall96b}\\\\\n&\\leq&\n\\lambda^2\n\\left\\{\n\\biggl\\|\n-(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\nQ_1 \\frac{{\\mit \\Gamma}(z)}{z} Q_1\n+\nQ_1\n\\biggr\\|\n+\n\\biggl\\|\n-(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\nQ_1 \\frac{d{\\mit \\Gamma}(z)}{dz} Q_1\n+\nQ_1\n\\biggr\\|\n(\\log z)\n\\right\\}\n\\nonumber \\\\\n&&\n\\times\n\\| ( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} \\|\n\\frac{1}{\\lambda^2 z \\log z}\n\\label{eqn:1stSmall96c}\\\\\n&=&\nO((z \\log z)^{-1}).\n\\label{eqn:1stSmall96d}\n\\end{eqnarray}\nThe second term in Eq. (\\ref{eqn:1stSmall95c}) also reads\n\\begin{eqnarray}\n&&\n\\biggl\\|\n-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\tilde{E}_{11}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n+\nz \\frac{1}{(\\lambda^2 z \\log z )^3}\n\\biggl(\n\\frac{d}{dz}\n\\lambda^2 z \\log z\n\\biggr)\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber\\\\\n&\\leq&\n\\|(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\\|\n\\biggl\\|\n-\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\nonumber \\\\\n&&\n\\times\n(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\bigl\\{ z Q_1\n+\\lambda^2 Q_1 \\{ A(z) -\\log z [{\\mit \\Gamma}(z) -z {\\mit \\Gamma}_1 ]\n+ \\pi i {\\mit \\Gamma}(z) \\} Q_1\n\\bigr\\}\n\\nonumber \\\\\n&&\n+\nz \\biggl(\n\\frac{d}{dz}\n\\lambda^2 z \\log z\n\\biggr)\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n\\biggr\\|\n\\frac{\\|( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} \\| }{(\\lambda^2 z \\log z )^{3}}\n\\label{eqn:1stSmall97b}\\\\\n&\\leq&\n\\|(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\\|\n\\biggl\\|\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n\\times\n\\bigl\\|\n-\n\\bigl\\{ z Q_1\n+\\lambda^2 Q_1 \\{ A(z) -\\log z [{\\mit \\Gamma}(z) -z {\\mit \\Gamma}_1 ]\n+ \\pi i {\\mit \\Gamma}(z) \\} Q_1\n\\bigr\\}\n\\nonumber \\\\\n&&\n+\nz (Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n\\bigr\\|\n\\frac{\\|( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} \\|}{ (\\lambda^2 z \\log z )^{3}}\n\\nonumber \\\\\n&&\n+\n\\|(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\\|\n\\biggl\\|\n-\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n+\n\\biggl(\n\\frac{d}{dz}\n\\lambda^2 z \\log z\n\\biggr)\nQ_1\n\\biggr\\|\n\\nonumber \\\\\n&&\n\\times\n\\bigl\\|\nz (Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n\\bigr\\|\n\\frac{\\|( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} \\|}{ (\\lambda^2 z \\log z )^{3}}\n\\label{eqn:1stSmall97c}\\\\\n&=&\nO((z \\log z)^{-1}),\n\\label{eqn:1stSmall97d}\n\\end{eqnarray}\nas $z \\to 0$.\nThe third term gives the same contribution to the order as the second one does.\nFurthermore, the last term in Eq. (\\ref{eqn:1stSmall95c}) comes from the estimations of\nthe second, third, and last terms in Eq. (\\ref{eqn:1stSmall95b}),\nwhere each contributes the same order as $O(z^{-2}(\\log z)^{-3} )$.\nTherefore, Eq. (\\ref{eqn:1stSmall95d}) is proved.\n\n\nOn the other hand, the third term in Eq. (\\ref{eqn:1stSmall80d}) reads\n\\begin{equation}\n\\biggl\\|\n\\bigl\\{ (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1} -Q_{11} \\bigr\\}\n\\frac{dE_{11}^{-1}}{dz}\n\\biggr\\|\n\\leq\n\\frac\n{\n\\bigl\\|\nE_{11}^{-1}E_{10} E_{00}^{-1} E_{01}\n\\bigr\\|}\n{\n1-\\bigl\\|\nE_{11}^{-1}E_{10} E_{00}^{-1} E_{01}\n\\bigr\\|\n}\n\\biggl\\|\n\\frac{dE_{11}^{-1}}{dz}\n\\biggr\\|\n=O(z^{-1}),\n\\label{eqn:1stSmall100b}\n\\end{equation}\nas $z \\to 0$,\nwhere Eqs. (\\ref{eqn:5.C.1.90b}), (\\ref{eqn:5.C.1.100}), (\\ref{eqn:5.C.1.130}),\nand (\\ref{eqn:1stSmall73c}) are used.\nIn the same way, the last term in Eq. (\\ref{eqn:1stSmall80d}) reads\n\\begin{eqnarray}\n&&\n\\hspace*{-5mm}\n\\biggl\\| E_{11}^{-1}\n\\frac{d}{dz}\n(Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}\n\\biggr\\|\n\\nonumber\\\\\n&\\leq&\n\\| E_{11}^{-1} \\|\n\\biggl\\|\n(Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}\n\\biggr\\|^2\n\\biggl[\n\\biggl\\|\n\\frac{dE_{11}^{-1}}{dz}\nE_{10} E_{00}^{-1} E_{01}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\\biggl\\|\nE_{11}^{-1}\\frac{dE_{10} }{dz}\nE_{00}^{-1} E_{01}\n\\biggr\\|\n+\\biggl\\|\nE_{11}^{-1}E_{10} E_{00}^{-1}\\frac{d E_{01}}{dz}\n\\biggr\\|\n+\\biggl\\|\nE_{11}^{-1}E_{10} \\frac{d E_{00}^{-1}}{dz} E_{01}\n\\biggr\\|\n\\biggr]\n\\label{eqn:1stSmall110a}\\\\\n&=&\nO(z^{-1} \\log z ),\n\\label{eqn:1stSmall110b}\n\\end{eqnarray}\nas $z \\to 0$.\nBy substituting Eqs. (\\ref{eqn:1stSmall90b}), (\\ref{eqn:1stSmall95d}), (\\ref{eqn:1stSmall100b}),\nand (\\ref{eqn:1stSmall110b}) into Eq. (\\ref{eqn:1stSmall80d}),\none finally obtains Eq. (\\ref{eqn:1stSmall80e}).\nWe can now show Eq. (\\ref{eqn:1stSmall7}) for $r=1$ by setting Eqs. (\\ref{eqn:1stSmall70})\nand (\\ref{eqn:1stSmall80e}) into Eq. (\\ref{eqn:1stSmall50}).\n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\nIf we start with expression (\\ref{eqn:5.C.1.125b}),\nwe obtain the following lemma instead of Lemma \\ref{lm:1stremainder+}.\n\n\n\n\\begin{lm} \\label{lm:1stremainder-}\n: Assume that $0$ is an exceptional point of the first kind for $H$.\nThen the $0$-th and the first derivative of $\\tilde{R} (z)$\nare approximated by those of a finite series\n\\begin{eqnarray}\n&&\n\\frac{1}{\\lambda^2 z (\\log z -2\\pi i)}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber\\\\\n&&\n+\n\\frac{1}{\\lambda^4 z (\\log z -2\\pi i)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 - \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} ,\n\\label{eqn:1stSmall5-}\n\\end{eqnarray}\nthat is, it is shown that\n\\begin{eqnarray}\n&&\\hspace{-10mm}\n\\biggl\\|\n\\frac{d^r}{dz^r}\n\\biggl[\n\\tilde{R}(z)\n-\n\\frac{1}{\\lambda^2 z (\\log z -2\\pi i)}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\n\\frac{1}{\\lambda^4 z (\\log z -2\\pi i)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 - \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr]\n\\biggr\\|\n\\nonumber \\\\\n&=&\nO(z^{-1} (\\log z -2\\pi i)^{-3}) ~~\\mbox{for}~~r=0,~~~\\mbox{or}~~~\nO(z^{-2} (\\log z -2\\pi i)^{-3}) ~~\\mbox{for}~~r=1,\n\\label{eqn:1stSmall7-}\n\\end{eqnarray}\nas $z \\to 0$.\n\\end{lm}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The reduced time evolution operator}\n\\label{sec:6}\n\n\n\nIn this section, we show that the reduced time evolution operator\nis expressed by the Fourier transform of the imaginary part of the reduced resolvent\nboth in the regular case and the exceptional case of the first kind.\nWe here define the reduced time evolution operator by the $N\\times N$ matrix $\\tilde{U}(t)$\nof the components\n$\\tilde{U}_{mn}(t):=\\langle m |Pe^{-itH}P| n \\rangle$,\nwhere $P=E((0, \\infty ))$ and $\\{ E(B) | B \\in \\mathbb{B} \\}$ is\nthe spectral measure of $H$, which is a family of the projection operator.\n$\\mathbb{B}$ is the Borel field of $\\mathbb{R}$.\n\n\n\n\\begin{lm}\n\\label{lm:reduced time evolution operator }\n:\nWe assume that Eq. (\\ref{eqn:.110}) holds so that there is no positive eigenvalue.\nThen, for the system with the rational form-factor (\\ref{eqn:formfactor1}),\nit holds that\n\\begin{equation}\n\\tilde{U}(t)\n=\n\\frac{1}{\\pi}\n\\int_{(0, \\infty )}\ne^{-it\\omega} {\\rm Im} \\tilde{R}^+ (\\omega) d\\omega\n=\n\\lim_{r \\to +0}\n\\frac{1}{\\pi}\n\\int_r^\\infty\ne^{-it\\omega} {\\rm Im} \\tilde{R}^+ (\\omega) d\\omega ,\n\\label{eqn:6.120}\n\\end{equation}\nboth in the regular case and the exceptional case of the first kind,\nwhere\n\\begin{eqnarray}\n{\\rm Im} \\tilde{R}^+ (\\omega)\n:=\n\\frac{1}{2 i} [\\tilde{R}^+(\\omega) -\\tilde{R}^-(\\omega)] ,\n\\label{eqn:6.65}\n\\end{eqnarray}\nwhich is sometimes called the spectral density.\n\\end{lm}\n\n\n\n\n\n\n{\\sl Proof} :\nLet us remember that the matrix $\\tilde{U}(t)$ is expressed by the spectral measure as\n\\begin{equation}\n\\tilde{U}_{mn}(t)\n=\n\\int_{(0, \\infty )}\ne^{-it\\lambda }\nd \\langle m |E(\\lambda )| n \\rangle\n=\n\\int_{(0, \\infty )}\ne^{-it\\lambda }\nd \\tilde{E}_{mn}(\\lambda ) ,\n\\label{eqn:6.110}\n\\end{equation}\nwhere $\\tilde{E}(B)$ is the matrix of the components $\\langle m |E(B)|n \\rangle $.\nTherefore, what we first should do is to clarify the relation between\n${\\rm Im} \\tilde{R}^+ (\\omega)$ and $\\tilde{E}(\\lambda )$.\nResorting to Stone's formula between $E(B)$ and $R(z)$,\nwe clearly see\n\\begin{equation}\n\\frac{1}{2}\n[\\tilde{E}([a,b])+\\tilde{E}((a,b))]\n=\\lim_{\\epsilon \\to +0}\n\\frac{1}{2\\pi i} \\int_a^b [\\tilde{R}(\\omega +i\\epsilon)\n-\\tilde{R}(\\omega -i\\epsilon)] d\\omega,\n\\label{eqn:6.30}\n\\end{equation}\nfor $a, b \\in \\mathbb{R}$ with $a<b$.\nUnder the assumption (\\ref{eqn:.110}),\nLebesgue's dominated convergence theorem and the proof of Lemma \\ref{lm:100} tell us that\nthe exchange between the limit and the integration in Eq. (\\ref{eqn:6.30}) is allowed\nfor $[a,b]\\subset (0,\\infty)$.\nIf $[a,b] \\subset (-\\infty, 0)\\backslash \\sigma(H)$,\nthen $\\tilde{R}^{\\pm} (\\omega) =\\tilde{R} (\\omega) $, and thus\n$\\tilde{E}([a,b])=\\tilde{E}((a,b))=0$.\nIn addition, by the continuity of $\\tilde{R}^{\\pm}(\\omega)$, Eq. (\\ref{eqn:6.30}) tells us\nthat $\\tilde{E}(\\{ a\\} )=0$ for all $a>0$, which leads to\n\\begin{equation}\n\\tilde{E}((a,b))\n=\\tilde{E}([a,b])=\n\\frac{1}{\\pi} \\int_a^b {\\rm Im} \\tilde{R}^+ (\\omega) d\\omega ,\n\\label{eqn:6.60}\n\\end{equation}\nfor all $a, b$ with $b>a>0$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nLet us now consider the regular case\nand in particular the validity of the expression (\\ref{eqn:6.60})\nfor the interval including the origin.\nIn this case, \n$\\tilde{R}^\\pm (0):=\\lim_{\\omega \\to +0}\\tilde{R}^\\pm (\\omega)$ exists to be finite.\nFurthermore, $\\lim_{\\omega \\to \\infty}\\tilde{R}^\\pm (\\omega)=0$\nfrom Lemma \\ref{lm:Large-omega}.\nThus, $\\tilde{R}^\\pm (\\omega)$ is uniformly continuous on $(0, \\infty)$.\nTherefore, we can take the limit of Eq. (\\ref{eqn:6.60}) as $a\\to +0$ to obtain \n$\\lim_{a\\to +0}E([a,b])=E((0,b])$.\nWe next see that all components of\n${\\rm Im} \\tilde{R}^+ (\\omega)$\nare integrable, i.e., belong to $L^1 ((0, \\infty ))$.\nSuppose that $| \\psi \\rangle \\in \\mathbb{C}^N$, then\n$\\langle \\psi | \\tilde{E}((0, \\lambda )) | \\psi \\rangle$\nis positive and a monotonically increasing function of $\\lambda$,\nand it is also differentiable in this case. Thus Eq. (\\ref{eqn:6.60}) tells us that\n$\\bra{\\psi} {\\rm Im} \\tilde{R}^+ (\\omega) \\ket{\\psi} \\geq 0$. In addition,\n\\begin{equation}\n\\| \\psi \\|^2\n\\geq \\lim_{\\lambda \\to \\infty }\n\\langle \\psi | \\tilde{E}((0, \\lambda )) | \\psi \\rangle\n= \\lim_{\\lambda \\to \\infty }\n\\frac{1}{\\pi} \\int_0^\\lambda\n\\bra{\\psi} {\\rm Im} \\tilde{R}^+ (\\omega) \\ket{\\psi} d\\omega\n=\n\\frac{1}{\\pi} \\int_0^\\infty\n\\bra{\\psi} {\\rm Im} \\tilde{R}^+ (\\omega) \\ket{\\psi} d\\omega .\n\\label{eqn:6.90}\n\\end{equation}\nHence, from the monotonic convergence theorem,\nwe see that $\\bra{\\psi} {\\rm Im} \\tilde{R}^+ (\\omega) \\ket{\\psi} \\in L^1 ((0, \\infty ))$.\nFrom this fact and the use of the polarization identity,\nwe can prove that all components of the matrix\n${\\rm Im} \\tilde{R}^+ (\\omega)$ are integrable.\nThus, extending the rhs of Eq. (\\ref{eqn:6.60}) to arbitrary\n$B \\in \\{ B \\in \\mathbb{B}| B \\subset (0,\\infty) \\}$,\n$\\int_B {\\rm Im} \\tilde{R}^+ (\\omega) d\\omega$ defines a measure.\nWe can now see from Eq. (\\ref{eqn:6.60}) and from E. Hopf's extension theorem that\n\\begin{equation}\n\\tilde{E}(B)\n=\n\\frac{1}{\\pi} \\int_B\n{\\rm Im} \\tilde{R}^+ (\\omega) d\\omega\n\\label{eqn:6.100}\n\\end{equation}\nholds for all $B \\in \\{ B \\in \\mathbb{B}| B \\subset (0,\\infty) \\}$.\nNote that this expression means that\nthe restriction of $\\tilde{E}_{mn}(B) $ to\n$\\{ B \\in \\mathbb{B}| B \\subset (0,\\infty) \\}$\nis absolutely continuous.\nTherefore, rewriting of $\\tilde{U}(t)$ in Eq. (\\ref{eqn:6.110}) into (\\ref{eqn:6.120})\nis straightforward.\n\n\nIn the exceptional case of the first kind,\nfrom the assumption (\\ref{eqn:.110}), $\\tilde{R}^\\pm (\\omega)$ is continuous on $(0, \\infty)$,\nwhile\n$\\tilde{R}^\\pm (\\omega)=O((\\omega \\log \\omega)^{-1})$ as $\\omega \\to +0$,\nso that it is not integrable around $0$.\nSee, Eq. (\\ref{eqn:5.C.1.120d}).\nHowever, ${\\rm Im} \\tilde{R}^+ (\\omega)$\nis of the order $O(\\omega^{-1} (\\log \\omega)^{-2})$ from Lemmas \\ref{lm:1stremainder+} and\n\\ref{lm:1stremainder-},\nand thus it is integrable around $0$.\nHence, Eq. (\\ref{eqn:6.100}) holds again, and\nEq. (\\ref{eqn:6.120}) is valid for this exceptional case.\n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\nWe remark that in the case of no negative eigenvalues (point spectrum) of $H$, \n\\cite{Miyamoto(2005)}\nthe restriction of $\\tilde{U}(t)$\nto the continuous energy spectrum is removed because in such a case $P=I$ the identity.\nFurthermore, the connection between $\\tilde{U}(t)$ and the observables is easily found, e.g.,\n$|\\bra{\\psi}\\tilde{U}(t)\\ket{\\psi}|^2\/\\| P\\ket{\\psi} \\|^4$\nfor $| \\psi \\rangle \\in \\mathbb{C}^N$ (or $\\mathbb{C}^N \\oplus \\{ 0\\}$)\nis the survival probability of $P\\ket{\\psi}$\nwhich is the probability of finding the system in the state $P\\ket{\\psi}$\nat the later time $t$,\nwhere $P\\ket{\\psi}$ is just the decaying component of the initial state $\\ket{\\psi}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The asymptotic expansion of the reduced time evolution operator}\n\\label{sec:7}\n\n\n\n\n\nWe can finally show the asymptotic formula for $\\tilde{U}(t)$ at long times\nfor the rational form factors satisfying our assumptions.\nIn the following, we assume that Eq. (\\ref{eqn:.110}) holds, i.e.,\nthere is no positive eigenvalue.\nHowever, this is not explicitly mentioned in the statements of the theorems.\nLet us first consider the regular case. For this purpose,\naccording to Lemma \\ref{lm:rffimremainder},\nwe introduce the remainder $F(\\omega)$ in the following way:\n\\begin{equation}\n\\frac{1}{\\pi}{\\rm Im} \\tilde{R}^+ (\\omega)\n=\n\\lambda^2\n(K(0))^{-1}\n\\omega^{n_b} ({\\mit \\Gamma}_{n_b} +\\omega {\\mit \\Gamma}_{n_b +1} + \\omega^2 {\\mit \\Gamma}_{n_b +2} )\n(K(0))^{-1} +F(\\omega),\n\\label{eqn:rffimbothLong110}\n\\end{equation}\nfor $\\omega >0$.\n\n\n\n\\begin{thm\n\\label{thm:rffimLong}\n: Assume that $0$ is a regular point for $H$.\nFor a system with the rational form factor (\\ref{eqn:formfactor1}) characterized by\nthe positive integers $n_a$ and $n_b$ that satisfy\nthat $n_b \\geq 2$ and $n_a =1$,\nthe reduced time evolution operator $\\tilde{U}(t)$ behaves asymptotically as\n\\begin{equation}\n\\tilde{U}(t)\n=\n\\lambda^2\n\\frac{\\Gamma(1+n_b)}{(it)^{n_b+1}}\n(K(0))^{-1}\n{\\mit \\Gamma}_{n_b} (K(0))^{-1}\n+O(t^{-n_b-2}) ,\n\\label{eqn:rffimbothLong50}\n\\end{equation}\nas $t\\to \\infty$.\nWhen $n_b=1$ and $n_a \\geq 1$, the error term is replaced by\n$O(t^{-3}\\log t)$.\n\\end{thm}\n\n\n\n\n{\\sl Proof} :\nWe first summarize the several properties of ${\\rm Im} \\tilde{R}^+ (\\omega) $.\nBy Lemma \\ref{lm:rffimremainder}, we see that the remainder $F(\\omega)$ in\nEq. (\\ref{eqn:rffimbothLong110}) is arbitrary-times differentiable.\nParticularly it holds that\n$\\lim_{\\omega \\to 0} d^r F(\\omega) \/ d \\omega^r = 0$\nfor all $r \\leq n_b$, and\n\\begin{eqnarray}\n\\left\\| \\frac{d^{n_b+1} F(\\omega)}{d\\omega^{n_b+1}} \\right\\|\n&&=\nO(\\log \\omega), ~\n\\mbox{or} ~\nO(1)\n,\n\\label{eqn:rffimbothLong120}\n\\end{eqnarray}\nfor $n_b=1$ and $n_a \\geq 1$, or $n_b \\geq 2$ and $n_a =1$, respectively, and\n\\begin{equation}\n\\left\\| \\frac{d^{n_b+2} F(\\omega)}{d\\omega^{n_b+2}} \\right\\|\n=\nO(\\omega^{-1}),\n\\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(1) & (n_b \\geq 3)\\\\\nO(\\log \\omega) & (n_b = 2)\n\\end{array}\n\\right. ,\n\\label{eqn:rffimbothLong130}\n\\end{equation}\nfor $n_b=1$ and $n_a \\geq 1$, or $n_b \\geq 2$ and $n_a =1$, respectively,\nas $\\omega \\to 0$.\nOn the other hand, we see from Lemma \\ref{lm:Large-omega} that\n$ (d \/ d \\omega )^r {\\rm Im} \\tilde{R}^+ (\\omega) = O(\\omega^{-r-1})$\nas $\\omega \\rightarrow \\infty $.\nIn particular, if $m \\geq 1$,\n$(d \/d \\omega )^m {\\rm Im} \\tilde{R}^+ (\\omega)$ is integrable\non $[\\delta , \\infty )$ for an arbitrary $\\delta >0$.\n\n\nLet us now split the integral in Eq. (\\ref{eqn:6.120})\ninto two parts by writing\n\\begin{equation}\n{\\rm Im} \\tilde{R}^+ (\\omega ) = \\phi (\\omega ) {\\rm Im} \\tilde{R}^+ (\\omega )\n+ (1- \\phi (\\omega ) ) {\\rm Im} \\tilde{R}^+ (\\omega ) ,\n\\label{eqn:rffimbothLong135}\n\\end{equation}\nwhere $\\phi \\in C_0 ^{\\infty} ([0,\\infty ) )$ and satisfies\n$\\phi (\\omega ) =1$ in a neighborhood of $\\omega = 0$.\nSuch a function is realized by $f (\\omega ) = 1- \\int_0 ^\\omega\ng(x) d x$, where $g(x) = h(x)\/\\int_{\\bf R} h(x) d x$ and\n$h(x) = \\exp(-1\/[a^2-(x-d)^2 ])$ ~($|x-d|<a$) or $0$ ($|x-d| \\geq a$)\nwith $d>a>0$.\n\n\nFrom Lemma 10.1 in Ref. \\makebox(0,1){\\Large \\cite{Jensen(1979)}}\\hspace{2mm}\nand the above-mentioned discussion, we see that\n$(1- \\phi (\\omega ) ) {\\rm Im} \\tilde{R}^+ (\\omega ) $ has a contribution of\n$O(t^{-m})$ to $\\tilde{U}(t)$ for an arbitrary $m\\geq 1$,\ni.e., this term decays faster than any negative power of $t$.\n\n\nOn the other hand, the contribution of\n$\\phi (\\omega ) {\\rm Im} \\tilde{R}^+ (\\omega ) $ to $\\tilde{U}(t)$\ngives\nthe main part of the asymptotic expansion.\nThen, the coefficient of ${\\mit \\Gamma}_{n_b}$, ${\\mit \\Gamma}_{n_b +1}$, and ${\\mit \\Gamma}_{n_b +2}$\nis given by the form \\cite{Copson}\n\\begin{eqnarray}\n\\int_0 ^{\\infty }\n\\phi (\\omega ) \\omega^q e^{-i t \\omega } ~ d \\omega\n& = &\n\\sum_{k=0}^{N-1}\n\\frac{1}{(it)^{k+1}}\n\\left. \\frac{d^k \\omega^q \\phi(\\omega)}{d\\omega^k} \\right|_{\\omega=0}\n+R_N (t)\n=\n\\frac{\\Gamma(1+q)}{(it)^{1+q}}\n+R_N (t) ,\n\\label{eqn:rffimbothLong140}\n\\end{eqnarray}\nfor all $N\\geq 1+q$, where $q$ takes the value $n_b$, $n_b+1$, or $n_b+2$.\nWe here used that\n\\begin{equation}\n\\frac{d^k \\omega^q \\phi(\\omega)}{d\\omega^k}\\biggr|_{\\omega=0}\n=\n\\sum_{j=0}^{\\min \\{ k,q \\}} {{k}\\choose{j}}\n\\frac{d^j \\omega^q }{d\\omega^j }\\biggr|_{\\omega=0}\n\\frac{d^{k-j} \\phi(\\omega)}{d\\omega^{k-j}}\\biggr|_{\\omega=0}\n=\\Gamma(1+q) \\delta_{kq},\n\\label{eqn:rffimbothLong145}\n\\end{equation}\nwhere $\\Gamma(1+n)=\\int_0^\\infty x^n e^{-x} dx$ is the gamma function.\nIn addition, the remainder $R_N (t)$ is bounded above by\n\\begin{equation}\n|R_N (t)|\n\\leq\n\\frac{1}{t^N} \\left| \\int_{0}^{\\infty}\n\\frac{d^N \\omega^q \\phi(\\omega)}{d\\omega^N} e^{-i\\omega t} d\\omega \\right|\n=\no(t^{-N}).\n\\label{eqn:rffimbothLong160}\n\\end{equation}\nNote that\nsince all\nderivatives of $\\phi(\\omega)$ vanish in the neighborhood of $\\omega =0$,\nEq. (\\ref{eqn:rffimbothLong140}) is valid for all $N\\geq q+1$ and thus\n$R_N (t)$ decays faster than any\nnegative power of $t$.\nFurthermore, we understand, by applying Eq. (\\ref{eqn:A.50a}) in\nLemma\\ \\ref{lm:Jensen and Kato, Lemma 10.2} directly to $\\phi (\\omega ) F(\\omega) $\nwith the discussion in the first part of this section, that\nthe contribution of\nthe Fourier transform of the remainder $\\phi (\\omega ) F(\\omega) $\nto $\\tilde{U}(t)$ is\n\\begin{equation}\nO(t^{-3}\\log t)~~\n\\mbox{or} ~~\nO(t^{-n_b-2}) ,\n\\label{eqn:rffimbothLong180}\n\\end{equation}\nfor $n_b=1$ and $n_a \\geq 1$, or $n_b \\geq 2$ and $n_a =1$, respectively, as $\\omega \\to 0$,\nwhere we used the formula of the indefinite integral that\n$\\int (\\log \\omega)^2 d\\omega =\\omega [(\\log \\omega)^2 -2\\log \\omega +2]$.\nSummarizing the above-noted results, we finally obtain that\n\\begin{eqnarray}\n&&\\hspace{-10mm}\n\\left\\| \\tilde{U}(t) -\n\\lambda^2\n(K(0))^{-1}\n\\left[\n\\frac{\\Gamma(1+n_b)}{(it)^{n_b+1}} {\\mit \\Gamma}_{n_b} + \\frac{\\Gamma(2+n_b)}{(it)^{n_b+2}} {\\mit \\Gamma}_{n_b +1} +\\frac{\\Gamma(3+n_b)}{(it)^{n_b+3}} {\\mit \\Gamma}_{n_b +2}\n\\right]\n(K(0))^{-1}\n\\right\\|\n\\nonumber  \\\\\n&&\\hspace{-10mm}\\leq\n\\left\\|\n\\frac{1}{\\pi}\\int_0^{\\infty}\n(1-\\phi(\\omega)){\\rm Im} \\tilde{R}^+ (\\omega ) e^{-i t \\omega } d \\omega\n\\right\\|\n+\n\\left\\|\n\\frac{1}{\\pi}\\int_0^{\\infty}\n\\phi(\\omega) F(\\omega) e^{-i t \\omega } d\\omega\n\\right\\| +O(t^{-N})\n\\nonumber  \\\\\n&&\\hspace{-10mm}=\nO(t^{-3}\\log t)~~\n\\mbox{or} ~~\nO(t^{-n_b-2}) ,\n\\label{eqn:rffimbothLong150}\n\\end{eqnarray}\nfor $n_b=1$ and $n_a \\geq 1$, or $n_b \\geq 2$ and $n_a =1$, respectively,\nas $t\\to \\infty$, where $O(t^{-N})$ is due to the contribution from $R_N (t)$.\nThis is just the asymptotic expansion of $\\tilde{U}(t)$\nin the statement.\n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\n\n\nIt is worth noting that if we resort to Lemma \\ref{lm:rffremainder},\ninstead of Eqs. (\\ref{eqn:rffimbothLong120}) and (\\ref{eqn:rffimbothLong130}),\nwe have\n\\begin{eqnarray}\n\\hspace*{-5mm}\n\\frac{d^r F(\\omega)}{d\\omega^r}\n&&=\nO(\\omega^{2-r} (\\log \\omega)^{1+\\theta (2-r)}), ~\n\\mbox{or} ~\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[2-r]^+}) & (0 \\leq r \\leq n_b)\\\\\nO(\\omega^{n_b +1-r} (\\log \\omega)^{\\theta (n_b +1 -r)}) & (r \\geq n_b +1)\n\\end{array}\n\\right.\n,\n\\label{eqn:rffimbothLong170}\n\\end{eqnarray}\nfor $n_b=1$ and $n_a \\geq 1$, or $n_b \\geq 2$ and $n_a =1$, respectively.\nHowever, in the latter case, we see that the Fourier transform of $\\phi(\\omega)F(\\omega)$\ngives the contribution of the order $O(t^{-n_b -1})$,\nwhich is just the same order as that coming from the dominant one.\nHence, we can only obtain an useless estimation.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe next show the asymptotic formula for $\\tilde{U}(t)$ at long times \nfor a system with an exceptional point of the first kind. To this end,\nwe write ${\\rm Im} \\tilde{R}^+(\\omega)$ with the remainder $F(\\omega)$\nagain as follows:\n\\begin{equation}\n\\frac{1}{\\pi}{\\rm Im} \\tilde{R}^+ (\\omega)\n=\n\\frac{1}{\\lambda^2 \\omega (\\log \\omega)^2} (Q_1 {\\mit \\Gamma}_1 Q_1)^{-1} +F(\\omega) .\n\\label{eqn:rffimLong1st110}\n\\end{equation}\n\n\n\n\\begin{thm\n\\label{thm:rffLong1st}\n: Assume that $0$ is an exceptional point of the first kind for $H$,\nwhich necessarily imposes that $n_b =1$.\nThen, the reduced time evolution operator $\\tilde{U}(t)$\nfor the rational form factor (\\ref{eqn:formfactor1}) behaves asymptotically as\n\\begin{equation}\n\\tilde{U}(t)\n=\n\\frac{1}{\\lambda^2 \\log t}\n(Q_1 {\\mit \\Gamma}_1 Q_1)^{-1} +O((\\log t)^{-2}),\n\\label{eqn:rffimLong1st50}\n\\end{equation}\nas $t \\to \\infty$.\n\\end{thm}\n\n\n\n\n{\\sl Proof} :\nLet us first look over the some properties of ${\\rm Im} \\tilde{R}^+ (\\omega) $ again.\nBy Lemmas \\ref{lm:1stremainder+} and \\ref{lm:1stremainder-},\nwe see that the remainder $F(\\omega)$ in\nEq. (\\ref{eqn:rffimLong1st110}) is arbitrary-times differentiable,\nsatisfies that $F(\\omega) = O(\\omega^{-1} (\\log \\omega)^{-3})$,\nand $d F(\\omega) \/ d \\omega = O(\\omega^{-2} (\\log \\omega)^{-3}) $,\nas $\\omega \\to +0$.\nOn the other hand, we see from Lemma \\ref{lm:Large-omega} that\n$ (d \/ d \\omega )^r {\\rm Im} \\tilde{R}^+ (\\omega) = O(\\omega^{-r-1})$\nas $\\omega \\rightarrow \\infty $.\nIn particular, if $m \\geq 1$,\n$(d \/d \\omega )^m {\\rm Im} \\tilde{R}^+ (\\omega)$ is integrable\non $[\\delta , \\infty )$ for an arbitrary $\\delta >0$.\n\n\nWe now split the integral in Eq. (\\ref{eqn:6.120})\ninto two parts as in Eq. (\\ref{eqn:rffimbothLong135}) again using\nthe $C_0 ^{\\infty}$-function $\\phi(\\omega )$.\nFrom Lemma 10.1 in Ref. \\makebox(0,1){\\Large \\cite{Jensen(1979)}}\\hspace{2mm}\nand the discussion mentioned above,\n$(1- \\phi (\\omega ) ) {\\rm Im} \\tilde{R}^+ (\\omega ) $ has a contribution of\n$O(t^{-m})$ to $\\tilde{U}(t)$ for an arbitrary $m\\geq 1$.\nOn the other hand, the contribution of\n$\\phi (\\omega ) {\\rm Im} \\tilde{R}^+ (\\omega ) $ to $\\tilde{U}(t)$\ngives\nthe dominant part of the asymptotic expansion.\nThen, the dominant time dependence of the asymptote of $\\tilde{U}(t)$ follows from\nLemma \\ref{lm:asymptote of inverse-lagarithmic-fourier-integral}, that is,\n\\begin{eqnarray}\n\\int_0 ^{\\infty }\n\\phi (\\omega ) (\\omega (\\log \\omega)^2)^{-1} e^{-i t \\omega } ~ d \\omega\n=\n(\\log t)^{-1} +O((\\log t)^{-2} ).\n\\label{eqn:rffimLong1st120}\n\\end{eqnarray}\nFurthermore,\nthe contribution of the Fourier transform of the remainder $\\phi (\\omega ) F(\\omega) $\nto $\\tilde{U}(t)$ can be estimated\nby the similar manner to Lemma\\ \\ref{lm:asymptote of inverse-lagarithmic-fourier-integral},\nrather than Lemma\\ \\ref{lm:Jensen and Kato, Lemma 10.2}.\nBy setting $\\sigma (\\omega)=F(\\omega) e^{-it\\omega}$\ninstead in the proof of Lemma \\ref{lm:asymptote of inverse-lagarithmic-fourier-integral},\nwe can apply it to this case,\nand we have\n\\begin{equation}\n\\int_0^\\infty F(\\omega) \\phi(\\omega) e^{-it\\omega} d\\omega\n=\n-\\lim_{\\omega \\to +0}(\\hat{I} \\sigma)(\\omega)\n+(-1)^N\n\\int_0^\\infty (\\hat{I}^N \\sigma)(\\omega) \\frac{d^N \\phi(\\omega)}{d\\omega^N} d\\omega .\n\\label{eqn:rffimLong1st155}\n\\end{equation}\nThen, corresponding to Eq. (\\ref{eqn:B.530}), we have\n\\begin{equation}\n\\|(\\hat{I}\\sigma)(\\omega)\\|\n=\n\\biggl\\|\ni e^{-it\\omega} \\int_0^\\infty F(\\omega-i\\eta) e^{-t\\eta} d\\eta\n\\biggr\\|\n+\nE(t)\n\\leq\nC\n\\int_0^\\infty \\bigr| (\\omega-i\\eta)^{-1} [\\log (\\omega-i\\eta)]^{-3} \\bigr| e^{-t\\eta} d\\eta\n+\nE(t),\n\\label{eqn:rffimLong1st160}\n\\end{equation}\nwith an appropriate constant $C$.\nNote that in this procedure, $\\tilde{R}^+ (\\omega)$ is analytically continued to\nthe lower plane of the second Riemann sheet, while $\\tilde{R}^- (\\omega)$ still remains\nin the lower plane of the first Riemann sheet.\nThen, both are ensured to contribute the remainders\nof the same order to $F(\\omega-i\\eta)$ in the above integral\nfrom Lemmas \\ref{lm:1stremainder+} and \\ref{lm:1stremainder-}.\nThe remainder term $E(t)$ in Eq. (\\ref{eqn:rffimLong1st160}) \nthat gives the order of $O(e^{-\\gamma t})$ for some $\\gamma >0$ \nis responsible for the possible poles\nof $\\tilde{R}^+ (\\omega)$ continued to the second Riemann sheet,\nthe number of which are guaranteed to be finite from\nthe analytic Fredholm theorem \\cite{the analytic Fredholm theorem}\nand Lemma \\ref{lm:Large-omega} for\nthe continued  $\\tilde{R}^+ (\\omega)$.\nThus, it follows from Lemma \\ref{lm:asymptote of inverse-lagarithmic-fourier-integral}\nthat $\\lim_{\\omega \\to +0} \\|(\\hat{I}\\sigma)(\\omega)\\| =O((\\log t)^{-2})$.\nBy the same argument as in Lemma \\ref{lm:asymptote of inverse-lagarithmic-fourier-integral},\none also sees that the remainder term in Eq. (\\ref{eqn:rffimLong1st155})\nis of the order of $O(t^{-N+1})$.\nSummarizing these arguments, we can finish the proof of the theorem.\n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Concluding remarks}\n\\label{sec:8}\n\n\n\n\nWe have rigorously derived the asymptotic formula of the reduced time evolution\noperator for the $N$-level Friedrichs model in the context of the zero energy resonance \n\\cite{Jensen(1979)}\nboth for the regular case and the exceptional case of the first kind.\nThen, in the latter case, the logarithmically slow decay proportional to $(\\log t)^{-1}$\nhas been found,\nand the expansion coefficient has been explicitly presented\nby the projection operator associated with the zero energy eigenstates\nof the total Hamiltonian, which is an extended state not belonging to the Hilbert space.\nWe note that the decay involving the logarithmic function \nexpressed by $t^{-j}(\\log t)^{k}$ ($j=1, 2, \\ldots$ and $k=0, \\pm 1, \\ldots$)\ncan occur in the short range potential systems in the even dimensional space. \\cite{Murata(1982)}\nIt should be noted that a realization of the exceptional cases require \nthe parameters, e.g., the coupling constant $\\lambda$, \nto take such special values \nthat the matrix $K(0)$ in Eq. (\\ref{eqn:.336}) has a zero eigenvalue. \nIn addition, some of the form factors $v_n(\\omega)$ have to behave as $|v_n(\\omega)|^2 \\sim c_n \\omega$ \naround $\\omega=0$. \nIn other words, if all of them behave as \n$|v_n(\\omega)|^2 \\sim c_n \\omega^{q_n}$ with $q_n \\geq 2$, \nthe exceptional case of the first kind never occurs \nthough that of the second kind could happen. \nThese circumstances explain how the exceptional cases are surely exceptional. \nThe presented results also enable us to calculate the asymptotic formula for the survival\nprobability of an arbitrary initial state $\\ket{\\psi}$ localized over the $N$ discrete levels.\nIf we choose the special initial state to satisfy\n${\\mit \\Gamma}_{n_b} (K(0))^{-1}\\ket{\\psi}=0$ in Eq. (\\ref{eqn:rffimbothLong50})\nor $(Q_1 {\\mit \\Gamma}_1 Q_1)^{-1}\\ket{\\psi}=0$ in Eq. (\\ref{eqn:rffimLong1st50}),\nour estimations are useless and other decay laws could appear. \\cite{Miyamoto(2004)}\nThe long time behavior of the reduced time evolution operator\nfor the exceptional case of the second and the third kind are not examined.\nAs is expected, in the former case, the non decaying component\nassociated with the localized zero energy eigenstate will appear\ndue to the divergent behavior of $Q_2\\tilde{R}(z)Q_2=O(z^{-1})$ in Eq. (\\ref{eqn:5.C.2.120d}).\nThe latter case can occur in the $N$-level cases of the model only for $N\\geq 3$,\nwhich yields a more complicated situation.\nIn the whole of the paper, we assumed that there is no bound eigenstate\nwith a positive eigenenergy. This situation is actually realized in the weak coupling cases.\n\\cite{Davies(1974),Miyamoto(2005)} However,\nits compatibility with the existence of the extended zero energy eigenstate\nis still not clear in the multilevel cases (except the single level case).\nThe emergence of the logarithmic decay $(\\log t)^{-1}$ is just due to\nthe logarithmic energy dependence of the self energy $S(\\omega)$\nand it comes from the assumption (\\ref{eqn:formfactor1}) \nwhere $|v_n(\\omega)|^2 \\sim c_n \\omega^{q_n}$ with a positive integer $q_n$ is required. \nTherefore, if we choose another type of form factor,\nit is not necessary for such a slow decay to occur even in the exceptional case.\n\\cite{Kofman(1994),Lewenstein(2000),Nakazato(2003)}\nHowever, we stress that our assumption is often satisfied by\nactual systems. \\cite{Facchi(1998),Antoniou(2001)}\nThe experimental realization of the exceptional case\nrequires\nthe setup of parameters like $\\omega_1 \\simeq \\lambda^2 \\Lambda $, \nwhere $\\Lambda$ is a typical cutoff constant. \nThis seems in a strong coupling region to be naturally satisfied\n\\cite{Jittoh(2005),Garcia-Calderon(2001)}, \nand hence it could be suggested to invoke the artificial quantum structures for a realization.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\n\nThe author would like to thank Professor I.\\ Ohba\nand Professor H.\\ Nakazato for useful comments.\nHe also would like to express his gratitude to the organizers\nof the International Workshop TQMFA2005, Palermo, Italy,\nNovember 11-13, 2005,\n{\\sl New Trends in Quantum Mechanics: Fundamental Aspects and Applications}.\nDiscussions during the YITP workshop YITP-W-05-21 on ``Fundamental Problems and Applications of\nQuantum Field Theory'' were useful in completing this work.\nThis research is partly supported by a Grant for The 21st Century COE Program\nat Waseda University from the Ministry of Education, Culture,\nSports, Science and Technology, Japan.\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA $k$-{\\it system} $\\mathcal{A}$ of arcs on a punctured surface $S$ is a collection of essential simple arcs on $S$ such that no two arcs of $\\mathcal{A}$ are homotopic or intersect more than $k$ times. We begin with the following observation.\n\n\\begin{remark}\\label{kequalszero}\nIf $k=0$, $S$ is an $n$-punctured sphere with $n \\geq 3$, and the arcs of $\\mathcal{A}$ all join a fixed pair of distinct punctures $p,q$ of $S$, then $|\\mathcal{A}| \\leq n-2$. To see this, fix a complete hyperbolic metric on $S$ of area $2\\pi(n-2)$ and realize the arcs of $\\mathcal{A}$ as geodesics on $S$. Cutting $S$ along $\\mathcal{A}$, we obtain a collection of hyperbolic punctured strips. Since the boundary of each strip consists of two arcs of $\\mathcal{A}$, and since each arc of $\\mathcal{A}$ appears twice as a boundary component of some strip, we count precisely $|\\mathcal{A}|$ strips. The bound on $|\\mathcal{A}|$ now follows from the fact that each of these strips has area at least $2\\pi$. Moreover, this bound is tight since we can easily devise a $0$-system on $S$ whose complement consists entirely of once-punctured strips (see Figure~\\ref{fig:zerosystem}). An area argument also shows that the maximum size of $\\mathcal{A}$ is $2n-5$ if we assume instead that $p=q$.\n\\end{remark}\n\n\\begin{figure}\n\\scalebox{.45}{\n\\includegraphics[trim=0 520 0 0, clip]{0system.eps}\n}\n\\captionof{figure}{A maximum-size $0$-system joining distinct punctures $p,q$ of the $7$-punctured sphere.}\n\\label{fig:zerosystem}\n\\end{figure}\n\nProblems involving bounding the size of a $k$-system of arcs, of which Remark~\\ref{kequalszero} serves as a trivial example, originated in similar problems for curves. Juvan, Malni\\v{c}, and Mohar introduced the term ``$k$-system\" and showed that the maximum size $N(k, \\Sigma)$ of a $k$-system of essential simple closed curves on a fixed compact surface $\\Sigma$ is finite \\cite{juvan1996systems}. Independently, Farb and Leininger inquired about $N(k, g) := N(k, \\Sigma)$ for $\\Sigma$ closed and oriented of genus $g$ and $k=1$. In response, Malestein, Rivin, and Theran provided an upper bound exponential in $g$, and showed that $N(1,2) = 12$ \\cite{malestein2014topological}. Also for $k=1$, Przytycki produced an upper bound on the order of $g^3$ \\cite[Theorem~1.4]{przytycki2015arcs}; since then, tighter bounds on $N(1,g)$ have been found by Aougab, Biringer, and Gaster \\cite{aougab2017packing}, and more recently, Greene \\cite{greene2018curves}. Moreover, Przytycki provided an upper bound on $N(k,g)$ for arbitrary $k$ that grows like $g^{k^2+k+1}$ \\cite[Cor.~1.6]{przytycki2015arcs}. This bound was subsequently improved by Greene to one that grows like $g^{k+1}\\log g$ \\cite{greene2018curvesii}. \n\nIn \\cite{przytycki2015arcs}, so as to prove the aforementioned results about $k$-systems of curves, Przytycki first proved stronger results about $k$-systems of arcs -- for example, that the maximum size of a $k$-system of arcs on a punctured surface $S$ of Euler characteristic $\\chi < 0$ (where distinct arcs are not required to have the same endpoints) grows like $|\\chi|^{k+1}$. In the same article, Przytycki proved the following:\n\n\\begin{theorem}\\cite[Theorem~1.7]{przytycki2015arcs}\\label{przytycki}\nLet $p,q$ be punctures of an $n$-punctured sphere $S$, where $n \\geq 3$. The maximum size of a $1$-system $\\mathcal{A}$ of arcs on $S$ joining $p$ and $q$ is $\\binom{n-1}{2}$. \n\\end{theorem}\n\nNote that $p$ and $q$ are not assumed to be distinct in the statement of Theorem~\\ref{przytycki}. More recently, Bar-Natan showed that for $S,p,q$ as in Theorem~\\ref{przytycki}, if $p=q$ then the maximum size of a $2$-system of arcs on $S$ joining $p$ and $q$ is $\\binom{n}{3}$ \\cite{bar2017arcs}. The main result of this article is that Bar-Natan's maximum holds for $p,q$ distinct:\n\n\\begin{theorem}\\label{maintheorem}\nLet $p,q$ be distinct punctures of an $n$-punctured sphere $S$, where $n \\geq 3$. The maximum size of a $2$-system $\\mathcal{A}$ of arcs on $S$ joining $p$ and $q$ is $\\binom{n}{3}$. \n\\end{theorem}\n\n\\subsection*{Organization} In Section \\ref{example}, we provide an example of a $2$-system of size $\\binom{n}{3}$ joining a fixed pair of distinct punctures of an $n$-punctured sphere for $n \\geq 3$. The remaining sections are concerned with proving that $\\binom{n}{3}$ is an upper bound on the size of such a $2$-system $\\mathcal{A}$. This is proved by induction on $n$; we prove that the number of arcs of $\\mathcal{A}$ that become homotopic after forgetting a puncture of $S$ is not too large. This, in turn, is proved by induction, via the following lemma:\n\n\\begin{lemma}\\label{relation}\nLet $S$ be an $n$-punctured sphere, and let $p,q,r$ be distinct punctures of $S$. Let $\\mathcal{P}, \\mathcal{Q}$ be $1$-systems of arcs starting at $r$ and ending at $p,q$, respectively, so that no arc of $\\mathcal{P}$ intersects an arc of $\\mathcal{Q}$. Suppose $\\mathcal{R} \\subset \\mathcal{P} \\times \\mathcal{Q}$ satisfies the following:\n\\begin{enumerate}[label=(\\roman*)]\n\\item there is at most one point of intersection between any two pairs of arcs in $\\mathcal{R}$;\n\\item the cyclic order around $r$ of any two intersecting pairs $(\\alpha, \\beta), (\\alpha', \\beta') \\in \\mathcal{R}$ with $\\alpha \\neq \\alpha', \\beta \\neq \\beta'$ is alternating.\n\\end{enumerate}\nThen $|\\mathcal{R}| \\leq \\binom{n-1}{2}$. \n\\end{lemma}\n\nSection \\ref{proofoflemma} is devoted to the proof of Lemma~\\ref{relation}. The inductive step again involves forgetting a puncture $s$ of $S$, but this time, we choose $s$ with care in order to control the subsequent behavior of the arcs of $\\mathcal{P}$ and $\\mathcal{Q}$. More precisely, we require that $s$ be $p$-{\\it isolated} (see Section \\ref{definitions} for definitions). \n\nTo guarantee that a puncture with this property exists, we take a detour into annular square diagrams. A $k$-{\\it system annular diagram} $A$ is an annular diagram whose dual curves constitute a $k$-system of arcs joining the boundary paths of $A$. Such a diagram arises as the dual square complex to a $k$-system $\\mathcal{A}$ on a punctured sphere with distinct prescribed endpoints. In Section \\ref{diagrams}, we prove the following:\n\n\n\\begin{theorem}\\label{corner}\nLet $A$ be a $1$-system annular diagram. Then either $A$ is a cycle, or $A$ has a corner on each of its boundary paths.\n\\end{theorem}\n\nRoughly speaking, a corner corresponds to an isolated puncture. Note that Theorem~\\ref{corner} does not hold for $k=2$; see Figure~\\ref{fig:diagrams} (bottom left) for a counterexample, suggested by Przytycki. \n\n\\subsection*{Acknowledgements} I thank my supervisor Piotr Przytycki for his crucial guidance and remarkable patience throughout the course of this article's preparation. I also thank Daniel Wise for providing helpful resources.\n\n\\section{Definitions}\\label{definitions}\n\n\\subsection{Arc systems} A {\\it puncture} is a topological end of a space $S$ obtained from a connected, oriented, compact surface $\\Sigma$ by removing finitely many points $p_1, \\ldots, p_n$ from $\\Sigma$. Note that the punctures of $S$ are in bijection with $p_1, \\ldots, p_n$, and that we allow punctures on the boundary of $\\Sigma$. If $p_1, \\ldots p_n$ are taken from the interior of $\\Sigma$, we call $S$ an $n$-{\\it punctured} $\\Sigma$. \n\nAn {\\it arc} on $S$ is a proper map $\\alpha: (0,1) \\rightarrow S$. A proper map induces a map between ends of topological spaces; in this sense, $\\alpha$ ``maps\" each endpoint of $(0,1)$ to a puncture $p$ of $S$. We call $p$ an {\\it end} of $\\alpha$. If $p,q$ are ends of $\\alpha$, we say $\\alpha$ {\\it starts} at $p$ and {\\it ends} at $q$, or that $\\alpha$ {\\it joins} $p$ and $q$. A {\\it segment} of $\\alpha$ is the restriction of $\\alpha$ to some positive-length subinterval of $(0,1)$. \n\nAn arc $\\alpha$ is {\\it simple} if it is an embedding, in which case we identify $\\alpha$ and its segments with their images in $S$. If $J$ is a subinterval of $(0,1)$ with endpoints $t_1, t_2$ and $\\alpha$ is a simple arc mapping $t_i$ to $x_i$ for $i=1,2$, we denote the segment $\\alpha \\bigr|_J$ by $(x_1x_2)_{\\alpha}$. If $R \\subset S$ is a subset and $p$ is an end of $\\alpha$ corresponding to an endpoint $t_0=0,1$ of $(0,1)$, we say the {\\it $p$-end of $\\alpha$ lies in $R$} if $\\alpha^{-1}(R)$ is a neighbourhood of $t_0$ in $(0,1)$.\n\nA {\\it homotopy} between arcs $\\alpha_1$ and $\\alpha_2$ is a proper map $(0,1) \\times [0,1] \\rightarrow S$ whose restrictions to $(0,1) \\times \\{0\\}$ and $(0,1) \\times \\{1\\}$ are $\\alpha_1$ and $\\alpha_2$, respectively. In particular, a homotopy preserves ends. If $r$ is a puncture of $S$, we say that two arcs on $S$ are {\\it $r$-homotopic} if they are homotopic on the surface $\\bar{S}$ obtained from $S$ by forgetting $r$. Two arcs are in {\\it minimal position} if the number of their intersection points cannot be decreased by a homotopy. Note that if a pair of arcs have a point of intersection that is not transversal, then they are not in minimal position. An arc $\\alpha$ is {\\it essential} if it cannot be homotoped into a puncture, in the sense that there is no proper map $(0,1) \\times [0,1) \\rightarrow S$ whose restriction to $(0,1) \\times \\{0\\}$ is $\\alpha$. Unless otherwise stated, all arcs in the article are simple and essential, and all intersections between arcs are transversal. Note that an arc joining distinct punctures of a punctured surface is automatically essential.\n\nLet $R$ be a closed disc with at most 2 punctures on its boundary and possibly with punctures in its interior. A {\\it region} between arcs $\\alpha_1, \\alpha_2$ on $S$ is a properly embedded $R \\subset S$ such that $\\partial R$ is a union of exactly two segments $\\sigma_1, \\sigma_2$, where $\\sigma_i$ is a segment of $\\alpha_i$ for $i=1,2$ (see Figure~\\ref{fig:region}). We say that $\\alpha_1, \\alpha_2$ (or, more specifically, $\\sigma_1, \\sigma_2$) {\\it form} or {\\it bound} $R$. If $R$ has no punctures in its interior, we say $R$ is {\\it empty}. If $R$ has exactly 0 (respectively, 1, 2) punctures on its boundary and $R \\cap (\\alpha_1 \\cup \\alpha_2) = \\partial R$, we call $R$ a {\\it bigon} (respectively, {\\it half-bigon}, {\\it strip}). We say $R$ is {\\it adjacent} to a puncture $p$ of $S$ if $p$ lies on the boundary of $R$. If $p,s$ are distinct punctures of $S$ and $\\mathcal{A}$ is a collection of arcs on $S$ with $s$  contained in a half-bigon or strip $H$ adjacent to $p$ formed by a pair of arcs of $\\mathcal{A}$ such that $H$ is a component of $S - \\bigcup \\mathcal{A}$, we say that $s$ is $p$-{\\it isolated by} $\\mathcal{A}$. \n\n\\begin{figure}\n\\scalebox{.5}{\n\\includegraphics[trim=0 550 0 0, clip]{region.eps}\n}\n\\captionof{figure}{Regions formed by arcs. The yellow region is a half-bigon. The pink and orange regions are bigons. The orange bigon is empty.}\n\\label{fig:region}\n\\end{figure}\n\nWe will make frequent use of the following lemma.\n\n\\begin{lemma}[The bigon criterion]\\label{bigoncriterion} \\cite[Proposition~1.7]{farb_margalit_2012}\nTwo intersecting arcs on a punctured surface are in minimal position if and only if they form no empty regions.\n\\end{lemma}\n\nSince an empty region bounded by intersecting arcs must contain an empty bigon or half-bigon, we immediately obtain the following corollary.\n\n\\begin{corollary}\\label{corollarybigon}\nTwo intersecting arcs on a punctured surface are in minimal position if and only if they form no empty bigons or half-bigons.  \n\\end{corollary}\n\nA {\\it $k$-system} of arcs on a punctured surface $S$ is a collection $\\mathcal{A}$ of essential simple arcs on $S$ such that no two arcs of $\\mathcal{A}$ are homotopic or have more than $k$ points of intersection. We will mainly consider the case where $S$ is a sphere punctured at least thrice and $\\mathcal{A}$ is a 2-system of arcs joining a fixed pair of distinct punctures $p,q$ of $S$. Note that for any two arcs $\\alpha_1, \\alpha_2$ of such a collection $\\mathcal{A}$, a region bounded by $\\alpha_1, \\alpha_2$ that contains neither $p$ nor $q$ must be a bigon, a half-bigon, or a strip.\n\nIf a punctured surface $S$ has Euler characteristic $\\chi < 0$, then $S$ admits a complete hyperbolic metric of area $2\\pi |\\chi|$. Under such a metric, the homotopy class of any arc contains a unique geodesic representative, and any two distinct geodesic arcs are in minimal position. Thus, for the purposes of determining the size of a $k$-system $\\mathcal{A}$ of arcs on $S$, we may assume that $\\mathcal{A}$ consists of geodesics. \n\n\n\n\n\\subsection{Combinatorial complexes} A map $X \\rightarrow Y$ between CW complexes $X,Y$ is {\\it combinatorial} if its restriction to each open cell of $X$ is a homeomorphism onto an open cell of $Y$. A CW complex $X$ is {\\it combinatorial} if the attaching map of each  cell in $X$ is combinatorial for some subdivision of the sphere. A cell of dimension 0 is a {\\it vertex} and a cell of dimension 1 is an {\\it edge}. The {\\it degree} of a vertex $v$ of $X$ is the number of edges in $X$ incident to $v$, with loops counted twice. \n\n \\subsection{Square complexes} An {$n$-cube} is a copy of $[-1,1]^n$. A {\\it square complex} $X$ is a combinatorial complex whose cells are $n$-cubes with $n \\leq 2$; that is, $X$ is a combinatorial 2-complex each of whose 2-cells is attached via a combinatorial map from a 4-cycle into the 1-skeleton of $X$. The cells of $X$ are called {\\it cubes}, and its 2-cells are called {\\it squares}. \n \n A {\\it midcube} is a subspace of a cube $[-1,1]^n$ obtained by restricting one coordinate to $0$. Let $U$ be a new square complex whose cells are midcubes of $X$ and whose attaching maps are restrictions of attaching maps in $X$ to midcubes. A {\\it dual curve} $\\alpha$ of a cube $c$ in $X$ is a connected component of $U$ containing a midcube of $c$. Note that if $c$ is a square, then it has exactly two dual curves. If $c$ is an edge, we say $\\alpha$ is {\\it dual} to $c$. We call $\\alpha$ an {\\it arc} if it is homeomorphic to an interval (possibly of length 0). There is a natural immersion $\\alpha \\rightarrow X$; if this map is an embedding, we say $\\alpha$ is {\\it simple}. In this case, we identify $\\alpha$ with its image in $X$.\n\n\\subsection{Annular diagrams} An {\\it annular diagram} $A$ is a finite combinatorial cell decomposition of $S^2$ minus two disjoint open 2-cells (see Figure~\\ref{fig:annulardiagram}). The attaching map of each of these 2-cells is a {\\it boundary path} of $A$. \n\\begin{figure}\n\\scalebox{.25}{\n\\includegraphics[trim=0 350 0 10, clip]{boundarypaths.eps}\n}\n\\captionof{figure}{An annular diagram. The boundary paths are indicated in red.}\n\\label{fig:annulardiagram}\n\\end{figure}\n\n\nWe call $A$ a {\\it square annular diagram}, or simply a {\\it diagram}, if it is also a square complex (see Figure~\\ref{fig:diagrams}). A {\\it corner} on a boundary path $P$ of a diagram $A$ is a vertex $v$ on $P$ of degree 2 that is contained in some square of $A$. \n\nLet $c$ be a square of a diagram $A$ with boundary path $P$, and let $x$ be the center of $c$. Suppose the dual curves $\\alpha, \\beta$ of $c$ are dual to consecutive edges $a,b$ on $P$ with shared vertex $v$. Let $\\gamma$ be the loop obtained from the subarcs of $\\alpha, \\beta$ joining $x$ and $P$ and the half-edges of $a,b$ containing $v$. If $\\gamma$ is homotopic in $A$ to a constant path, then we call $c$ a {\\it cornsquare} with {\\it outerpath} $ab$. \n\n\\begin{figure}\n\\scalebox{0.55}{\n\\includegraphics[trim=0 330 0 0, clip]{annulardiagrams2.eps}\n}\n\\captionof{figure}{Square annular diagrams. Dual curves are dashed and colored. Corners are colored red. The bottom left (respectively, bottom right) diagram is a $2$-system (respectively, $1$-system) annular diagram. Note that the square at which the blue and orange dual curves meet in the bottom right diagram is a cornsquare with an outerpath on each boundary path, while the square at which the green and purple dual curves meet is not a cornsquare, even though the latter two dual curves are dual to consecutive edges on each boundary path.} \n\\label{fig:diagrams}\n\\end{figure}\n\nA {\\it hexagon move} on a diagram $A$ is the replacement of three squares forming a subdivided hexagon by an alternate three squares forming a subdivided hexagon (see Figure \\ref{fig:hexagon}). A hexagon move can be visualized as a benign ``sliding\" operation on one of the dual curves of $A$, so that if $A'$ is obtained from $A$ by a hexagon move, there is a natural correspondence between the dual curves of $A$ and those of $A'$. Note that the number of squares of $A$ is preserved under hexagon moves.\n\n\\begin{figure}\n\\scalebox{.3}{\n\\includegraphics[trim=0 570 0 0, clip]{hexagon.eps}\n}\n\\captionof{figure}{A hexagon move and its effect on dual curves.}\n\\label{fig:hexagon}\n\\end{figure}\n\nA square annular diagram $A$ is a {\\it $k$-system annular diagram} if its dual curves are simple arcs joining the boundary paths of $A$ and pairwise intersecting at most $k$ times in $A$. Note that the number of intersections between any pair of dual curves of $A$ is preserved under a hexagon move. Thus, if $A'$ is obtained from a $k$-system annular diagram $A$ by a hexagon move, then $A'$ is also a $k$-system annular diagram. \n\n\n\n\n\n\n\\section{A 2-system of maximum size}\\label{example}\n\nWe provide an example of a 2-system of arcs of size $\\binom{n}{3}$ joining a fixed pair of distinct punctures of an $n$-punctured sphere $S$. This collection was independently discovered by Assaf Bar-Natan.\n\nWe think of $S$ as $\\mathbb{R}^2$ punctured at $p = (-1,0)$ and at the points $r_i = (i - \\frac{1}{2}, 0)$ for $i = 1, \\ldots, n-2$. We construct a 2-system $\\mathcal{A}$ joining $p$ and the puncture $q$ at infinity. \n\nLet $\\alpha_{< -1}$ be the arc given by the ray $\\{(x,0) : \\> x < -1\\}$. For $a, b, c \\in \\{0, 1, \\ldots, n-2\\}$ with $a < b <c$ or $0 < a < b = c = n-2$, let $\\alpha_{abc}$ be the graph of the polynomial function $f_{abc}: (-1, \\infty) \\rightarrow \\mathbb{R}$ given by $x \\mapsto (x+1)(x-a)(x-b)(x-c)$. \n\nNote that for distinct triples $(a,b,c), (a', b',c')$, the difference $f_{abc}-f_{a'b'c'}$ is a cubic polynomial, one of whose roots is $-1$. Thus, the $\\alpha_{abc}$ pairwise intersect at most twice. Furthermore, the $\\alpha_{abc}$ are pairwise non-homotopic \\cite[proof~of~Lemma~4.2]{bar2017arcs}, and $\\alpha_{<-1}$ is not homotopic to any of the $\\alpha_{abc}$ since the complement of $\\alpha_{<-1} \\cup \\alpha_{abc}$ is a pair of punctured strips. \n\nNow fix $M > 0$ such that $M > |f_{abc}(x)|$ for all $x \\in (-1, n-2]$. For each $i,j \\in \\{1, \\ldots, n-2\\}$ with $i < j$, let $\\alpha_{ij}$ be the union of the following horizontal and vertical segments: the segment joining $(-1,0)$ and $(-1, M)$, the segment joining $(-1,M)$ and $(i - \\frac{1}{2} + \\frac{1}{4}, M)$, the segment joining $(i - \\frac{1}{2} + \\frac{1}{4}, M)$ and $(i - \\frac{1}{2} + \\frac{1}{4}, -M)$, the segment joining $(i - \\frac{1}{2} + \\frac{1}{4}, -M)$ and $(-2, -M)$, the segment joining $(-2, -M)$ and $(-2, M+1)$, the segment joining $(-2, M+1)$ and $(j - \\frac{1}{2} - \\frac{1}{4}, M+1)$, and the vertical ray travelling down from $(j - \\frac{1}{2} - \\frac{1}{4}, M+1)$. Note that each $\\alpha_{ij}$ intersects $\\alpha_{<-1}$ exactly once and each $\\alpha_{abc}$ exactly twice (see Figure \\ref{fig:maxfamily}). Furthermore, each $\\alpha_{ij}$ is in minimal position with $\\alpha_{<-1}$ by Corollary~\\ref{corollarybigon}; since $\\alpha_{<-1}$ is disjoint from the $\\alpha_{abc}$, this shows that none of the $\\alpha_{ij}$ is homotopic to any of the $\\alpha_{abc}$. \n\nWe claim that the $\\alpha_{ij}$ are pairwise non-homotopic. Indeed, for $k \\in \\{1, \\ldots, n-3\\}$, let $\\gamma_k$ be the horizontal arc joining the punctures at $x = k- \\frac{1}{2}$ and $x = k + \\frac{1}{2}$, and note that $\\alpha_{ij}$ and $\\gamma_k$ are in minimal position by Corollary~\\ref{corollarybigon}. Since no two of the $\\alpha_{ij}$ share the same number of intersection points with each of the $\\gamma_k$, the $\\alpha_{ij}$ must be pairwise non-homotopic.\n\nWe claim further that the $\\alpha_{ij}$ pairwise intersect at most twice. Indeed, if $i,j,i',j' \\in \\{1, \\ldots, n-2\\}$ with $i \\leq i'$, then the number of intersection points between $\\alpha_{ij}, \\alpha_{i'j'}$ is determined by the order of $j, i', j'$. If $i' < j \\leq j'$, then $\\alpha_{ij}$ and $\\alpha_{i'j'}$ are disjoint (see Figure \\ref{fig:twistingcurves}, top). If $i' < j' < j$, then $\\alpha_{ij}$ and $\\alpha_{i'j'}$ intersect once (see Figure \\ref{fig:twistingcurves}, middle). Otherwise, $j \\leq i'$, and there are two points of intersection between $\\alpha_{ij}$ and $\\alpha_{i',j'}$ (see Figure \\ref{fig:twistingcurves}, bottom). Thus, the family $\\mathcal{A}$ consisting of $\\alpha_{<-1}$, the $\\alpha_{abc}$, and the $\\alpha_{ij}$ is a 2-system of size $1 + (n-3) + \\binom{n-1}{3} + \\binom{n-2}{2} = \\binom{n}{3}$. \n\n\\begin{figure}\n \\begin{tikzpicture}[scale=1]\n\n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n[->] (-2,0)--(4,0); \n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n    [->] (0,-3)--(0,5);\n   \n    \n\\begin{scope}\n\\clip (-2,-3) rectangle (4,5);\n\n\n\n\\draw[thick, green](0.5,0)--(1.5,0);\n\\draw [thick, red, domain=-1:4 , samples=500] plot (\\x , {0.1*(\\x+1)*\\x*(\\x-1)*(\\x-2)});\n\\draw [thick, pink, domain=-1:4 , samples=500] plot (\\x , {0.1*(\\x+1)*\\x*(\\x-1)*(\\x-3)});\n\\draw [thick, orange, domain=-1:4 , samples=500] plot (\\x , {0.1*(\\x+1)*\\x*(\\x-2)*(\\x-3)});\n\\draw [thick, brown, domain=-1:4 , samples=500] plot (\\x , {0.1*(\\x+1)*(\\x-1)*(\\x-2)*(\\x-3)});\n\\draw [thick, magenta, domain=-1:4 , samples=500] plot (\\x , {0.1*(\\x+1)*(\\x-1)*(\\x-3)^2});\n\\draw [thick, gray, domain=-1:4 , samples=500] plot (\\x , {0.1*(\\x+1)*(\\x-2)*(\\x-3)^2});\n\\draw[thick, blue](-1,0)--(-2,0);\n\\draw[thick, violet](-1,0)--(-1,2.1);\n\\draw[thick, violet](-1,2.1)--(1.75,2.1);\n\\draw[thick, violet](1.75,2.1)--(1.75,-2.2);\n\\draw[thick, violet](1.75,-2.2)--(-1.5,-2.2);\n\\draw[thick, violet](-1.5,-2.2)--(-1.5,2.6);\n\\draw[thick, violet](-1.5,2.6)--(2.25,2.6);\n\\draw[thick, violet](2.25,2.6)--(2.25,-3);\n\n\\draw [fill=white] (-1,0) circle [radius=0.05];\n\\draw [fill= black] (0,0) circle [radius=0.03];\n\\draw [fill= white] (0.5,0) circle [radius=0.05];\n\\draw [fill= black] (1,0) circle [radius=0.03];\n\\draw [fill= white] (1.5,0) circle [radius=0.05];\n\\draw [fill= black] (2,0) circle [radius=0.03];\n\\draw [fill= white] (2.5,0) circle [radius=0.05];\n\\draw [fill= black] (3,0) circle [radius=0.03];\n\\node at(-1.1,-0.6)[label=$p$]{};\n\\end{scope} \n\\end{tikzpicture}\n\\captionof{figure}{The arcs $\\alpha_{abc}$ on the 5-punctured sphere, together with arc $\\alpha_{<-1}$, drawn in blue, arc $\\alpha_{23}$, drawn in violet, and arc $\\gamma_1$, drawn in green.}\n\\label{fig:maxfamily}\n\\end{figure}\n\n\\begin{figure}\n \\begin{tikzpicture}[scale=1]\n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n[->] (-2,0)--(4.25,0); \n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n    [->] (0,-1)--(0,1.5);\n   \n    \n\\begin{scope}\n\n\n\\draw[thick, violet](-1,0)--(-1,0.5);\n\\draw[thick, violet](-1,0.5)--(0.75,0.5);\n\\draw[thick, violet](0.75,0.5)--(0.75,-0.5);\n\\draw[thick, violet](0.75,-0.5)--(-1.5,-0.5);\n\\draw[thick, violet](-1.5,-0.5)--(-1.5,0.75);\n\\draw[thick, violet](-1.5,0.75)--(2.25,0.75);\n\\draw[thick, violet](2.25,0.75)--(2.25,-1);\n\\draw[thick, orange](-1.05,0)--(-1.05,0.55);\n\\draw[thick, orange](-1.05,0.55)--(1.75,0.55);\n\\draw[thick, orange](1.75,0.55)--(1.75,-0.55);\n\\draw[thick, orange](1.75,-0.55)--(-1.55,-0.55);\n\\draw[thick, orange](-1.55,-0.55)--(-1.55,0.8);\n\\draw[thick, orange](-1.55,0.8)--(3.25,0.8);\n\\draw[thick, orange](3.25,0.8)--(3.25,-1);\n\n\n\n\\draw [fill=white] (-1,0) circle [radius=0.07];\n\\draw [fill= black] (0,0) circle [radius=0.03];\n\\draw [fill= white] (0.5,0) circle [radius=0.07];\n\\draw [fill= black] (1,0) circle [radius=0.03];\n\\draw [fill= white] (1.5,0) circle [radius=0.07];\n\\draw [fill= black] (2,0) circle [radius=0.03];\n\\draw [fill= white] (2.5,0) circle [radius=0.07];\n\\draw [fill= black] (3,0) circle [radius=0.03];\n\\draw [fill= white] (3.5,0) circle [radius=0.07];\n\\node at(-1.1,-0.6)[label=$p$]{};\n\\end{scope} \n\\end{tikzpicture}\n\n \\begin{tikzpicture}[scale=1]\n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n[->] (-2,0)--(4.25,0); \n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n    [->] (0,-1)--(0,1.5);\n   \n    \n\\begin{scope}\n\n\n\\draw[thick, violet](-1,0)--(-1,0.5);\n\\draw[thick, violet](-1,0.5)--(0.75,0.5);\n\\draw[thick, violet](0.75,0.5)--(0.75,-0.5);\n\\draw[thick, violet](0.75,-0.5)--(-1.5,-0.5);\n\\draw[thick, violet](-1.5,-0.5)--(-1.5,0.75);\n\\draw[thick, violet](-1.5,0.75)--(3.25,0.75);\n\\draw[thick, violet](3.25,0.75)--(3.25,-1);\n\\draw[thick, orange](-1.05,0)--(-1.05,0.55);\n\\draw[thick, orange](-1.05,0.55)--(1.75,0.55);\n\\draw[thick, orange](1.75,0.55)--(1.75,-0.55);\n\\draw[thick, orange](1.75,-0.55)--(-1.55,-0.55);\n\\draw[thick, orange](-1.55,-0.55)--(-1.55,0.8);\n\\draw[thick, orange](-1.55,0.8)--(2.25,0.8);\n\\draw[thick, orange](2.25,0.8)--(2.25,-1);\n\n\n\n\\draw [fill=white] (-1,0) circle [radius=0.07];\n\\draw [fill= black] (0,0) circle [radius=0.03];\n\\draw [fill= white] (0.5,0) circle [radius=0.07];\n\\draw [fill= black] (1,0) circle [radius=0.03];\n\\draw [fill= white] (1.5,0) circle [radius=0.07];\n\\draw [fill= black] (2,0) circle [radius=0.03];\n\\draw [fill= white] (2.5,0) circle [radius=0.07];\n\\draw [fill= black] (3,0) circle [radius=0.03];\n\\draw [fill= white] (3.5,0) circle [radius=0.07];\n\\node at(-1.1,-0.6)[label=$p$]{};\n\\end{scope} \n\\end{tikzpicture}\n\n \\begin{tikzpicture}[scale=1]\n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n[->] (-2,0)--(4.25,0); \n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n    [->] (0,-1)--(0,1.5);\n   \n    \n\\begin{scope}\n\n\n\\draw[thick, violet](-1,0)--(-1,0.5);\n\\draw[thick, violet](-1,0.5)--(0.75,0.5);\n\\draw[thick, violet](0.75,0.5)--(0.75,-0.5);\n\\draw[thick, violet](0.75,-0.5)--(-1.5,-0.5);\n\\draw[thick, violet](-1.5,-0.5)--(-1.5,0.75);\n\\draw[thick, violet](-1.5,0.75)--(1.25,0.75);\n\\draw[thick, violet](1.25,0.75)--(1.25,-1);\n\\draw[thick, orange](-1.05,0)--(-1.05,0.55);\n\\draw[thick, orange](-1.05,0.55)--(1.75,0.55);\n\\draw[thick, orange](1.75,0.55)--(1.75,-0.55);\n\\draw[thick, orange](1.75,-0.55)--(-1.55,-0.55);\n\\draw[thick, orange](-1.55,-0.55)--(-1.55,0.8);\n\\draw[thick, orange](-1.55,0.8)--(3.25,0.8);\n\\draw[thick, orange](3.25,0.8)--(3.25,-1);\n\n\n\n\\draw [fill=white] (-1,0) circle [radius=0.07];\n\\draw [fill= black] (0,0) circle [radius=0.03];\n\\draw [fill= white] (0.5,0) circle [radius=0.07];\n\\draw [fill= black] (1,0) circle [radius=0.03];\n\\draw [fill= white] (1.5,0) circle [radius=0.07];\n\\draw [fill= black] (2,0) circle [radius=0.03];\n\\draw [fill= white] (2.5,0) circle [radius=0.07];\n\\draw [fill= black] (3,0) circle [radius=0.03];\n\\draw [fill= white] (3.5,0) circle [radius=0.07];\n\\node at(-1.1,-0.6)[label=$p$]{};\n\\end{scope} \n\\end{tikzpicture}\n\n\n\\captionof{figure}{The $\\alpha_{ij}$ pairwise intersect at most twice.}\n\\label{fig:twistingcurves}\n\\end{figure}\n\n\\section{Properties of $r$-homotopic arcs intersecting at most twice}\n\nLet $p,q,r$ be distinct punctures of a punctured sphere $S$, and let $\\mathcal{A}$ be a $2$-system of arcs on $S$ joining $p$ and $q$. Let $\\bar{S}$ be the surface obtained from $S$ by forgetting $r$, and for each arc $\\alpha \\in \\mathcal{A}$, let $\\bar{\\alpha}$ be the homotopy class of the corresponding arc on $\\bar{S}$. In order to bound the size of $\\mathcal{A}$ from above, we will need to examine to what extent the map $\\alpha \\mapsto \\bar{\\alpha}$ is injective. In this section, we collect some facts about the fibers of this map. Together, the results of this section show that we can extend $\\mathcal{A}$ so that the size of each fiber is $1$ larger than the number of pairs of disjoint arcs in that fiber. \n\nThe main results of this section are Lemmas~\\ref{rhomotopic}, \\ref{rhomotopicbigon}, and \\ref{extend}. The proofs are rather technical and may be skipped on an initial reading.\n\n\n\n\\begin{lemma}\\label{configurations}\nLet $p,q,r$ be distinct punctures of a punctured sphere $S$, and let $\\alpha_1, \\alpha_2$ be a pair of $r$-homotopic arcs joining $p$ and $q$ and intersecting at most twice. If the $\\alpha_i$ are in minimal position, then they are in one of the configurations shown in Figure \\ref{fig:rhomotopicpairs}, up to relabeling $p$ and $q$. \n\\end{lemma}\n\n\n\\begin{proof}\nIf $\\alpha_1, \\alpha_2$ are disjoint, then they bound a strip whose only puncture is $r$ (see Figure \\ref{fig:rhomotopicpairs}, top left). Otherwise, by Corollary~\\ref{corollarybigon}, $\\alpha_1, \\alpha_2$ bound a half-bigon or bigon $R$ whose only puncture is $r$. If $\\alpha_1, \\alpha_2$ intersect exactly once, then $R$ is a half-bigon and, since the $\\alpha_i$ are $r$-homotopic, all punctures of $S$ distinct from $p,q,r$ lie in the other half-bigon formed by $\\alpha_1, \\alpha_2$ (see Figure \\ref{fig:rhomotopicpairs}, top right). \n\nIf the $\\alpha_i$ intersect twice and $R$ is a half-bigon, then the $\\alpha_i$ do not form a bigon, since otherwise they would not be $r$-homotopic (see Figure~\\ref{fig:nobigonallowed}). Thus, in this case, the $\\alpha_i$ must be as in the bottom right diagram of Figure \\ref{fig:rhomotopicpairs}, and since the $\\alpha_i$ are $r$-homotopic, all punctures of $S$ distinct from $p,q,r$ must lie in the other half-bigon formed by the $\\alpha_i$.\n\nOtherwise, $R$ is a bigon, and the $\\alpha_i$ also bound a pair of punctured half-bigons. These half-bigons must contain all the remaining punctures of $S$ since the $\\alpha_i$ are $r$-homotopic (see Figure \\ref{fig:rhomotopicpairs}, bottom left).\n\\end{proof}\n\n\nThe corollary of the following lemma will be useful in the proofs of Lemmas~\\ref{rhomotopic} and \\ref{rhomotopicbigon}. The former tells us that, in a particular context, if we have a portion of an arc then we can trace out the remainder of that arc.\n\n\n\\begin{figure}\n\\scalebox{.4}{\n\\includegraphics[trim=0 250 0 0, clip]{rhomotopicpairs2.eps}\n}\n\\captionof{figure}{The possible configurations of a pair of $r$-homotopic arcs in minimal position and intersecting at most twice, up to relabeling $p$ and $q$.}\n\\label{fig:rhomotopicpairs}\n\\end{figure}\n\n\\begin{figure}\n\\scalebox{.45}{\n\\includegraphics[trim=0 520 0 0, clip]{nobigonallowed.eps}\n}\n\\captionof{figure}{If $\\alpha_1, \\alpha_2$ are in minimal position, intersect exactly twice, and form a bigon that does not contain $r$, then they cannot be $r$-homotopic. The fact that the bigon and half-bigons bounded by the $\\alpha_i$ prior to forgetting $r$ are punctured, and the fact that the arcs on the right are in minimal position, are consequences of Corollary~\\ref{corollarybigon}.}\n\\label{fig:nobigonallowed}\n\\end{figure}\n\n\n\\begin{lemma}\\label{punctureddisc}\nLet $D$ be a disc with at least $2$ punctures in its interior and at least $1$ puncture on its boundary, and let $\\alpha$ be an arc joining an interior puncture $p$ of $D$ to a puncture $x$ on $\\partial D$. If $\\beta$ is another arc joining $p$ and $x$ such that $\\alpha$ and $\\beta$ bound a strip containing all interior punctures of $D$ distinct from $p$, then $\\beta$ is homotopic to exactly one of the arcs $\\alpha_1, \\alpha_2$ shown in Figure \\ref{fig:punctureddisc} (left).\n\\end{lemma}\n\n\\begin{proof}\nSuppose that the $x$-end of $\\beta$ lies to the right of $\\alpha$. Then we may homotope $\\alpha_1$ so that it bounds an empty strip with $\\beta$, as in Figure \\ref{fig:punctureddisc} (right). Thus, in this case, $\\beta$ is homotopic to $\\alpha_1$. Similarly, if the $x$-end of $\\beta$ lies to the left of $\\alpha$, then $\\beta$ is homotopic to $\\alpha_2$.\n\\end{proof}\n\n\\begin{figure}\n\\scalebox{.4}{\n\\includegraphics[trim=0 520 0 0, clip]{punctureddisc.eps}\n}\n\\captionof{figure}{}\n\\label{fig:punctureddisc}\n\\end{figure}\n\n\\begin{corollary}\\label{uniquecontinuation}\nLet $p,q,r$ be distinct punctures of an $n$-punctured sphere $S$, with $n \\geq 4$, and let $\\alpha, \\beta$ be $r$-homotopic arcs in minimal position joining $p,q$ and intersecting once or twice. Let $x_1, \\ldots, x_m$ be the points of intersection of $\\alpha, \\beta$ in the order that $\\beta$ traverses them as $\\beta$ travels from $p$ to $q$, and set $x_0=p$, $x_{m+1}=q$. For $i=0, \\ldots, m$, let $\\beta_i$ be the segment of $\\beta$ joining $x_i$ and $x_{i+1}$. If $m=2$, then the homotopy types of $\\beta_0$ and $\\beta_1$ determine that of $\\beta$. If $\\alpha$ and $\\beta$ do not bound a bigon, then the homotopy type of $\\beta_0$ determines that of $\\beta$ for $m=1,2$.\n\\end{corollary}\n\n\\begin{proof} For $i = 0, \\ldots, m$, let $\\alpha_i$ be the segment of $\\alpha$ joining $x_i$ and $x_{i+1}$. We puncture $S$ at $x_1, \\ldots, x_m$. \\\\\n\n\\paragraph{\\textbf{Case 1: $\\alpha$ and $\\beta$ intersect exactly once.}} Cutting $S$ along $\\alpha_0, \\beta_0$ yields two punctured strips. Let $D$ be the strip containing $q$. Note that $x_1$ is now a puncture on $\\partial D$, and that $\\alpha_1$ and $\\beta_1$ are arcs joining $q$ and $x_1$ and bounding a strip containing all the interior punctures of $D$ distinct from $q$. Thus, by Lemma~\\ref{punctureddisc}, the homotopy type of $\\beta_1$ is uniquely determined, since only one of the arcs described in Lemma~\\ref{punctureddisc} produces a $\\beta$ that intersects $\\alpha$ transversally at $x_1$ (in fact, the only other candidate homotopy class of $\\beta_1$ produces a $\\beta$ that is homotopic to $\\alpha$). \\\\\n\n\\begin{figure}\n\\scalebox{.4}{\n\\includegraphics[trim=0 520 0 0, clip]{bigoncontinuation.eps}\n}\n\\captionof{figure}{By Lemma~\\ref{punctureddisc}, given segments $\\beta_0$ and $\\beta_1$ of $\\beta$, there are at most $2$ homotopy classes of arcs joining $q$ and $x_2$ to which $\\beta_2$ can belong. One such homotopy class produces a $\\beta$ that intersects $\\alpha$ non-transversally at $x_2$, as shown above.}\n\\label{fig:bigoncontinuation}\n\\end{figure}\n\n\\paragraph{\\textbf{Case 2: $\\alpha$ and $\\beta$ form a bigon.}} Let $D$ be the square containing the puncture $q$ obtained by cutting $S$ along $\\alpha_0, \\beta_0, \\alpha_1, \\beta_1$. Now $x_2$ is a puncture on $\\partial D$, and $\\alpha_2, \\beta_2$ are arcs joining $q$ and $x_2$ and bounding a strip containing all the interior punctures of $D$ distinct from $q$, so we may apply Lemma~\\ref{punctureddisc} as in Case 1. Again, only one of the two homotopy classes to which $\\beta_2$ must belong by Lemma~\\ref{punctureddisc} produces a $\\beta$ that intersects $\\alpha$ transversally at $x_2$ (see Figure~\\ref{fig:bigoncontinuation}).  \\\\\n\n\\paragraph{\\textbf{Case 3: $\\alpha$ and $\\beta$ intersect exactly twice but do not form a bigon.}} Let $D$ be the strip containing $q$ obtained by cutting $S$ along $\\alpha_0, \\beta_0$. Since $x_1$ is a puncture on $\\partial D$, and $\\alpha_1, \\beta_1$ are arcs joining $x_1, x_2$ and bounding a strip containing all the interior punctures of $D$ distinct from $x_2$, the homotopy type of $\\beta_1$ is uniquely determined by Lemma~\\ref{punctureddisc} as in the previous cases. Now let $D'$ be the strip containing $q$ obtained by cutting $D$ along $\\alpha_1, \\beta_1$. Since $x_2$ is a puncture on $\\partial D'$, and $\\alpha_2, \\beta_2$ are arcs joining $q, x_2$ and bounding a strip containing all interior punctures of $D'$ distinct from $q$, the homotopy type of $\\beta_2$ is uniquely determined by Lemma~\\ref{punctureddisc}. \n\\end{proof} \n\n\n\\begin{lemma}\\label{rhomotopic}\nLet $p,q,r$ be distinct punctures of an $n$-punctured sphere $S$, with $n \\geq 4$. Let $\\mathcal{A}_r$ be a maximal $2$-system of $r$-homotopic arcs on $S$ joining $p$ and $q$. If $\\mathcal{A}_r$ contains intersecting arcs $\\alpha_1, \\alpha_2 \\in \\mathcal{A}_r$ in minimal position that do not form a bigon, then $\\mathcal{A}_r$ is as in Figure \\ref{fig:halfbigonmaximal}, up to homotopy and relabeling $p$ and $q$.\n\\end{lemma}\n\n\n\n\\begin{proof}\nLet $H$ be the half-bigon formed by the $\\alpha_i$ containing $r$, and assume that $H$ is adjacent to $p$ (see Figures \\ref{fig:case1nobigon}, \\ref{fig:case2nobigon}, left). Let $\\beta \\in \\mathcal{A}_r$. \\\\ \n\n\n\\item \\paragraph{\\textbf{Case 1: The $\\alpha_i$ intersect exactly once.}} Let $x$ be their unique point of intersection. If $\\beta$ is disjoint from the $\\alpha_i$, then $\\beta$ is homotopic to arc $\\beta_1$ in Figure \\ref{fig:case1nobigon} (right). Now suppose $\\beta$ is not disjoint from the $\\alpha_i$, and let $z$ be the first point of intersection of $\\beta$ and the $\\alpha_i$ as $\\beta$ travels from $p$ to $q$. Suppose that $z$ lies on $\\alpha_1$, and that $\\alpha_1, \\beta$ are in minimal position. If the $p$-end of $\\beta$ lies in $H$, then $\\beta$ forms a half-bigon with $\\alpha_1$ whose only puncture is $r$, since otherwise $\\alpha_1$ and $\\beta$ would form an empty half-bigon, contradicting our assumption that $\\alpha_1, \\beta$ are in minimal position (Corollary~\\ref{corollarybigon}). Thus, by Lemma~\\ref{configurations} and Corollary~\\ref{uniquecontinuation}, $\\beta$ is either homotopic to $\\alpha_2$ or to arc $\\beta_2$ in Figure \\ref{fig:case1nobigon} (right). If the $p$-end of $\\beta$ lies outside $H$, then $z$ cannot lie on the segment $(px)_{\\alpha_1}$, since otherwise $\\beta$ and $\\alpha_1$ would form an empty half-bigon. Thus, $z$ lies on $(xq)_{\\alpha_1}$, and so $\\beta$ again forms a half-bigon with $\\alpha_1$ whose only puncture is $r$. Thus, by Corollary~\\ref{uniquecontinuation}, $\\beta$ is again homotopic to one of $\\alpha_2, \\beta_2$. Note that, by the above, $\\mathcal{A}_r$ cannot contain an additional arc $\\beta'$ in minimal position with $\\alpha_2$ and intersecting $\\alpha_2$ first as $\\beta'$ travels from $p$ to $q$. This is because the reflection of $\\beta_2$ across the vertical diameter in Figure \\ref{fig:case1nobigon} (right) intersects $\\beta_2$ thrice. \n\n\\item \\paragraph{\\textbf{Case 2: The $\\alpha_i$ intersect exactly twice.}} Let $x,y$ be the points of intersection of the $\\alpha_i$ in the order that $\\alpha_1$ traverses them as it travels from $p$ to $q$. In this case, $\\beta$ intersects at least one of the $\\alpha_i$ since $p,q$ are in distinct components of the complement of $\\alpha_1 \\cup \\alpha_2$. Let $z$ be the first point of intersection of $\\beta$ and the $\\alpha_i$ as $\\beta$ travels from $p$ to $q$. We suppose again that $z$ lies on $\\alpha_1$, and that $\\alpha_1, \\beta$ are in minimal position. If the $p$-end of $\\beta$ lies in $H$, then, as in Case 1, $\\beta$ forms a half-bigon with $\\alpha_1$ whose only puncture is $r$. Thus, by Corollary \\ref{uniquecontinuation}, $\\beta$ is either homotopic to $\\alpha_2$ or to arc $\\beta_1$ in Figure \\ref{fig:case2nobigon} (right). If the $p$-end of $\\beta$ lies outside $H$, then, as in Case 1, $z$ cannot lie on the segment $(px)_{\\alpha_1}$. Thus, $z$ lies on $(xy)_{\\alpha_1}$. But then $\\beta$ again forms a half-bigon with $\\alpha_1$ whose only puncture is $r$, and so $\\beta$ is either homotopic to $\\alpha_2$ or to $\\beta_1$ as before. Similarly, if $\\beta$ is in minimal position with $\\alpha_2$ and intersects $\\alpha_2$ first as it travels from $p$ to $q$, then $\\beta$ is either homotopic to $\\alpha_1$ or to arc $\\beta_2$ in Figure \\ref{fig:case2nobigon} (right). \n\\end{proof}\n\n\\begin{figure}\n\\scalebox{.5}{\n\\includegraphics[trim=0 520 0 0, clip]{halfbigonmaximal.eps}\n}\n\\captionof{figure}{}\n\\label{fig:halfbigonmaximal}\n\\end{figure}\n\n\\begin{figure}\n\\scalebox{.4}{\n\\includegraphics[trim=0 520 0 0, clip]{case1nobigon.eps}\n}\n\\captionof{figure}{}\n\\label{fig:case1nobigon}\n\\end{figure}\n\n\\begin{figure}\n\\scalebox{.4}{\n\\includegraphics[trim=0 520 0 0, clip]{case2nobigon.eps}\n}\n\\captionof{figure}{}\n\\label{fig:case2nobigon}\n\\end{figure}\n\n\n\\begin{lemma}\\label{rhomotopicbigon}\nLet $p,q,r$ be distinct punctures of an $n$-punctured sphere $S$, with $n \\geq 4$. Let $\\mathcal{A}_r$ be a maximal $2$-system of $r$-homotopic arcs on $S$ joining $p$ and $q$. If $\\mathcal{A}_r$ contains arcs $\\alpha_1, \\alpha_2 \\in \\mathcal{A}_r$ in minimal position that form a bigon, then $\\mathcal{A}_r$ is as in Figure \\ref{fig:bigon}, up to homotopy and relabeling $p$ and $q$.\n\\end{lemma}\n\n\\begin{proof}\nLet $H$ be the half-bigon adjacent to $p$ formed by the $\\alpha_i$. Let $x,y$ be the points of intersection of the $\\alpha_i$ in the order that $\\alpha_1$ traverses them as it travels from $p$ to $q$. Let $\\beta \\in \\mathcal{A}_r$, and assume $\\alpha_1, \\alpha_2, \\beta$ are in minimal position. \n\nIf $\\beta$ is disjoint from the $\\alpha_i$, then $\\beta$ is homotopic to the blue arc in Figure \\ref{fig:bigon}. Now suppose $\\beta$ is not disjoint from the $\\alpha_i$, and let $z_1, z_2, \\ldots$ be the points of intersection of $\\beta$ and the $\\alpha_i$ in the order that $\\beta$ traverses them as it travels from $p$ to $q$. We assume that $z_1$ lies on $\\alpha_1$. Note that, by Lemma \\ref{rhomotopic}, $\\beta$ forms a bigon (containing only the puncture $r$) with each of the $\\alpha_i$ that it intersects. \\\\\n\n\\item \\paragraph{\\textbf{Case 1: $z_1$ lies on the segment $(yq)_{\\alpha_1}$.}} In this case, $\\alpha_1$ and $\\beta$ form a half-bigon whose only puncture is $r$. As remarked above, this is impossible.\n\n\\item \\paragraph{\\textbf{Case 2: $z_1$ lies on the segment $(xy)_{\\alpha_1}$.}} In this case, $z_2$ does not lie on $(xy)_{\\alpha_2}$. Otherwise, since $\\alpha_2$ and $\\beta$ are in minimal position, they would form a half-bigon whose only puncture is $r$ (as in Case 1 of Lemma~\\ref{rhomotopic}), but this is impossible. Thus, $z_2$ lies on $(xy)_{\\alpha_1}$, and so $\\beta$ is homotopic to $\\alpha_2$ by Corollary~\\ref{uniquecontinuation}. \n\n\n\n\\item \\paragraph{\\textbf{Case 3: $z_1$ lies on the segment $(px)_{\\alpha_1}$.}} In this case, since $\\alpha_1, \\beta$ are in minimal position, $\\beta$ forms a half-bigon $H'$ with $\\alpha_1$ adjacent to $p$ and containing at least one of the punctures of $H$. \n\nObserve that $z_2$ cannot lie on the segment $(yq)_{\\alpha_2}$, since otherwise $\\alpha_2$ and $\\beta$ would form a half-bigon containing $r$. We also have that $z_2$ cannot lie on $(xy)_{\\alpha_1}$, since otherwise $\\alpha_1$ and $\\beta$ would not form a bigon. Furthermore, if $z_2$ lies on $(px)_{\\alpha_2}$, then so must $z_3$, since otherwise $\\alpha_1$ and $\\beta$ would not form a bigon. But if $z_2$ and $z_3$ both lie on $(px)_{\\alpha_2}$, then $\\alpha_2$ and $\\beta$ form a bigon that does not contain $r$, which is impossible. \n\nNow, if $H'$ contains all the punctures of $H$, then $z_2$ cannot lie on $(xy)_{\\alpha_2}$ since $\\alpha_2, \\beta$ are in minimal position, and if $z_2$ lies on $(yq)_{\\alpha_1}$ then $\\beta$ is homotopic to $\\alpha_2$ by Corollary~\\ref{uniquecontinuation}. Thus, we may assume that $H'$ contains some but not all of the punctures of $H$. \n\nUnder this assumption, $z_2$ cannot lie on $(yq)_{\\alpha_1}$, since otherwise $\\beta$ would be homotopic to the purple arc Figure~\\ref{fig:bigoncases} (left) by Corollary~\\ref{uniquecontinuation}, and so $\\beta$ would intersect $\\alpha_2$ thrice. For the same reason, $z_3$ cannot lie on $(xy)_{\\alpha_1}$ if $z_2$ lies on $(xy)_{\\alpha_2}$. The only case left to consider is that $z_2$ and $z_3$ both lie on $(xy)_{\\alpha_2}$. But then $\\beta$ is homotopic to the orange arc in Figure~\\ref{fig:bigoncases} (right) by Corollary~\\ref{uniquecontinuation}, and so $\\beta$ intersects $\\alpha_1$ thrice. \n\\end{proof}\n\n\\begin{figure}\n\\scalebox{.5}{\n\\includegraphics[trim=0 520 0 0, clip]{bigon.eps}\n}\n\\captionof{figure}{}\n\\label{fig:bigon}\n\\end{figure}\n\n\\begin{figure}\n\\scalebox{.4}{\n\\includegraphics[trim=0 520 0 0, clip]{bigoncases.eps}\n}\n\\captionof{figure}{}\n\\label{fig:bigoncases}\n\\end{figure}\n\n\n\\begin{lemma}\\label{extend}\nLet $p,q$ be distinct punctures of an $n$-punctured sphere $S$, with $n \\geq 4$. Let $\\alpha_1, \\alpha_2, \\beta$ be arcs on $S$ joining $p,q$ in one of the configurations shown in Figure~\\ref{fig:extend}. Then an arc $\\gamma$ joining $p,q$ that is in minimal position with $\\beta$ and intersects $\\beta$ at least thrice must intersect $\\alpha_1$ or $\\alpha_2$ at least thrice.\n\\end{lemma}\n\n\\begin{proof}\nSet $x_0= p$, and let $x_1, x_2, x_3$ be the first 3 points of intersection of $\\beta$ and $\\gamma$ in the order that $\\gamma$ traverses them as $\\gamma$ travels from $p$ to $q$. For $i=0,1,2$, let $\\beta_i = (x_ix_{i+1})_\\beta, \\gamma_i = (x_ix_{i+1})_\\gamma$, and let $R_i$ be the region not containing $p$ bounded by $\\beta_i, \\gamma_i$. If $\\gamma_0$ does not intersect the $\\alpha_i$, then $R_0$ contains no punctures, and so $\\beta, \\gamma$ are not in minimal position by Lemma~\\ref{bigoncriterion}, contradicting our assumption. Thus, $\\gamma_0$ has at least one point of intersection with the $\\alpha_i$. Similarly, $R_1, R_2$ must each contain at least one puncture of $S$, and so each of $\\gamma_1, \\gamma_2$ has at least 2 points of intersection with the $\\alpha_i$. Thus, $\\gamma$ has at least 3 points of intersection with $\\alpha_1$ or $\\alpha_2$. \n\\end{proof}\n\n\\begin{figure}\n\\scalebox{.5}{\n\\includegraphics[trim=0 600 0 0, clip]{extend.eps}\n}\n\\captionof{figure}{}\n\\label{fig:extend}\n\\end{figure}\n\n\n\n\n\n\n\n \n\n\n\n\\section{1-System annular diagrams}\\label{diagrams}\n\nIn this section, we prove Theorem~\\ref{corner}, which will be useful in the inductive step of the proof of Lemma~\\ref{relation}.\n\n\\begin{lemma}\nLet $S$ be a twice-punctured sphere and $p,q$ its punctures. Let $\\mathcal{A}$ be a finite collection of simple arcs joining $p$ and $q$ and pairwise intersecting at most once. If there is a pair of intersecting arcs of $\\mathcal{A}$, then there is a pair of arcs of $\\mathcal{A}$ forming a half-bigon $H$ adjacent to $p$ such that no other arc of $\\mathcal{A}$ has its $p$-end in $H$. \n\\end{lemma}\n\n\\begin{proof}\nPick $\\alpha, \\beta \\in \\mathcal{A}$ such that $\\alpha$ and $\\beta$ intersect, and let $H$ be the half-bigon adjacent to $p$ formed by $\\alpha, \\beta$. If $H$ is as in the statement of the lemma, then we are done. Otherwise, there is an arc $\\beta' \\in \\mathcal{A}$ whose $p$-end lies in $H$. Since the $q$-end of $\\beta'$ is outside $H$, the arc $\\beta'$ must intersect one of $\\alpha, \\beta$, say $\\alpha$. We now repeat the above steps with arcs $\\alpha, \\beta'$. Since there are finitely many arcs in $\\mathcal{A}$, this process must terminate. \n\\end{proof}\n\nThe following corollary follows immediately.\n\n\\begin{corollary}\\label{cornsquare}\nLet $A$ be a $1$-system annular diagram, and $P$ a boundary path of $A$. If $A$ has at least one square, then $A$ has a cornsquare with outerpath on $P$.\n\\end{corollary}\n\nWe now proceed to the proof of Theorem~\\ref{corner}. \n\n\\begin{proof}[Proof of Theorem~\\ref{corner}]\nWe proceed by induction on the number of squares of $A$. If $A$ has no squares, then $A$ is a cycle. Now suppose $A$ has at least one square, that there is a boundary path $P$ of $A$ without a corner, and that the theorem holds for any annular square complex with fewer squares than $A$. Since $A$ contains a square, $A$ contains a cornsquare with outerpath on $P$ by Corollary~\\ref{cornsquare}. Thus, we may produce a corner on $P$ via a series of hexagon moves \\cite[Figure~3.17]{wise2012riches}. Note that a single hexagon move cannot produce two corners on $P$; otherwise there would be a dual curve beginning and terminating at $P$ (see Figure \\ref{fig:singlecorner}). \n\n\\begin{figure}\n\\scalebox{.3}{\n\\includegraphics[trim=0 580 0 0, clip]{singlecorner.eps}\n}\n\\captionof{figure}{If a single hexagon move produces two corners on $P$, then there is a dual curve beginning and ending at $P$.}\n\\label{fig:singlecorner}\n\\end{figure}\n\nWe perform hexagon moves until the first corner $v$ on $P$ is produced. Note that each neighbor of $v$ has degree at least 4. Indeed, since $P$ had no corners, we had to have performed at least one hexagon move to obtain $v$, but a neighbor of $v$ of degree 3 would correspond to a corner prior to performing that move, contradicting our assumption that $v$ is the first corner produced on $P$ (see Figure~\\ref{fig:vertexdegree}). Thus, by deleting $v$ as well as the two edges and the square incident to $v$, we obtain a 1-system annular diagram with one fewer square than $A$ and without any corners on one of its boundary paths, contradicting the induction hypothesis. \n\\end{proof}\n\n\\begin{figure}\n\\scalebox{.3}{\n\\includegraphics[trim=0 580 0 0, clip]{vertexdegree.eps}\n}\n\\captionof{figure}{If a corner $v$ produced by a hexagon move has a neighbor of degree 3, then that neighbor had to have been a corner prior to performing that move.}\n\\label{fig:vertexdegree}\n\\end{figure}\n\n\n\\begin{corollary}\\label{isolatedpuncture}\nLet $p,q$ be punctures of a punctured sphere $S$, and let $\\mathcal{A}$ be a 1-system of arcs joining $p$ and $q$ such that $|\\mathcal{A}| \\geq 2$ and the arcs of $\\mathcal{A}$ are pairwise in minimal position. There is a puncture $s$ of $S$ distinct from $p,q$ that is $p$-isolated by $\\mathcal{A}$. If $\\mathcal{A}$ contains a pair of intersecting arcs, then $s$ can be chosen so that the component $H$ of $S - \\bigcup \\mathcal{A}$ containing $s$ is a half-bigon.\n\\end{corollary}\n\n\\begin{proof}\nThe dual square complex $A$ to $\\mathcal{A}$ is an annular square complex as in Theorem~\\ref{corner}. Let $P$ be the boundary path corresponding to $p$. Note that $A$ has at least 2 vertices since $|\\mathcal{A}| \\geq 2$, and that $A$ has at least one square if and only if $\\mathcal{A}$ contains at least one pair of intersecting arcs. If $A$ is a cycle, then we may take $H$ to be the strip corresponding to any vertex of $A$. Otherwise, $A$ has a corner $v$ on $P$, and we may take $H$ to be the half-bigon corresponding to $v$. In either case, $H$ is punctured since the arcs of $\\mathcal{A}$ are pairwise non-homotopic and in minimal position.\n\\end{proof}\n\n\n\n\n\\section{Proof of Lemma~\\ref{relation}}\\label{proofoflemma}\n\nIn this section, we prove Lemma~\\ref{relation}, which essentially constitutes the inductive step in the proof of Theorem~\\ref{maintheorem}. We will need the following:\n\n\\begin{lemma}\\label{erdos}\\cite{erdos1946sets}\nA set of pairwise intersecting straight line segments between $\\ell$ points on a circle in $\\mathbb{R}^2$ has size at most $\\ell$. \n\\end{lemma}\n\n\n\n\\begin{proof}[Proof of Lemma~\\ref{relation}] We fix a complete hyperbolic metric on $S$ of area $2\\pi(n-2)$. We may assume that $\\mathcal{P}, \\mathcal{Q}$ are nonempty, and that the arcs of $\\mathcal{P} \\cup \\mathcal{Q}$ are pairwise in minimal position. We divide the proof into steps:\\\\\n\n\n\\item \\paragraph{\\textbf{Step 0. The arcs of $\\mathcal{P}$ (and hence the arcs of $\\mathcal{Q}$) are consecutive at $r$.}}\nIndeed, suppose $\\alpha, \\alpha' \\in \\mathcal{P}$ are distinct, and suppose there is an arc $\\beta \\in \\mathcal{Q}$ whose $r$-end lies in the strip or half-bigon $H$ bounded by $\\alpha, \\alpha'$ and adjacent to $r$. Since $\\beta$ does not intersect $\\alpha, \\alpha'$, the puncture $q$ must lie in $H$. Since no arc of $\\mathcal{Q}$ intersects $\\alpha, \\alpha'$, it follows that the $r$-end of every arc of $\\mathcal{Q}$ must also lie in $H$. \n\n\n\n\\item \\paragraph{We fix an orientation on $S$.} This induces a cyclic order $C$ of the arcs of $\\mathcal{P} \\cup \\mathcal{Q}$ around $r$. By Step 0, this order in turn induces a linear order $<$ on $\\mathcal{P}$, where the minimum and maximum arcs of $\\mathcal{P}$ are those with a successor or predecessor in $\\mathcal{Q}$ under $C$. \n\nWe proceed by induction. If $n=3$, then, up to homotopy, there is a unique arc joining $r$ to each of $p,q$, and the statement of the lemma holds. Now let $n \\geq 4$, and assume the lemma holds if $S$ has fewer punctures. If $\\mathcal{P}$ consists of a single arc then the lemma is trivially satisfied since $|\\mathcal{Q}| \\leq \\binom{n-1}{2}$ by Theorem~\\ref{przytycki}.  Thus, we may assume that $|\\mathcal{P}| \\geq 2$.\n\n\\item \\paragraph{\\textbf{Step 1. There is a puncture $s$ of $S$ distinct from $p,q,r$ that is $p$-isolated by $\\mathcal{P} \\cup \\mathcal{Q}$.}} Indeed, if the arcs of $\\mathcal{P}$ are pairwise disjoint, then since $|\\mathcal{P}| \\geq 2$, we have that $S - \\mathcal{P}$ consists of at least two punctured strips adjacent to $p$ and $r$, and so we may take $s$ to be a puncture of any such strip that does not contain $q$ (see Figure~\\ref{fig:step1}, left). Otherwise, by Corollary~\\ref{isolatedpuncture}, there is a puncture $s$ distinct from $p,r$ that is $p$-isolated by $\\mathcal{P}$ such that the component of $S - \\bigcup\\mathcal{P}$ containing $s$ is a half-bigon (see Figure~\\ref{fig:step1}, right). In this case, $s$ is necessarily distinct from $q$ since we are assuming $\\mathcal{Q}$ to be nonempty, and so there is at least one arc disjoint from the arcs of $\\mathcal{P}$ joining $q$ and $r$.\n\n\\begin{figure}\n\\scalebox{.4}{\n\\includegraphics[trim=0 520 0 0, clip]{step1.eps}\n}\n\\captionof{figure}{An illustration of Step 1.}\n\\label{fig:step1}\n\\end{figure}\n\n\\item \\paragraph{Let $\\bar{S}$ be the surface obtained from $S$ by forgetting the puncture $s$ (endowed with a complete, finite-area hyperbolic metric), and for each arc $\\alpha \\in \\mathcal{P} \\cup \\mathcal{Q}$, let $\\bar{\\alpha}$ be the corresponding arc on $\\bar{S}$.} Let $\\bar{\\mathcal{P}}, \\bar{\\mathcal{Q}}$ be the collection of all $\\bar{\\alpha}$ for $\\alpha \\in \\mathcal{P}, \\mathcal{Q}$, respectively. We tighten the arcs of $\\bar{\\mathcal{P}}\\cup \\bar{\\mathcal{Q}}$ to geodesics, thereby identifying arcs that correspond to $s$-homotopic arcs on $S$.\n\nThe orientation on $S$ induces an orientation on $\\bar{S}$. As above, this gives us a linear order $\\prec$ on $\\bar{\\mathcal{P}}$.\n\n\\item \\paragraph{\\textbf{Step 2. Two distinct arcs in $\\mathcal{Q}$ cannot be $s$-homotopic.}} Otherwise, they would form a strip adjacent to $q,r$ or a half-bigon adjacent to one of $q,r$ whose only puncture is $s$, which cannot happen since the component of $S - \\bigcup(\\mathcal{P}\\cup\\mathcal{Q})$ containing $s$ is adjacent to $p$. Thus, we may identify $\\bar{\\mathcal{Q}}$ with $\\mathcal{Q}$. \n\n\\item \\paragraph{\\textbf{Step 3. Arcs in $\\mathcal{P}$ that are $s$-homotopic must be consecutive at $r$.}} Indeed, suppose that $\\alpha, \\alpha' \\in \\mathcal{P}$ are $s$-homotopic. If $\\alpha, \\alpha'$ bound a strip whose only puncture is $s$, then the $r$-end of any arc $\\beta \\in \\mathcal{P}$ distinct from $\\alpha, \\alpha'$ cannot lie inside this strip, since otherwise $\\beta$ and one of $\\alpha, \\alpha'$ would necessarily form a half-bigon adjacent to $r$ whose only puncture is $s$, contradicting the fact that $s$ is $p$-isolated. \n\nOtherwise, $\\alpha$ and $\\alpha'$ form a half-bigon adjacent to $p$ whose only puncture is $s$, and a half-bigon adjacent to $r$ containing all punctures of $S$ except $p,r,s$. Thus, any arc $\\alpha''$ in $\\mathcal{P}$ distinct from $\\alpha, \\alpha'$ with $r$-end outside the latter half-bigon must be disjoint from $\\alpha,\\alpha'$, in which case $\\alpha, \\alpha', \\alpha''$ are in fact $s$-homotopic. \n\n\\item \\paragraph{\\textbf{Step 4. For any $\\alpha, \\alpha' \\in \\mathcal{P}$, if $\\alpha < \\alpha'$ then $\\bar{\\alpha} \\preceq \\bar{\\alpha'}$.}} Indeed, if $\\bar{\\alpha} \\succ \\bar{\\alpha'}$, then $\\alpha, \\alpha'$ form a half-bigon adjacent to $r$ whose only puncture is $s$. This cannot happen since $s$ is $p$-isolated.\n\n\n\\item \\paragraph{We now extend $\\mathcal{P}$ and $\\mathcal{R}$ as follows.} Suppose $\\mathcal{P}$ contains intersecting $s$-homotopic arcs $\\alpha, \\alpha'$. Then, as discussed in Step 3, an arc $\\alpha''$ disjoint from $\\alpha, \\alpha'$ joining $p$ and $r$ is $s$-homotopic to $\\alpha, \\alpha'$ and is between $\\alpha, \\alpha'$ with respect to the linear order on $\\mathcal{P}$. If $\\alpha''$ is not homotopic to an arc in $\\mathcal{P}$, then we add $\\alpha''$ to $\\mathcal{P}$ (otherwise, name the latter arc $\\alpha''$). If there is an arc $\\beta \\in \\mathcal{Q}$ such that $(\\alpha, \\beta), (\\alpha', \\beta) \\in \\mathcal{R}$ but $(\\alpha'', \\beta) \\notin \\mathcal{R}$, then we add the pair $(\\alpha'', \\beta)$ to $\\mathcal{R}$. Since $\\alpha''$ is disjoint from all arcs in $\\mathcal{P}\\cup \\mathcal{Q}$, we have not violated any of the conditions of the lemma. We do this for each pair of intersecting $s$-homotopic arcs $\\alpha, \\alpha'$.\n\nLet $\\bar{\\mathcal{R}}$ be the image of $\\mathcal{R}$ under the map $\\mathcal{P} \\times \\mathcal{Q} \\rightarrow \\bar{\\mathcal{P}} \\times \\mathcal{Q}, (\\alpha, \\beta) \\mapsto (\\bar{\\alpha}, \\beta)$.\nIt is clear that $\\bar{\\mathcal{R}}$ satisfies condition $(i)$, and it follows from Step 4 that $\\bar{\\mathcal{R}}$ satisfies $(ii)$ as well, so that $|\\bar{\\mathcal{R}}| \\leq \\binom{n-2}{2}$. It remains to show that $|\\mathcal{R}| - |\\bar{\\mathcal{R}}| \\leq n-2$. \n\nLet $\\mathcal{I}$ be the subset of $\\bar{\\mathcal{R}}$ consisting of elements with more than one pre-image under the map $\\mathcal{R} \\rightarrow \\bar{\\mathcal{R}}$. \n\n\\item \\paragraph{\\textbf{Step 5. If $(\\bar{\\alpha}, \\beta), (\\bar{\\alpha'}, \\beta') \\in \\mathcal{I}$ with $\\beta \\neq \\beta'$, then $\\beta, \\beta'$ are disjoint.}} Indeed, let $(\\alpha_1, \\beta), (\\alpha_2, \\beta)$ be pre-images of $(\\bar{\\alpha}, \\beta)$, and $(\\alpha_1', \\beta'), (\\alpha_2', \\beta')$ pre-images of $(\\bar{\\alpha'}, \\beta')$ with $\\alpha_1 \\neq \\alpha_2, \\alpha_1' \\neq \\alpha_2'$. Suppose that $\\beta$ and $\\beta'$ intersect. Note that we cannot have $\\{\\alpha_1, \\alpha_2\\} = \\{\\alpha_1', \\alpha_2'\\}$, since otherwise either $(\\alpha_1, \\beta), (\\alpha_2, \\beta')$ or $(\\alpha_1, \\beta'), (\\alpha_2, \\beta)$ would be two intersecting pairs of arcs in $\\mathcal{R}$ whose cyclic order around $r$ is not alternating, contradicting assumption $(ii)$. \n\nNow suppose $\\{\\alpha_1, \\alpha_2\\} \\cap \\{\\alpha_1', \\alpha_2'\\} \\neq \\emptyset$. Then there are 3 distinct arcs among $\\alpha_1, \\alpha_2, \\alpha_1', \\alpha_2'$ that are $s$-homotopic. Assume without loss of generality that these arcs are $\\alpha_1, \\alpha_2, \\alpha_1'$. Observe that $\\alpha_1'$ cannot intersect $\\alpha_i$ for $i=1,2$, since otherwise $(\\alpha_i, \\beta), (\\alpha_1', \\beta')$ would intersect more than once, contradicting assumption $(i)$. Thus, $\\alpha_1$ intersects $\\alpha_2$, but then $\\alpha_1'$ lies between $\\alpha_1$ and $\\alpha_2$ at $r$, and so there is an $i\\in \\{1,2\\}$ such that $(\\alpha_i, \\beta), (\\alpha_1', \\beta')$ contradict assumption $(ii)$.\n\nWe conclude that $\\alpha_1, \\alpha_2, \\alpha_1', \\alpha_2'$ are distinct; in particular, since at most 3 distinct arcs in $\\mathcal{P}$ can be $s$-homotopic, we must have $\\bar{\\alpha} \\neq \\bar{\\alpha'}$. Thus, by Step 3, the order of $\\alpha_1, \\alpha_2, \\alpha_1', \\alpha_2'$ at $r$ is neither alternating nor nested. By the latter, we mean that $\\alpha_1, \\alpha_2$ do not both lie between $\\alpha_1', \\alpha_2'$ in the linear order on $\\mathcal{P}$, and vice versa. Since each of the pairs of arcs $\\alpha_1, \\alpha_2$ and $\\alpha_1', \\alpha_2'$ bound a strip or half-bigon containing $s$, we must have that $\\alpha_i, \\alpha_j'$ intersect for some $i,j \\in \\{1,2\\}$, but then $(\\alpha_i, \\beta), (\\alpha_j', \\beta')$ intersect more than once, contradicting assumption $(i)$.\n\n\n\n\\item \\paragraph{\\textbf{Step 6. If $(\\bar{\\alpha}, \\beta), (\\bar{\\alpha'}, \\beta') \\in \\mathcal{I}$ with $\\bar{\\alpha} \\neq \\bar{\\alpha'}, \\beta \\neq \\beta'$, then the cyclic order of $(\\bar{\\alpha}, \\beta), (\\bar{\\alpha'}, \\beta')$ at $r$ is alternating.}} Indeed, suppose otherwise, and let $(\\alpha_1, \\beta), (\\alpha_2, \\beta)$ be pre-images of $(\\bar{\\alpha}, \\beta)$, and $(\\alpha_1', \\beta'), (\\alpha_2', \\beta')$ pre-images of $(\\bar{\\alpha'}, \\beta')$ with $\\alpha_1 \\neq \\alpha_2, \\alpha_1' \\neq \\alpha_2'$. Then, by Step 3, the order of $\\alpha_1, \\alpha_2, \\alpha_1', \\alpha_2'$ at $r$ is neither alternating nor nested. Thus, as in Step 5, $\\alpha_i, \\alpha_j'$ must intersect for some $i,j \\in \\{1,2\\}$, but then $(\\alpha_i, \\beta), (\\alpha_j', \\beta')$ are two intersecting pairs of arcs in $\\mathcal{R}$ whose cyclic order around $r$ is not alternating, contradicting assumption $(ii)$ (see Figure~\\ref{fig:nestedalternating}).\n\n\\begin{figure}\n\\scalebox{.5}{\n\\includegraphics[trim=0 520 0 0, clip]{nestedalternating.eps}\n}\n\\captionof{figure}{An illustration of Step 6. Here, the pairs $(\\alpha_1, \\beta)$ and $(\\alpha_1', \\beta')$ contradict assumption $(ii)$.}\n\\label{fig:nestedalternating}\n\\end{figure}\n\n\\item \\paragraph{Let $\\mathcal{H}_q$ be the image of $\\mathcal{I}$ under the projection map $\\bar{\\mathcal{P}} \\times \\mathcal{Q} \\rightarrow \\mathcal{Q}$.} Let $\\mathcal{H}_s$ be the collection of all geodesic arcs $a$ joining $r$ and $s$ such that $a$ is contained in a strip bounded by a pair of distinct, disjoint $s$-homotopic arcs in $\\mathcal{P}$. \n\n Let $\\mathcal{H} = \\mathcal{H}_s \\cup \\mathcal{H}_q$, and let $\\mathcal{I}' \\subset \\mathcal{H}_s \\times \\mathcal{H}_q$ be the set of all $(a, \\beta) \\in \\mathcal{H}_s \\times \\mathcal{H}_q$ such that $(\\alpha, \\beta) \\in \\mathcal{I}$ for an arc $\\alpha$ bounding a strip corresponding to $a$. We extended $\\mathcal{R}$ so that the map $\\mathcal{R} \\rightarrow \\bar{\\mathcal{R}}$ is injective outside a set of cardinality $|\\mathcal{I}'|$. Thus, to complete the proof, it suffices to show that $|\\mathcal{I}'| \\leq n-2$. \n\n\\item \\paragraph{\\textbf{Step 7. The complement of $\\mathcal{H}$ consists of punctured strips and a single square, possibly with no punctures.}} Indeed, the arcs of $\\mathcal{H}_s$ are disjoint by construction \\cite[proof~of~Theorem~1.7]{przytycki2015arcs}, and the arcs of $\\mathcal{H}_q$ are disjoint by Step 5, so that the complement of each of $\\mathcal{H}_s, \\mathcal{H}_q$ consists of punctured strips. By Step 0, the arcs in $\\mathcal{H}_q$ (and hence the arcs in $\\mathcal{H}_s$) are consecutive at $r$. Thus, $\\mathcal{H}_s$ is contained in a single strip of $S - \\mathcal{H}_q$ and vice versa. Let $\\beta, \\beta'$ be the arcs in $\\mathcal{H}_q$ bounding the unique strip of $S - \\mathcal{H}_q$ containing $\\mathcal{H}_s$, and let $\\gamma, \\gamma'$ be the arcs in $\\mathcal{H}_s$ bounding the unique strip of $S - \\mathcal{H}_s$ containing $\\mathcal{H}_q$ (note that we do not exclude the possibility that $\\beta = \\beta'$ or $\\gamma = \\gamma'$). Then the complement of $\\mathcal{H}$ consists of the remaining strips of $S - \\mathcal{H}_q$, $S - \\mathcal{H}_s$ and a square bounded by $\\beta, \\beta', \\gamma, \\gamma'$. \n\n\\item \\paragraph{\\textbf{Step 8. $|\\mathcal{H}| \\leq n-1$.}} Indeed, $|\\mathcal{H}|$ is $2$ larger than the number of strips of $S - \\mathcal{H}$, so it suffices to show that there are at most $n-3$ of these strips. This is true by Step 7 since $S$ has area $2\\pi (n-2)$, and a punctured strip and a square each have area at least $2\\pi$. \n\n\\item \\paragraph{\\textbf{Step 9. $|\\mathcal{I}'| \\leq n-2$.}} To show this, we intersect $\\mathcal{H}$ with a small circle $C$ centered at $r$. Each element of $\\mathcal{I}'$ is determined by a pair of points of this intersection, and we connect them by a straight line segment. We also draw a line segment between the outermost points on $C$ corresponding to elements of $\\mathcal{H}_q$. By Step 6, these line segments are pairwise intersecting, so by Lemma~\\ref{erdos}, $|\\mathcal{I}'|+1 \\leq |\\mathcal{H}| \\leq n-1$.\n\\end{proof}\n\n\n\n\n\\section{Proof of Theorem~\\ref{maintheorem}}\n\n\\begin{figure}\n\\scalebox{.5}{\n\\includegraphics[trim=0 520 0 0, clip]{alternating.eps}\n}\n\\captionof{figure}{}\n\\label{fig:alternating}\n\\end{figure}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{maintheorem}]\nWe proceed by induction on $n$. The case $n=3$ is trivial. Now let $n \\geq 4$, and assume the theorem holds if $S$ has fewer punctures. Let $r$ be a puncture of $S$ distinct from $p,q$. Let $\\bar{S}$ be the $(n-1)$-punctured sphere obtained from $S$ by forgetting $r$. For each arc $\\alpha \\in \\mathcal{A}$, let $\\bar{\\alpha}$ be the corresponding arc on $\\bar{S}$, and let $\\bar{\\mathcal{A}} = \\{\\bar{\\alpha} \\> : \\> \\alpha \\in \\mathcal{A}\\}$. We tighten the arcs of $\\bar{\\mathcal{A}}$ to geodesics. Note that $\\bar{\\mathcal{A}}$ is a $2$-system on $\\bar{S}$, and so $|\\bar{\\mathcal{A}}| \\leq \\binom{n-1}{3}$ by the induction hypothesis. Thus, it is enough to show that $|\\mathcal{A}| - |\\bar{\\mathcal{A}}| \\leq \\binom{n-1}{2}$. To that end, we examine the extent to which the map $\\pi: \\mathcal{A} \\rightarrow \\bar{\\mathcal{A}}$, $\\alpha \\mapsto \\bar{\\alpha}$ is injective. \n\nBy Lemmas~\\ref{rhomotopic}, \\ref{rhomotopicbigon}, and \\ref{extend}, we may add arcs to $\\mathcal{A}$ so that for each $\\alpha \\in \\mathcal{A}$,\n\\[\n|\\pi^{-1}(\\bar{\\alpha})| -1 = |\\{ \\{\\alpha_1, \\alpha_2\\} \\in \\pi^{-1}(\\bar{\\alpha}) \\> : \\> \\alpha_1, \\alpha_2 \\text{ distinct  and disjoint} \\}|\n\\]\nLet $\\mathcal{P}$ (resp., $\\mathcal{Q}$) be the collection of all geodesic arcs $\\alpha$ on $S$ starting at $r$ and ending at $p$ (resp., ending at $q$) such that $\\alpha$ is contained entirely in a strip bounded by a pair of distinct, disjoint $r$-homotopic arcs in $\\mathcal{A}$. Let $\\mathcal{R} \\subset \\mathcal{P} \\times \\mathcal{Q}$ be the relation consisting of all pairs $(\\alpha, \\beta)$ such that both $\\alpha$ and $\\beta$ lie in a single such strip. We claim that $\\mathcal{P}, \\mathcal{Q}, \\mathcal{R}$ satisfy the conditions of Lemma~\\ref{relation}, so that $|\\mathcal{R}| \\leq \\binom{n-1}{2}$. Since $|\\mathcal{R}| = |\\mathcal{A}| - |\\bar{\\mathcal{A}}|$, this completes the proof.\n\nWe first note that for $(\\alpha, \\beta), (\\alpha', \\beta') \\in \\mathcal{R}$ corresponding to pairs of disjoint $r$-homotopic arcs $\\gamma_1, \\gamma_2 \\in \\mathcal{A}$ and $\\gamma_1', \\gamma_2' \\in \\mathcal{A}$, respectively, we have for some $i, j \\in \\{1,2\\}$ that $r$ produces at least one point of intersection between $\\gamma_i, \\gamma_j'$. Each point of intersection between the arcs $\\alpha, \\beta, \\alpha', \\beta'$ produces an additional point of intersection between $\\gamma_i, \\gamma_j'$. It follows that there is at most one point of intersection between any two pairs of arcs in $\\mathcal{R}$. \n\nWe now show that no arc in $\\mathcal{P}$ intersects an arc in $\\mathcal{Q}$. Indeed, suppose we have $(\\alpha, \\beta), (\\alpha', \\beta') \\in \\mathcal{R}$ such that $\\alpha$ intersects $\\beta'$. By the above, $\\alpha$ intersects $\\beta'$ exactly once and that is the only point of intersection between the pairs of arcs $(\\alpha, \\beta), (\\alpha', \\beta')$. But then we can find two arcs in $\\mathcal{A}$ that intersect thrice, as shown in Figure~\\ref{fig:alternating} (left).\n\nFinally, if there are two intersecting pairs of arcs in $\\mathcal{R}$ whose cyclic order around $r$ is not alternating, then we can also find two arcs in $\\mathcal{A}$ that intersect thrice, as shown in Figure~\\ref{fig:alternating} (right).\n\\end{proof}\n\n\n\n\n\\bibliographystyle{amsalpha}\n\n\\section*{This is an unnumbered first-level section head}\nThis is an example of an unnumbered first-level heading.\n\n\\specialsection*{THIS IS A SPECIAL SECTION HEAD}\nThis is an example of a special section head%\n\\footnote{Here is an example of a footnote. Notice that this footnote\ntext is running on so that it can stand as an example of how a footnote\nwith separate paragraphs should be written.\n\\par\nAnd here is the beginning of the second paragraph.}%\n.\n\n\\section{This is a numbered first-level section head}\nThis is an example of a numbered first-level heading.\n\n\\subsection{This is a numbered second-level section head}\nThis is an example of a numbered second-level heading.\n\n\\subsection*{This is an unnumbered second-level section head}\nThis is an example of an unnumbered second-level heading.\n\n\\subsubsection{This is a numbered third-level section head}\nThis is an example of a numbered third-level heading.\n\n\\subsubsection*{This is an unnumbered third-level section head}\nThis is an example of an unnumbered third-level heading.\n\n\\begin{lemma}\nLet $f, g\\in  A(X)$ and let $E$, $F$ be cozero\nsets in $X$.\n\\begin{enumerate}\n\\item If $f$ is $E$-regular and $F\\subseteq E$, then $f$ is $F$-regular.\n\n\\item If $f$ is $E$-regular and $F$-regular, then $f$ is $E\\cup\nF$-regular.\n\n\\item If $f(x)\\ge c>0$ for all $x\\in E$, then $f$ is $E$-regular.\n\n\\end{enumerate}\n\\end{lemma}\n\nThe following is an example of a proof.\n\n\\begin{proof} Set $j(\\nu)=\\max(I\\backslash a(\\nu))-1$. Then we have\n\\[\n\\sum_{i\\notin a(\\nu)}t_i\\sim t_{j(\\nu)+1}\n  =\\prod^{j(\\nu)}_{j=0}(t_{j+1}\/t_j).\n\\]\nHence we have\n\\begin{equation}\n\\begin{split}\n\\prod_\\nu\\biggl(\\sum_{i\\notin\n  a(\\nu)}t_i\\biggr)^{\\abs{a(\\nu-1)}-\\abs{a(\\nu)}}\n&\\sim\\prod_\\nu\\prod^{j(\\nu)}_{j=0}\n  (t_{j+1}\/t_j)^{\\abs{a(\\nu-1)}-\\abs{a(\\nu)}}\\\\\n&=\\prod_{j\\ge 0}(t_{j+1}\/t_j)^{\n  \\sum_{j(\\nu)\\ge j}(\\abs{a(\\nu-1)}-\\abs{a(\\nu)})}.\n\\end{split}\n\\end{equation}\nBy definition, we have $a(\\nu(j))\\supset c(j)$. Hence, $\\abs{c(j)}=n-j$\nimplies (5.4). If $c(j)\\notin a$, $a(\\nu(j))c(j)$ and hence\nwe have (5.5).\n\\end{proof}\n\n\\begin{quotation}\nThis is an example of an `extract'. The magnetization $M_0$ of the Ising\nmodel is related to the local state probability $P(a):M_0=P(1)-P(-1)$.\nThe equivalences are shown in Table~\\ref{eqtable}.\n\\end{quotation}\n\n\\begin{table}[ht]\n\\caption{}\\label{eqtable}\n\\renewcommand\\arraystretch{1.5}\n\\noindent\\[\n\\begin{array}{|c|c|c|}\n\\hline\n&{-\\infty}&{+\\infty}\\\\\n\\hline\n{f_+(x,k)}&e^{\\sqrt{-1}kx}+s_{12}(k)e^{-\\sqrt{-1}kx}&s_{11}(k)e^\n{\\sqrt{-1}kx}\\\\\n\\hline\n{f_-(x,k)}&s_{22}(k)e^{-\\sqrt{-1}kx}&e^{-\\sqrt{-1}kx}+s_{21}(k)e^{\\sqrt\n{-1}kx}\\\\\n\\hline\n\\end{array}\n\\]\n\\end{table}\n\n\\begin{definition}\nThis is an example of a `definition' element.\nFor $f\\in A(X)$, we define\n\\begin{equation}\n\\mathcal{Z} (f)=\\{E\\in Z[X]: \\text{$f$ is $E^c$-regular}\\}.\n\\end{equation}\n\\end{definition}\n\n\\begin{remark}\nThis is an example of a `remark' element.\nFor $f\\in A(X)$, we define\n\\begin{equation}\n\\mathcal{Z} (f)=\\{E\\in Z[X]: \\text{$f$ is $E^c$-regular}\\}.\n\\end{equation}\n\\end{remark}\n\n\\begin{example}\nThis is an example of an `example' element.\nFor $f\\in A(X)$, we define\n\\begin{equation}\n\\mathcal{Z} (f)=\\{E\\in Z[X]: \\text{$f$ is $E^c$-regular}\\}.\n\\end{equation}\n\\end{example}\n\n\\begin{xca}\nThis is an example of the \\texttt{xca} environment. This environment is\nused for exercises which occur within a section.\n\\end{xca}\n\nThe following is an example of a numbered list.\n\n\\begin{enumerate}\n\\item First item.\nIn the case where in $G$ there is a sequence of subgroups\n\\[\nG = G_0, G_1, G_2, \\dots, G_k = e\n\\]\nsuch that each is an invariant subgroup of $G_i$.\n\n\\item Second item.\nIts action on an arbitrary element $X = \\lambda^\\alpha X_\\alpha$ has the\nform\n\\begin{equation}\\label{eq:action}\n[e^\\alpha X_\\alpha, X] = e^\\alpha \\lambda^\\beta\n[X_\\alpha X_\\beta] = e^\\alpha c^\\gamma_{\\alpha \\beta}\n \\lambda^\\beta X_\\gamma,\n\\end{equation}\n\n\\begin{enumerate}\n\\item First subitem.\n\\[\n- 2\\psi_2(e) =  c_{\\alpha \\gamma}^\\delta c_{\\beta \\delta}^\\gamma\ne^\\alpha e^\\beta.\n\\]\n\n\\item Second subitem.\n\\begin{enumerate}\n\\item First subsubitem.\nIn the case where in $G$ there is a sequence of subgroups\n\\[\nG = G_0, G_1, G_2, \\ldots, G_k = e\n\\]\nsuch that each subgroup $G_{i+1}$ is an invariant subgroup of $G_i$ and\neach quotient group $G_{i+1}\/G_{i}$ is abelian, the group $G$ is called\n\\textit{solvable}.\n\n\\item Second subsubitem.\n\\end{enumerate}\n\\item Third subitem.\n\\end{enumerate}\n\\item Third item.\n\\end{enumerate}\n\nHere is an example of a cite. See \\cite{A}.\n\n\\begin{theorem}\nThis is an example of a theorem.\n\\end{theorem}\n\n\\begin{theorem}[Marcus Theorem]\nThis is an example of a theorem with a parenthetical note in the\nheading.\n\\end{theorem}\n\n\\begin{figure}[tb]\n\\includegraphics{lion.png}\n\\caption{This is an example of a figure caption with text.}\n\\label{firstfig}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\includegraphics{lion.png}\n\\caption{}\\label{otherfig}\n\\end{figure}\n\n\\section{Some more list types}\nThis is an example of a bulleted list.\n\n\\begin{itemize}\n\\item $\\mathcal{J}_g$ of dimension $3g-3$;\n\\item $\\mathcal{E}^2_g=\\{$Pryms of double covers of $C=\\openbox$ with\nnormalization of $C$ hyperelliptic of genus $g-1\\}$ of dimension $2g$;\n\\item $\\mathcal{E}^2_{1,g-1}=\\{$Pryms of double covers of\n$C=\\openbox^H_{P^1}$ with $H$ hyperelliptic of genus $g-2\\}$ of\ndimension $2g-1$;\n\\item $\\mathcal{P}^2_{t,g-t}$ for $2\\le t\\le g\/2=\\{$Pryms of double\ncovers of $C=\\openbox^{C'}_{C''}$ with $g(C')=t-1$ and $g(C'')=g-t-1\\}$\nof dimension $3g-4$.\n\\end{itemize}\n\nThis is an example of a `description' list.\n\n\\begin{description}\n\\item[Zero case] $\\rho(\\Phi) = \\{0\\}$.\n\n\\item[Rational case] $\\rho(\\Phi) \\ne \\{0\\}$ and $\\rho(\\Phi)$ is\ncontained in a line through $0$ with rational slope.\n\n\\item[Irrational case] $\\rho(\\Phi) \\ne \\{0\\}$ and $\\rho(\\Phi)$ is\ncontained in a line through $0$ with irrational slope.\n\\end{description}\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:introduction}\n\n\\myparagraph{Motivation} Question answering (QA)\naims to compute\ndirect answers to information needs posed as natural language (NL)\nutterances~\\cite{lu2019answering,sun2019pullnet,clark2018simple,diefenbach2019qanswer,kaiser2021reinforcement,jia21complex,saxena2020improving,he2021improving,saharoy2021question}.\nWe focus on the class of \\textit{factoid questions}\nthat are objective in\nnature and have one or more named entities as answers~\\cite{abujabal2019comqa,dubey2019lc,vakulenko2019message,qiu2020stepwise,christmann2019look,plepi2021context}. \nEarly research~\\cite{ravichandran2002learning,voorhees1999trec} used patterns to \nextract text passages with candidate answers, or had sophisticated\npipelines like the\nproprietary \\textsc{IBM Watson} system~\\cite{ferrucci2010building,ferrucci2012introduction}.\nthat won the Jeopardy! quiz show. \nWith the rise of large knowledge graphs (KGs)\nor knowledge bases (KBs)\nlike YAGO~\\cite{suchanek2007yago}, DBpedia~\\cite{auer2007dbpedia}, \nFreebase~\\cite{bollacker2008freebase}, and\nWikidata~\\cite{vrandevcic2014wikidata}, \nthe focus shifted from text corpora as inputs to these \nstructured RDF data sources,\nrepresented as subject-predicate-object (SPO) triples.\nWe refer to such answering of questions over knowledge graphs as \\textit{KG-QA}.\n\nWhile KGs capture a large\npart of the world's encyclopedic knowledge, they are inherently\nincomplete. This is because they cannot stay up-to-date with\nthe latest information, so that emerging and ephemeral facts \n(e.g., teams losing in semi-finals of sports leagues, or celebrities \ndating each other) are not included. \nAlso, user interests go way beyond the predicates that are modeled in KGs\nlike Wikidata.\nAs a result, answering over text from the open Web, like websites of newspapers or magazines,\nis an absolute necessity. \nWe refer to this paradigm as question answering over text,\n{\\em Text-QA} for short.\n\n\\myparagraph{Limitations of the state-of-the-art} This fragmented landscape\nof QA research has led to a state where methods from one paradigm are\ncompletely incompatible with the other.\nSystems operating over knowledge graphs\nare not equipped at all\nto compute answers from text passages. \nThe reason behind this is two-fold. \nFirst, QA over KGs assumes one of two major paradigms:\n(i) an explicit structured query (in a SPARQL-like language) is constructed \nthat, when executed over the KG, retrieves the answer~\\cite{tanon2018demoing,diefenbach2019qanswer,bhutani2019learning,shen2019multi,abujabal2018never}.\nSuch structured queries cannot be executed over a textual source;\n(ii) they perform approximate graph search without an explicit query,\nusing graph algorithms~\\cite{vakulenko2019message,christmann2019look},\ngraph embeddings~\\cite{saxena2020improving,huang2019knowledge},\nor graph convolutions~\\cite{sun2018open,sun2019pullnet}.\nThe basic idea here is to learn to traverse the graph in symbolic or latent space,\nand these cannot work over text inputs as there is no natural underlying graph representation.\n\nOn the other hand, powerful\ndeep learning methods for text-QA~\\cite{asai2020learning,zhao2021multi,clark2018simple,izacard2021leveraging,chen2017reading}\nare not geared at all\nfor tapping into KGs and other RDF data.\nThis is because these algorithms are basically\nclassifiers trained to predict the start and end position of the most\nlikely answer span in a given piece of text, a sequence-based setup that does not agree\nwith RDF triples.\n\nWorks on {\\em heterogeneous QA}~\\cite{savenkov2016knowledge,xu2016question,sun2018open,sun2019pullnet,oguz2021unikqa,sun2015open,das2017question}, \nincorporate text (typically articles in Wikipedia\nor search results) as a supplement to RDF triples, or\nthe other way around.\nHowever, usually KG and text are combined merely to\nimprove the rankings of candidate answers through additional evidence,\nbut the answers themselves come either from the KG alone or\nfrom text alone.\nMoreover, none of these methods is geared to handle the extreme, but\nrealistic situations, where the input is either only a KG or only a text corpus.\nState-of-the-art methods in heterogeneous QA,\n\\textsc{GRAFT-Net}\\xspace~\\cite{sun2018open} and \\textsc{PullNet}\\xspace~\\cite{sun2019pullnet},\nthat we compare against, suffer from shortcomings that involve\nan indispensable reliance on the KG source.\nSpecifically, they\nrequire trained KG entity embeddings,\nKG entity-linked text,\nshortest paths in KGs for creating training data,\nand can only provide such KG entities as answers. Thus, they cannot operate\nover arbitrary text corpora with an open vocabulary\nwhere ad-hoc phrases could be answers.\nA very recent method, \\textsc{UniK-QA}\\xspace~\\cite{oguz2021unikqa}, takes an approach of unifying sources via verbalization of every knowledge repository~\\cite{thorne2020neural}.\nWe show that our proposal outperforms \\textsc{UniK-QA}\\xspace, that, while being a commendable effort towards an unified view, still cannot deal with more complex information needs.\nSuch cases of complexity are a primary focus in this work, as discussed below.\n\nProblems are particularly exacerbated\nfor complex questions with multiple entities and predicates.\nA running example in this paper is:\n\\begin{Snugshade}\n\t\\textit{\\textbf{Question:} director of the western for which Leo won an Oscar? [\\textbf{Answer:} Alejandro I\\~{n}\\'{a}rritu]}\n\\end{Snugshade}\nHere, SPARQL queries become particularly brittle\nfor KG-QA systems,\ngraph search becomes more complicated,\nor text evidence for Text-QA systems has to \nbe joined from more than one document. \nFor example, the correct query for our running example\nwould be\nsomething like:\n\\begin{Snugshade}\n\\struct{SELECT ?x WHERE\n\\{?y director ?x .\n?y castMember LeonardoDiCaprio .\\\\\n?y genre western .\nLeonardoDiCaprio awarded AcademyAward [forWork ?y] .\\}}\n\\end{Snugshade}\n\nHardly any KG-QA system would get this mapping onto\nKG predicates and entities perfectly right\\footnote{Here, the triple pattern involving \\texttt{[forWork ?y]} shows a simplified representation for matching a fact with qualifiers~\\cite{christmann2022beyond}. In practice this would require matching over a reified fact~\\cite{hernandez2015reifying}.}.\nEven though graph-search methods do not formulate explicit queries, \nthey still need to get the mapping onto KB entities and predicates right.\nFor Text-QA, computing answers requires stitching together\ninformation from several texts, as it is not easy\nto find a single Web page that\ncontains all relevant cues in\nthe same zone.\n\n\\begin{figure} [t]\n\t\\centering\n\t\\begin{subfigure}[h]{0.49\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/xg-kg.pdf}\n\t\t\\caption{$XG(q)$ example for KG as input.}\n\t\t\\label{fig:xg-kg}\n\t\\end{subfigure}\n\\hfill\t\n\n\n\t\\begin{subfigure}[h]{0.49\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/xg-text.pdf}\n\t\t\\caption{$XG(q)$ example for text as input.}\n\t\t\\label{fig:xg-text}\n\t\\end{subfigure}\n\t\\\\\n\t\\begin{subfigure}[h]{\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/xg-kg-text.pdf}\n\t\t\\caption{$XG(q)$ example for KG and text as input.}\n\t\t\\label{fig:xg-hetero}\n\t\\end{subfigure}\n\t\\caption{Context graphs (XG) built by \\textsc{Uniqorn}\\xspace in each of the answering setups for\n\t\tthe question $q=$\n\t\t\\utterance{director of the western for which Leo won an Oscar?}\n\tAnchors are nodes with (partly) underlined labels; answers are in\nbold. Orange subgraphs are Group Steiner Trees.}\n\\label{fig:xg}\n\\vspace*{-0.5cm}\n\\end{figure}\n\n\n\\myparagraph{Approach} To overcome these limitations, we propose \\textsc{Uniqorn}\\xspace, \na \\underline{Uni}fied framework\nfor \\underline{q}uestion answering \\underline{o}ver\n\\underline{R}DF knowledge graphs and \\underline{N}atural language text,\nthat addresses these limitations.\nOur proposal hinges on two key ideas: \n\\squishlist\n\\item Instead of attempting to compute perfect translations of questions\ninto crisp SPARQL queries (which seems elusive for complex questions), \nwe relax the search strategy over KGs by devising graph algorithms\nfor Group Steiner Trees~\\cite{ding2007finding,sun2021finding}, this way connecting question-relevant cues\nfrom which candidate answers can be extracted and ranked.\n\\item Handling KG-QA, Text-QA and heterogeneous setups with the same unified method,\nby building a noisy KG-like context graph from KG or text inputs on-the-fly\nfor each question,\nusing named entity disambiguation (NED), open information extraction (Open IE) and BERT fine-tuning.\n\\squishend\n\nIn a nutshell, \\textsc{Uniqorn}\\xspace works as follows. \nGiven an input question, we first retrieve question-relevant \\textit{evidences} from one or more knowledge sources\nusing a fine-tuned BERT model for sequence pair classification.\nFrom these evidences, that are either KG facts or text fragments, \\textsc{Uniqorn}\\xspace\nconstructs a {\\em context graph (XG)} that contains\nquestion-specific\nentities, predicates, types and candidate answers.\nDepending upon the input source, this XG thus either consists of:\n(i) KG facts\ndefining the neighborhood of the question entities,\nor, (ii) a quasi-KG\ndynamically built by joining Open IE triples extracted from\ntext\nsnippets,\nor, (iii) both, in case of retrieval from heterogeneous sources.\nTriples in the XG originate from evidences that are deemed question-relevant by a fine-tuned BERT model.\nWe identify {\\em anchor nodes} in the XG \nthat match\nphrases in \nthe question.\nTreating the anchors as terminals, Group Steiner Trees \n(GST) are computed \nthat contain candidate answers. \nThese GSTs establish a joint context for \ndisambiguating mentions of\nentities, predicates and types\nin the question.\nCandidate answers are ranked by simple statistical measures rewarding redundancy within the top-$k$\nGSTs. Fig.~\\ref{fig:xg}\nillustrates this unified approach for the settings of KG-QA, Text-QA and heterogeneous-QA.\n\\textsc{Uniqorn}\\xspace thus belongs to the family of methods that locate an answer using graph search and traversal,\nand does not build an explicit structured query~\\cite{saharoy2021question}.\n\n\\myparagraph{Contributions} This work makes the following salient \ncontributions:\n(i) proposing the first unified method\nfor answering\ncomplex\nfactoid questions over knowledge graphs, or text corpora, or both; \n(ii) applying Group Steiner Trees as\na strategy for locating answers to complex questions involving\nmultiple entities and relations;\n(iii) experimentally comparing \\textsc{Uniqorn}\\xspace on\nseveral benchmarks of complex questions against state-of-the-art baselines on KGs, text\nand heterogeneous sources.\nAll code, data, and results for this project are available at:\n\\url{https:\/\/uniqorn.mpi-inf.mpg.de}.\n\n\\myparagraph{Improvements over \\textsc{Quest}\\xspace} \\textsc{Uniqorn}\\xspace is an extension of \\textsc{Quest}\\xspace~\\cite{lu2019answering}, that used GSTs for question answering over quasi KGs induced from text, and was published in SIGIR 2019 as a long paper. \\textsc{Uniqorn}\\xspace contains substantial improvements over \\textsc{Quest}\\xspace, the key factors being the following:\n(i) Unlike \\textsc{Quest}\\xspace that was limited to text inputs, \\textsc{Uniqorn}\\xspace works over KGs, text, and heterogeneous sources;\n(ii) \\textsc{Uniqorn}\\xspace incorporates a novel BERT fine-tuning model for assessing relevance of heterogeneous evidences with respect to the question;\n(iii) \\textsc{Quest}\\xspace was evaluated on a small dataset of $300$ questions. \\textsc{Uniqorn}\\xspace, on the other hand, uses LC-QuAD 2.0~\\cite{dubey2019lc}, a large and recent benchmark for complex KG-QA with about $5000$ test questions (the text corpus counterpart was completely curated by us), along with five other smaller datasets;\nand\n(iv) \\textsc{Uniqorn}\\xspace, over the three setups, includes ten baselines, whereas \\textsc{Quest}\\xspace used just three of them.\n\n\\section{Experimental Results}\n\\label{sec:exp-res}\n\n\\subsection{Key Findings}\n\\label{subsec:keyres}\n\n\\begin{table}[t]\n\t\\centering\n\n\t\\setlength{\\tabcolsep}{0.5em}\n\t\\begin{tabular}{l c c c}\t\\toprule\n\t\n       \n        \\textbf{Method}        \t\t\t\t\t& \\textbf{KG}           & \\textbf{Text}         & \\textbf{KG+Text}      \\\\ \\midrule\n\t\t\\textsc{Uniqorn}\\xspace                                & $\\boldsymbol{0.315}$* & $0.110$               & $\\boldsymbol{0.331}$* \\\\ \\midrule\n\t\t\\textsc{QAnswer}\\xspace~\\cite{diefenbach2020towards}   & $0.308$               & -                     & -                     \\\\\n\t\t\\textsc{Platypus}\\xspace~\\cite{tanon2018demoing}       & $0.036$               & -                     & -                     \\\\ \\midrule\n\t\t\\textsc{PathRetriever}\\xspace~\\cite{asai2020learning}  & -                     & $\\boldsymbol{0.230}$  & -                     \\\\\n\t\t\\textsc{DocumentQA}\\xspace~\\cite{clark2018simple}           & -                     & $0.162$               & -                     \\\\\n\t\t\\textsc{DrQA}\\xspace~\\cite{chen2017reading}            & -                     & $0.112$               & -                     \\\\ \\midrule\n\t\t\\textsc{UniK-QA}\\xspace~\\cite{oguz2021unikqa}           & $0.005$               & $\\boldsymbol{0.240}$  & $0.107$               \\\\ \n\t\t\\textsc{PullNet}\\xspace~\\cite{sun2019pullnet}          & $0.011$               & -                     & $0.010$               \\\\\n\t\t\\textsc{GRAFT-Net}\\xspace~\\cite{sun2018open}            & $0.129$               & -                     & $0.175$               \\\\ \\midrule\n\t\t\\textsc{BFS}\\xspace~\\cite{kasneci2009star}             & $0.179$               & $0.046$               & $0.116$               \\\\\n\t\t\\textsc{ShortestPaths}\\xspace                          \t    & $0.131$               & $0.074$               & $0.110$               \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\caption{Comparison of \\textsc{Uniqorn}\\xspace and baselines over the LC-QuAD $2.0$ test set, as measured by P@1 performance. \n\tThe best value per column is in \\textbf{bold}. An asterisk (*) indicates \n\tstatistical significance of \\textsc{Uniqorn}\\xspace over the best baseline in that column.\n\tA hyphen (`-') indicates that the corresponding baseline cannot be applied to that particular setting. An `x' denotes the particular\n\tmetric cannot be computed.}\n\t\\label{tab:main-res}\n\t\\vspace*{-0.7cm}\n\\end{table}\n\n\\begin{table}[t]\n\t\\centering\n\t\\resizebox*{\\textwidth}{!}{\n\t\t\\setlength{\\tabcolsep}{0.5em}\n\t\t\\begin{tabular}{l ccccc c ccccc c ccccc}\t\\toprule\n\t\t \\multirow{2}{*}{\\textbf{Method}} \n\t\t & \\multicolumn{5}{c}{\\textbf{KG (P@1)}}   \n\t\t && \\multicolumn{5}{c}{\\textbf{Text (P@1)}} \n\t\t && \\multicolumn{5}{c}{\\textbf{KG+Text (P@1)}} \\\\ \n\t\t \\cmidrule{2-6}\\cmidrule{8-12}\\cmidrule{14-18}\n\t\t\t& \\textbf{\\begin{tabular}[c]{@{}c@{}}LCQ1\\end{tabular}}     & \\textbf{ComQA}     & \\textbf{CQ-W}     & \\textbf{CQ-T}     & \\textbf{QALD}       \n\t\t\t&& \\textbf{\\begin{tabular}[c]{@{}c@{}}LCQ1\\end{tabular}}     & \\textbf{ComQA}     & \\textbf{CQ-W}     & \\textbf{CQ-T}     & \\textbf{QALD}      \n\t\t\t&& \\textbf{\\begin{tabular}[c]{@{}c@{}}LCQ1\\end{tabular}}     & \\textbf{ComQA}     & \\textbf{CQ-W}     & \\textbf{CQ-T}     & \\textbf{QALD} \\\\ \\bottomrule\n\t\t\t\\textsc{Uniqorn}\\xspace       & $0.171$    &   $0.099$\t    &    $0.187$    &  $0.107$     &\t$0.114$              && $0.213$ &\t$0.183$ &\t$0.213$ &\t$0.360$ &\t$0.271$  && $0.267$      & $0.208$    & $0.313$        & $0.380$  & $0.171$      \\\\ \\midrule\n\t\t\tQAnswer       & $0.164$\t& $0.238$\t& $0.293$ & $0.147$ & \t$0.114$ && -                      & -                       & -                      & -                      & -                      && -                         & -              & -                  & -            & -               \\\\ \n\t\t\tPlatypus      & $0.021$ & $0.010$ &\t$0.007$\t & $0.007$\t& $0.100$           && -                      & -                       & -                      & -                      & -                      && -                         & -              & -                  & -            & -               \\\\ \\midrule\n\t\t\tPath Retriever         & -                 & -                       & -                      & -                     & -                    && $0.373$ &\t$0.594$ &\t$0.560$ &\t$0.367$ &\t$0.357$             && -                         & -              & -                  & -            & -               \\\\\n\t\t\tDocQA         & -                 & -                       & -                      & -                     & -                    && $0.314$ & $0.609$\t& $0.667$\t& $0.573$\t& $0.229$          && -                         & -              & -                  & -            & -               \\\\ \n\t\t\tDrQA          & -                 & -                       & -                      & -                     & -                    && $0.182$ &\t$0.193$ & $0.300$\t& $0.430$ & $0.057$           && -                         & -              & -                  & -            & -               \\\\ \\midrule\n\t\t\tPullNet       &$0.016$ & $0.040$               & $0.000$                & $0.000$               & $0.000$              && -                      & -                       & -                      & -                      & -                      && $0.040$                   & $0.101$        & $0.014$            & $0.020$      & $0.006$              \\\\ \n\t\t\tGRAFT-Net     & $0.053$           & $0.045$                 & $0.033$                & $0.007$               & $0.000$              && -                      & -                       & -                      & -                      & -                      && $0.116$                   & $0.104$        & $0.087$            & $0.031$      & $0.022$              \\\\\n\t\t\tUniK-QA     & $0.042$ & $0.045$ & $0.013$ & $0.000$ & $0.029$           && $0.207$ & $0.267$ & $0.140$ & $0.087$ & $0.171$       && $0.145$ & $0.149$ & $0.047$ & $0.047$ & $0.086$              \\\\\n\t\t\t\\midrule\n\t\tBFS           &     $0.117$     &       $0.055$     &\t$0.107$     &\t$0.080$     &\t$0.014$              && $0.096$ &\t$0.129$ &\t$0.147$ &\t$0.187$ &\t$0.100$                && $0.118$                   & $0.134$        & $0.083$            & $0.079$      & $0.000$              \\\\ \n\t\t\tSP            &     $0.077$      &   $0.074$  &     $0.133$  &     $0.047$   &   $0.057$   && $0.149$ & $0.243$ &\t$0.193$ &\t$0.273$ & $0.129$                && $0.208$                   & $0.266$        & $0.173$            & $0.291$      & $0.104$              \\\\ \\bottomrule\n\n\t\t\n\t\t\\end{tabular}\n\t}\n\t\\caption{Comparison of \\textsc{Uniqorn}\\xspace and baselines for LC-QuAD 1.0, ComQA, CQ-W, CQ-T, and QALD datasets.\n\t\tThe best value per column is in \\textbf{bold}. An asterisk (*) indicates \n\tstatistical significance of \\textsc{Uniqorn}\\xspace over the best baseline in that column.\n\t\t`-' indicates that the corresponding baseline cannot be applied to this setting.}\n\t\\label{tab:other-res}\n\t\\vspace*{-0.9cm}\n\\end{table}\n\nOur main results are presented in Tables~\\ref{tab:main-res} and~\\ref{tab:other-res}.\nThe following discusses our key findings.\nAll tests of statistical significance hereafter correspond to the two-tailed paired $t$-test,\nwith the $p$-value $\\leq 0.05$.\n\n\\textbf{(i) \\textsc{Uniqorn}\\xspace yields competitive results for most of the settings\nconsidered:} This performance is observed across-the-board,\nby knowledge source, benchmark, baseline (including neural models\nlike PullNet with large-scale training), or metric.\nAchieving this for both\nsettings is a success, given that the competitors\nfor most paradigms do not have any entries in the others\n(as observed by the numerous hyphens in the table).\nNote that PullNet and GRAFT-Net cannot deal with unseen KG entities at test time,\nwhich is very often the case for benchmarks.\nBut to be fair to these strong baselines, we allowed them to have\naccess to questions in the test set during training, so that appropriate\nentity embeddings to be learnt. Additionally, we have only considered questions \nwhere their final classifier receives correct input (i.e. with ground truth in the entity set) \nfrom the previous classification steps.\nIf this is not allowed, their\nperformance would drop drastically: MRRs of $0.062$\nand $0.049$ for PullNet, the better method of the two, for KG+Text and KG, respectively.\nTable~\\ref{tab:other-res} test open-domain performance: ability\nto answer questions on benchmarks (here, the multiple smaller ones) that are different from\nthe one trained on (the largest one).\n\nUnlike many other systems, GRAFT-Net and PullNet can indeed take into\naccount information in qualifiers (these were originally designed for\nFreebase CVTs, we used the analogous ideas when reimplementing for\nWikidata qualifiers).\nWe attribute the relatively poor performance\nof the complex QA system PullNet\nin some scenarios to its tackling of certain classes of complex questions\nonly: namely, the multi-hop ones\n(\\utterance{brother of director of Inception?}).\nHowever, there are also other common classes\n(\\utterance{coach who managed both liverpool and barcelona?}).\n\\textsc{Uniqorn}\\xspace's ability to handle arbitrary types of questions with respect\nto multiple entities and relations contributes this improved numbers.\nThis is a reason why GRAFT-Net systematically shows a better \nperformance than PullNet, even though the latter is an extension of\nthe former.\n\n\\textbf{(ii) Computing GSTs on a question-specific context graph is\na preferable alternative to SPARQL-based systems over RDF graphs:}\nThis is seen with the Hit@5 of $0.414$ for \\textsc{Uniqorn}\\xspace versus $0.318$\nfor QAnswer and $0.109$ for Platypus, the two SPARQL-based systems).\nLet us examine this in more detail.\nFor KGs, QAnswer is clearly the stronger baseline.\nIt operates by mapping names and phrases in the question to KG concepts,\nenumerating all possible SPARQL queries connecting these concepts\nthat return a non-empty answer, followed by a query ranking phase.\nThis approach works really well for simple questions with\nrelatively few keywords. However, the query ranking phase becomes\na bottleneck\nfor complex questions with multiple entities and relations, resulting\nin too many possibilities. This is the reason why \nreliance on the best SPARQL query may be a bad choice.\nAdditionally, unlike \\textsc{Uniqorn}\\xspace, QAnswer cannot leverage\nany qualifier information in Wikidata.\nThis is both due to the complexity of the SPARQL query necessary\nto tap into such records,\nas well as an explosion in the number of query candidates\nif qualifier attributes are allowed into the picture.\nOur GST establishes this common question context \nby a joint disambiguation of all question phrases, and smart answer\nranking helps to cope with the noise and irrelevant information in the XG.\nWhile QAnswer is completely syntax-agnostic, Platypus is the other\nextreme: it relies on accurate dependency parses of the NL question\nand hand-crafted question templates to come up with the best logical\nform. This performs impressively when questions fit the supported\nset of syntactic patterns, but is brittle when exposed to\na multitude of open formulations from an ensemble of QA benchmarks.\nGRAFT-Net performing better than QAnswer also attests to the superiority\nof graph-based search over SPARQL for complex questions.\n\n\\textbf{(iii) \\textsc{Uniqorn}\\xspace's results are better than the MRC systems:}\nIt is \nremarkable that \\textsc{Uniqorn}\\xspace \nsignificantly outperforms the reading comprehension systems DocQA and DrQA.\nBoth systems use sophisticated neural networks with extensive\ntraining and leverage large resources like Wikipedia, SQuAD or\nTriviaQA, whereas \\textsc{Uniqorn}\\xspace is completely unsupervised and could be\nreadily applied to other benchmarks with no further tuning. \n\\textsc{Uniqorn}\\xspace's success comes from its unique ability to stitch\nevidence from multiple documents to faithfully answer a question.\nThe Text-QA baselines can also handle multiple passages, but the\nassumption is still that a single passage or sentence contains\nthe correct answer and\n\\textit{all} question conditions. This is flawed: as an aftermath,\nthe accuracy of these systems is often gained by only satisfying part\nof the question (for \\utterance{Who directed Braveheart and Paparazzi?},\nonly spotting the director of Braveheart suffices).\nAlso, DocQA and DrQA are not able to address such levels of complexity\nas demanded by our challenging benchmarks.\n\n\\textbf{(iv) Graph-based methods open up the avenue for this unified\nanswering process:}\nWhat is especially noteworthy is that\nusing the the joint optimization of interconnections between\nanchors via GSTs is essential and powerful.\nUse of GSTs is indeed required for best performance, and cannot be easily approximated\nwith simpler graph methods like BFS and SP.\nIt is worthwhile to note that in the open-domain experiments (Table~\\ref{tab:other-res}),\nat least for the text setting, BFS performs notably well (ComQA,\nCQ-W, CQ-T, QALD).\nBFS tree search is actually a building block for many Steiner Tree\napproximation algorithms~\\cite{kasneci2009star}.\nShortestPaths (SP) also performs respectably in these settings, \nas notions of shortest paths are also implicit in GSTs.\nThe brittleness of these ad hoc techiques are exposed in\nthe larger and more challenging settings of KG and KG+Text.\n\nWe show representative examples from the various benchmarks in Table~\\ref{tab:anecdotes},\nto give readers a feel of the complexity of information needs that \\textsc{Uniqorn}\\xspace\ncan handle.\n\n\\begin{table} [t]\n\t\\centering\n\t\\resizebox*{\\textwidth}{!}{\n\t\t\\setlength{\\tabcolsep}{0.5em}\n\t\t\\begin{tabular}{p{7cm} p{7cm} p{7cm}} \n\t\t\t\\toprule\n\t\t\t\\textbf{KG}\t\t\t\t\t\t\t\t\t\t\t\t  & \\textbf{Text}\t& \\textbf{KG+Text}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\ \\toprule\n\t\t\t\\utterance{Which charitable organization runs Wikipedia ?} (LC-QuAD 2.0) \t                                                                                        & \\utterance{Apollo 13 was nominated for and won which awards?} (LC-QuAD 2.0)  & \\utterance{Who is the master of Leonardo da Vinci, who was employed as a sculptor?}  (LC-QuAD 2.0)   \\\\\t\\midrule\n\t\t\t\n\t\t\t\\utterance{Which research library archives the works of James Joyce?} (LC-QuAD 2.0)\t                                                                        & \\utterance{Who is the singer in the series of Bone Palace Ballet?} (LC-QuAD 2.0)\t                  & \\utterance{What is the name of the reservoir the Grand Coulee Dam made?}\t(LC-QuAD 2.0)\t\\\\\t\\midrule\n\t\t\t\n\t\t\t\\utterance{What church building was constructed by Dutch East India Company?} (LC-QuAD 2.0)\t& \\utterance{In what mountain range is Mt. Anhui located?} (LC-QuAD 2.0)      & \\utterance{What Empire used to have Istanbul as its capital?} (LC-QuAD 2.0) \\\\\t\\midrule\n\t\t\t\n\t\t\t\\utterance{At what university did Jacques Barzun get the doctorate of philosophy?} (LC-QuAD 2.0)  \t                                                                                & \\utterance{What is the location of the formation that is named after Lynyrd Skynyrd?}  (LC-QuAD 2.0)                                     & \\utterance{What is the jurisdiction of the area of East Nusa Tenggara Province?} \t(LC-QuAD 2.0)\t\t\t\t\t\t\t\\\\ \\midrule\n\t\t\t\n\t\t\t\\utterance{What award did David Geffen receive in the year 1990?}  (LC-QuAD 2.0)\t\t                                                                        & \\utterance{What are the video game console model which letter starts with x?} (LC-QuAD 2.0)               & \\utterance{Tell me which logographic writing system is used by the Japanese.}\t(LC-QuAD 2.0)\t\t\t\t\t\t\\\\ \n\t\t\t\\bottomrule\n\t\t\\end{tabular} }\n\t\\caption{Anecdotal examples from all the datasets where \\textsc{Uniqorn}\\xspace was able to compute a correct answer at the top rank (P@1 $= 1$), but none of the baselines could.}\n\t\\label{tab:anecdotes}\n\t\\vspace*{-1cm} \n\\end{table}\n\n\\subsection{Insights and Analysis} \n\\label{subsec:analysis}\n\n\\begin{table} [t]\n\t\\centering\n\t\\setlength{\\tabcolsep}{0.5em}\n\t\\begin{tabular}{l c c c} \\toprule\n\t\t\t\\textbf{Configuration}\t& \\textbf{KG}\t\t\t& \\textbf{Text}\t\t\t& \\textbf{KG+Text}\t\\\\ \\toprule\n\t\t\tFull\t\t        & $0.266$\t                        &  $0.108$\t        &   $\\boldsymbol{0.292}$\t\\\\ \\midrule\n\t\t\tNo types \t\t& $\\boldsymbol{0.279}$                       &  $\\boldsymbol{0.114}$                      &   $0.289$\t\t        \\\\\n\t\t\tNo entity alignment \t&     -                         &  $0.101$                     &   $0.289$ \t\t        \\\\\t\n\t\t\tNo predicate alignment  &     -                         &  $0.097$                      &   $0.240$ \t\t        \\\\\n\t\t\tNo type alignment  &     -                         &  $0.110$                      &   $0.286$ \t\t        \\\\  \\bottomrule \n\t\\end{tabular}\n\t\\caption{Pipeline ablation results on the full LC-QuAD 2.0 dev set with P@1. \n\tBest values per column are in \\textbf{bold}.\n\tStatistically significant differences from the full configuration are marked with *.}\n\t\\label{tab:ablation}\n \t\\vspace*{-0.7cm}\n\\end{table}\n\n\\textbf{Ablation experiments.} To get a proper\nunderstanding of \\textsc{Uniqorn}\\xspace's successes and failures, it is important to\nsystematically ablate its configurational pipeline\n(Table~\\ref{tab:ablation}). We note the following:\n\\squishlist\n\\item \\textbf{[Row 1]} The full pipeline of \\textsc{Uniqorn}\\xspace achieves the best performance, showing that all cogs\nin its wheel play a vital role, as detailed below.\n\\item \\textbf{[Row 2]} ``No NED expansion'' implies directly using entities from NED for KG-lookups.\nWe find that\nexpanding lookups for XG construction with NER + lexicons\ncompensates for NED errors substantially;\n\\item \\textbf{[Row 3]} Using entity types is useful, as seen by the significant\ndrops of $0.327$ to $0.288$\nfor KG+Text and $0.279$ to $0.234$ for text;\n\\item \\textbf{[Rows 4 and 5]} On-the-fly entity and predicate alignments are crucial for addressing\nvariability in surface forms for text-QA (also, as a result, for KG+Text).\n\\squishend\n\n\\begin{table} [t]\n\t\\centering\n\n\t\\setlength{\\tabcolsep}{0.5em}\n\t\\begin{tabular}{l c c c}\n\t\t\\toprule\n\t\t\\textbf{Answer ranking}\t\t& \\textbf{KG}\t\t\t& \\textbf{Text}\t\t\t& \\textbf{KG+Text}\t\\\\ \\toprule\n \t\tGST counts\t\t\t\t\t&   $\\boldsymbol{0.266}$        &   $\\boldsymbol{0.108}$        &   $\\boldsymbol{0.292}$        \\\\ \n\t\tGST costs \t\t\t\t\t&   $0.266$                     &   $0.094$        &   $0.282$        \\\\\n\t\tGST node weights \t\t\t&   $0.264$                       &   $0.098$        &   $0.284$        \\\\\t\t\t\n\t\tAnchor distances        \t&   $0.250$                       &   $0.082$                     &   $0.225$\t \\\\    \n\t\tWeighted anchor distances\t&   $0.260$                     &   $0.082$                     &   $0.227$        \\\\ \\bottomrule\n\t\\end{tabular}\n\n\t\\caption{Different answer ranking results on the LC-QuAD 2.0 dev set on P@1. \n\t\tBest values per column are in \\textbf{bold}.\n\t\tStatistically significant differences from the first row are marked with *.}\n\t\\label{tab:ranking}\n\t\\vspace*{-0.9cm}\n\\end{table}\n\n\\textbf{Answer ranking.} \\textsc{Uniqorn}\\xspace ranks answers by the number of different\nGSTs that they occur in Row 1. However, there arise some natural variants\nthat leverage a weighted sum rather than a simple count\n(Table~\\ref{tab:ranking}). This weight\ncan come from the following sources: (i) the total cost of the GST,\nthe less the better [Row 2]; (ii) the sum of node weights in the GST,\nreflecting question relevance, the more the better [Row 3];\n(iii) total distances of answers to the anchors in the GST, the less\nthe better [Row 4]; and, (iv) a weighted version of anchor distances,\nwhen edge weights are available, the less the better [Row 5].\nWe find that using plain\nGST counts works surprisingly well. Zooming in to the GST and\nexamining anchor proximities is not necessary\n(actually hurts significantly for KG+Text).\n\n\\begin{table} [t]\n\t\\centering\n\t\\setlength{\\tabcolsep}{0.5em}\n\t\\begin{tabular}{l c c c}\n\t\t\\toprule\n\t\t\\textbf{Error scenario}\t                             & \\textbf{KG}\t \t     & \\textbf{Text}      & \\textbf{KG+Text} \\\\ \\toprule\n\t\tAnswer not in corpus\t                             &  $57.99\\%$          \t     \t\t\t& $24.92\\%$\t    & $34.32\\%$\t   \\\\\n\t\tAnswer in corpus but not in XG                       &  $22.59\\%$                     & $24.49\\%$            & $29.63\\%$          \\\\\n\t\tAnswer in XG but not in top-$k$ GSTs \t     \t     &  $2.67\\%$                     & $23.44\\%$             & $7.64\\%$          \\\\\t\t\t\n\t\tAnswer in top-$k$ GSTs but not in top-$5$          &  $6.37\\%$                     & $23.92\\%$             & $28.41\\%$           \\\\    \n\t\tAnswer in top-$5$ but not in top-$1$                 &  $10.38\\%$                     & $3.24\\%$             & $0.00\\%$          \\\\ \\bottomrule\n\t\t\\end{tabular}\n\t\\caption{Percentages of different error scenarios where the answer is not in top-$1$, averaged over the full LC-QuAD 2.0 test set.}\n\t\\label{tab:error}\n\t\\vspace*{-0.9cm}\n\\end{table}\n\n\\textbf{Error analysis.} In Table~\\ref{tab:error}, we extract all questions for which \\textsc{Uniqorn}\\xspace\nproduces an imperfect ranking ($P@1 = 0$), and discuss cases in\na cascaded style. Each column adds up to $100\\%$, and reports the distribution of errors\nfor the specific setting. Note that values across rows are not comparable.\nWe make the following observations:\n\n(i) \\textbf{[Row 1]} indicates \nthe sub-optimal ranking of Google's retrieval from the Web\nwith respect to complex questions.\nStrictly speaking, this is out-of-scope for \\textsc{Uniqorn}\\xspace; nevertheless,\nan ensemble of search engines (Bing+Google)\nmay help improve answer coverage. This cell is null for KGs as all\nthe sampled LC-QuAD $2.0$ questions were guaranteed to\nhave answers in Wikidata.\n\n(ii) \\textbf{[Row 2]} indicates answer presence\nin $KG$ or $KG^D$, but not in the XG, and indicates NED\/NER errors for KG,\nand incorrect phrase matches for text. The largest connected component (LCC)\nis also a likely culprit at this stage, pruning away\nthe question-relevant entities in favor of more popular ones.\nReducing reliance on NERD systems and computing larger neighborhood joins\nas opposed to LCCs could be useful here.\n\n(iii) \\textbf{[Row 3]} Presence of an answer in the XG but not in top-$500$ GSTs\nusually indicates an incorrect anchor matching. This could be due to \nproblems in the similarity functions for entities (currently based\non Jaccard overlaps of mention-entity lexicons, that are inherently\ncomplete) and predicates.\nFor example, incorrect predicate matching due to shortcomings\nof embedding-based similarity (\\phrase{married} and \\phrase{divorced}\nhaving a very high similarity of $0.82$, making the latter an anchor too\nwhen the former is in the question) can happen.\nWe had conducted extensive experiments on hyperparameter-based\nlinear combinations of various similarity measures for both entities\nand predicates (word Jaccard, mention-entity lexical Jaccard, word2vec,\nlexicon priors based on anchor links)\nto see what worked best: the single choices mentioned above performed\nbest and were used. But clearly, there is a big room for improvement\nwith respect to these similarity functions, as both rows 2 and 3\nare affected by this.\n\n(iv) \\textbf{[Rows 4 and 5]} represent cases\nwhen the answer is in the top-$k$ GSTs but languish at lower ranks\nin the candidates.\nExploring weighted rank aggregation by tuning on the development set\nwith variants in Table~\\ref{tab:ranking} is\na likely remedy. A high volume of errors in this bucket is actually one of\npositive outlook: the core GST algorithm generally works well,\nand significant performance gains can be obtained\nby fine-tuning the ranking function with additional parameters.\n\n\\begin{table} [t]\n\t\\centering\n\t\\resizebox*{\\textwidth}{!}{\n\t\t\\setlength{\\tabcolsep}{0.5em}\n\t\t\\begin{tabular}{c cc c cc c cc}\n\t\t\\toprule\n\t\t& \\multicolumn{2}{c}{\\textbf{\\textbf{KG}}} &&  \\multicolumn{2}{c}{\\textbf{\\textbf{Text}}}  &&  \\multicolumn{2}{c}{\\textbf{\\textbf{KG+Text}}} \\\\ \n                \\cmidrule{2-3}\\cmidrule{5-6}\\cmidrule{8-9}\n\t\t\t\n \t\t\tGST Ranks            & \\begin{tabular}[c]{@{}c@{}}Avg. \\#Docs\\\\ in GST\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}\\% of Questions with\\\\ Answers in GST\\end{tabular} && \\begin{tabular}[c]{@{}c@{}}Avg. \\#Docs\\\\ in GST\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}\\% of Questions with\\\\ Answers in GST\\end{tabular} && \\begin{tabular}[c]{@{}c@{}}Avg. \\#Docs\\\\ in GST\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}\\% of Questions with\\\\ Answers in GST\\end{tabular} \\\\ \\bottomrule\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n  \t\t\t  1              &       -                                                       & $37.80\\%$                                                                    && $1.557$                                                      & $31.97\\%$                                                                     && $1.715$                                                  & $46.92\\%$                                                                  \\\\\n\t\t\t2              &       -                                                       & $39.42\\%$                                                                    && $1.602$                                                      & $32.37\\%$                                                                     && $1.724$                                                  & $47.41\\%$                                                                  \\\\\n\t\t\t3               &       -                                                       & $40.28\\%$                                                                    && $1.640$                                                      & $32.78\\%$                                                                     && $1.732$                                                  & $47.82\\%$                                                                  \\\\\n\t\t\t4            &       -                                                       & $41.13\\%$                                                                    && $1.679$                                                      & $33.25\\%$                                                                     && $1.742$                                                  & $48.12\\%$                                                                   \\\\\n\t\t\t5            &       -                                                       & $41.90\\%$                                                                    && $1.712$                                                      & $33.63\\%$                                                                     && $1.754$                                                  & $48.53\\%$                                                                  \\\\\\midrule\n\t\t\t\n\t\t\t\\#Evidences in GST        & \\begin{tabular}[c]{@{}c@{}}Avg. Rank \\\\ of GST\\end{tabular}  & \\begin{tabular}[c]{@{}c@{}}\\% of Questions with\\\\ Answers in GST\\end{tabular} && \\begin{tabular}[c]{@{}c@{}}Avg. Rank \\\\ of GST\\end{tabular}  & \\begin{tabular}[c]{@{}c@{}}\\% of Questions with\\\\ Answers in GST\\end{tabular} && \\begin{tabular}[c]{@{}c@{}}Avg. Rank \\\\ of GST\\end{tabular}& \\begin{tabular}[c]{@{}c@{}}\\% of Questions with\\\\ Answers in GST\\end{tabular} \\\\ \\midrule\n\t\t\t1                    & -                                                            & -                                                                             && $1.875$                                                    & $7.62\\%$                                                                      && $4.422$                                                  & $18.63\\%$                                                                 \\\\\n\t\t\t2                    & -                                                            & -                                                                             && $1.964$                                                    & $13.21\\%$                                                                     && $2.895$                                                  & $13.03\\%$                                                                  \\\\\n\t\t\t3                    & -                                                            & -                                                                             && $2.030$                                                    & $10.38\\%$                                                                     && $2.679$                                                  & $10.95\\%$                                                                  \\\\\n\t\t\t4                    & -                                                            & -                                                                             && $2.109$                                                    & $4.37\\%$                                                                     && $1.814$                                                  & $5.51\\%$                                                                   \\\\ \n\t\t\t5                    & -                                                            & -                                                                             && $1.992$                                                    & $0.73\\%$                                                                      && $2.112$                                                  & $1.54\\%$                                                                     \\\\ \\bottomrule\n\t\t\\end{tabular}\n\t}\n\t\\caption{Effect of multi-document evidence shown via edge contributions by distinct documents to GSTs for the LC-QuAD $2.0$ test set.}\n\t\\label{tab:gst-rank}\n\t\\vspace*{-0.9cm} \n\\end{table}\n\n\\begin{figure} [t]\n\t\\centering\n\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/gst-k-kg2.pdf}\n\t\t\\caption{KG - GSTs.}\n\t\t\\label{fig:gst-kg}\n\t\\end{subfigure}\n\t\\hspace{0.1cm}\n\t\n\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/TEXT_align_tune2.pdf}\n\t\t\\caption{Text - Alignments.}\n\t\t\\label{fig:align-text}\n\t\\end{subfigure}\n\t\\hspace{0.1cm}\n\t~\n\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/gst-k-text2.pdf}\n\t\t\\caption{Text - GSTs.}\n\t\t\\label{fig:gst-text}\n\t\\end{subfigure}\n\t\\hspace{0.1cm}\n\t\n\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/KG_TEXT_align_tune2.pdf}\n\t\t\\caption{KG+Text - Alignments.}\n\t\t\\label{fig:align-kg-text}\n\t\\end{subfigure}\n\t\\hspace{0.1cm}\n\t~\n\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/gst-k-kg-text2.pdf}\n\t\t\\caption{KG+Text - GSTs.}\n\t\t\\label{fig:gst-kg-text}\n\t\\end{subfigure}\n\t\\caption{Results of parameter tuning for different settings on the LC-QuAD 2.0 development set of \\textsc{Uniqorn}\\xspace.}\n\t\\label{fig:parameters}\n\t\\vspace*{-0.5cm}\n\\end{figure}\n\n\\textbf{Usage of multi-document evidence.} For text-QA, \\textsc{Uniqorn}\\xspace improved over\nDrQA and DocQA on all benchmarks, even\nthough the latter are supervised deep learning methods\ntrained on the large SQuAD and TriviaQA datasets.\nThis is because\nreading comprehension (RC) QA systems\nsearch for the best answer span within a passage, and will not\nwork well unless the passage\nmatches the question tokens and contains the answer.\nWhile DrQA and DocQA\ncan additionally select a good set of documents from a collection,\nit still relies\non the best document to extract the answer from. \\textsc{Uniqorn}\\xspace,\nby joining fragments of evidence across documents via GSTs, thus improves\nover\nthese methods without any training.\n\\textsc{Uniqorn}\\xspace benefits two-fold from multi-document evidence:\n\\squishlist\n\\item Confidence in an answer increases when all conditions for correctness\nare indeed satisfiable (and found) only when looking at multiple documents.\nThis increases\nthe answer's likelihood of appearing in \\textit{some GST}.\n\\item Confidence in an answer increases when it is spotted\nin multiple documents. This increases its likelihood of appearing in\nthe \\textit{top-k} GSTs, as presence in multiple documents increases\nweights\nand lowers costs of the corresponding edges.\n\\squishend\nA detailed investigation of the use of multi-document information is\npresented in Table~\\ref{tab:gst-rank}. We make the following observations:\n(i) Looking at the ``Avg. \\#Docs in GST'' columns in the upper half,\nwe see that \nconsidering the top-$500$ GSTs is worthwhile as all the bins combine\nevidence from multiple ($2+$ on average) documents.\nThis is measured by labeling edges in GSTs with\ndocuments (identifiers) that contribute the corresponding edges.\n(ii) Moreover, they also contain\nthe correct answer uniformly often\n(corresponding\n``\\#Questions with Answers in GST'' columns).\n(iii) The bottom half of the table inspects\nthe inverse phenomenon, and finds that\nconsidering only the top few GSTs is not sufficient\nfor aggregating multi-document evidence.\n(iv) Finally,\nthere is a sweet spot for GSTs aggregating nuggets from multiple documents\nto contain correct answers, and this turns out to be around three\ndocuments (see corresponding ``\\#Questions with Answers in GST'' columns).\nThis, however, is an effect of our questions\nin our benchmarks,\nthat are not complex enough to require stitching evidence\nacross more than three documents.\nDeep-learning-based RC methods over text can handle syntactic complexity\nvery well, but are typically restricted to identifying answer spans from\na single text passage. DrQA and DocQA could not properly tap the answer evidence that\ncomes from combining cues spread across multiple documents.\nAnalogous results for KG and KG+Text are provided for completeness.\n\n\\textbf{Parameter tuning.}\nThere are only five parameters in \n\\textsc{Uniqorn}\\xspace: the alignment edge insertion thresholds on\nnode-node similarity (one for entities $\\tau_{align}^E$ and\none for predicates $\\tau_{align}^P$),\nanalogous anchor selection thresholds on node weight\n($\\tau_{anchor}^E$ and $\\tau_{anchor}^P$),\nand the number of GSTs $k$.\nVariation of\nthese parameters are shown with MRR \nin Fig.~\\ref{fig:parameters} on the LC-QuAD $2.0$ development set.\nWe observe that:\n(i) having several alignment edges in the graph\n(valid only for text- and KG+text-setups)\nfor entities (corresponding to low thresholds)\nactually helps improve\nperformance though apparently inviting noise\n(dark zones indicate good performance).\nFor predicates, relatively high thresholds work\nbest to avoid spurious matches. Note that \nthe exact values of the $x$- and $y$-axes in the heat maps\nare not comparable due to the different similarity functions\nused for the two cases.\nThe predicate threshold exploration is confined to\nthe zone for which the graph algorithm was tractable\n(corresponding to a high cut-off time of thirty minutes\nto produce the answer);\n(ii) There is a similar situation for anchor thresholds:\nlow thresholds for entities work best, while \n\\textsc{Uniqorn}\\xspace is not really sensitive to the exact choice\nfor predicates.\nThe specifics vary across the setups, but as a general\nguideline, most non-extreme choices should work out fine.\nThe white zones in the top right corners\ncorrespond to \nsetting both thresholds to very high values\n(almost no anchors qualify to be in top-$5$, resulting in zero performance);\n(iii) going beyond\nthe chosen value of $k=500$ GSTs\n(for all setups) gives only diminishing returns.\n\n\n\\textbf{Effect of Open IE.} \\textsc{Uniqorn}\\xspace's noisy triple extractor\nresults in quasi KGs that contain the correct answer $85.2\\%$ of\nthe times for CQ-W ($82.3\\%$ for CQ-T). If we use triples\nfrom Stanford OpenIE~\\cite{angeli2015leveraging} to build the\nquasi KG instead, the answer is found only $46.7\\%$ and $35.3\\%$\ntimes for CQ-W and CQ-T, respectively. Thus, in the context\nof QA, losing information with precision-oriented, fewer triples,\ndefinitely hurts more than the adding potentially many noisy ones.\n\n\\begin{figure} [t]\n\t\\centering\n\t\\begin{subfigure}[b]{\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.3\\textwidth]{images\/gst-1.pdf}\n\t\n\t\t\\caption{Example GST for \\utterance{Which aspiring model split with Chloe Moretz and is dating Lexi Wood?}}\n\t\t\\label{fig:gst-1}\n\t\\end{subfigure}\n\t\n\t\\begin{subfigure}[b]{\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.7\\textwidth]{images\/gst-2.pdf}\n\t\t\\caption{Example GST for \\utterance{Who played for FC Munich and was born in Karlsruhe?}}\n\t\t\\label{fig:gst-2}\n\t\\end{subfigure}\n\t\n\t\\begin{subfigure}[b]{\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.7\\textwidth]{images\/gst-2.pdf}\n\t\t\\caption{Example GST for \\utterance{Which actor is married to Kaaren Verne and played in Casablanca?}}\n\t\t\\label{fig:gst-3}\n\t\\end{subfigure}\n\t\\caption{GSTs construct interpretable evidence for complex questions.}\n\t\\label{fig:gst-eg}\n\n\\end{figure}\n\n\\textbf{GSTs contribute to explainability.} Finally, we posit that\nGroup Steiner Trees over question-focused context graphs help\nin understanding the process of answer derivation for an end-user.\nWe illustrate\nthis using three anecdotal examples of GSTs for text-QA, in\nFigs.~\\ref{fig:gst-1} through~\\ref{fig:gst-3}. The corresponding\nquestion is in the subfigure caption. Anchor nodes are in orange,\nand answers in light blue. Entities, predicates, and class nodes\nare in rectangles, ellipses, and diamonds, respectively.\nClass edges are dashed, while triple and alignments are in solid lines.\n\n\\begin{figure} [t]\n\t\\centering\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/KG_TEXT_XG_Time.pdf}\n\t\t\\caption{XG construction time.}\n\t\t\\label{fig:xg-time}\n\t\\end{subfigure}\n\t\\hspace{0.1cm}\n\t~\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/KG_TEXT_GST_Time.pdf}\n\t\t\\caption{GST computation time.}\n\t\t\\label{fig:gst-time}\n\t\\end{subfigure}\n\t~\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/KG_TEXT_TOTAL_Time.pdf}\n\t\t\\caption{Total time.}\n\t\t\\label{fig:tot-time}\n\t\\end{subfigure}\n\t\\hspace{0.1cm}\n\t\\caption{(a) XG contruction time, (b) GST computation time, and (c) Total time,\n\t\ttaken by \\textsc{Uniqorn}\\xspace on the LC-QuAD 2.0 test set.}\n\t\\label{fig:time} \n\t\\vspace*{-0.5cm}\n\\end{figure}\n\n\\textbf{Runtime analysis.} To conclude our detailed introspection into\nthe inner workings of \\textsc{Uniqorn}\\xspace, we provide a distribution of runtimes.\nThere are two main components that contribute to the total runtime:\nthe construction of the context graph XG, and the computation of the GST\nover the XG. We find that the first step is usually the more expensive\nof the two, as it involves a large number of similarity computations.\nThe use of the fixed-parameter tractable exact algorithm for GSTs\nhelps achieve interactive response times for a large number of questions.\nAs expected, the efficiency for the KG-only setups is much better than\nthe noisier text and heterogeneous setups, as the graphs are much\nbigger in the latter cases. Nevertheless, we find that our runtimes\ncan get quite large for several questions, and this represents the most\npromising direction for future effort for us.\n\n\\myparagraph{Implementation details} All code is in \\texttt{Python}, making use of the popular PyTorch library\\footnote{\\url{https:\/\/pytorch.org}}.\nWhenever a neural model was used, code was run on a GPU (single GPU, NVIDIA Quadro RTX 8000, 48 GB GDDR6).\n\n\n\\section{Conclusions and Future Work}\n\\label{sec:confut}\n\nWe show that \\textsc{Uniqorn}\\xspace\ncan reach performance comparable to state-of-the-art baselines\non benchmarks of complex questions.\n\\textsc{Uniqorn}\\xspace is geared to work over RDF KGs,\ntext corpora, or combinations.\nThrough our proposal of computing Group Steiner Trees on dynamically\nconstructed context graphs, we aim to unify\nthe fragmented paradigms of KG-QA and text-QA.\nEven for canonicalized KGs as input, sole reliance\non accurate SPARQL queries is inferior when it \ncomes to answering complex questions with multiple entities or predicates.\nWe show that relaxing crisp\nSPARQL-style querying to an approximate graph pattern search is vital\nfor question answering over KGs,\nand in the process identify the bridge to text-QA.\nFinally, \\textsc{Uniqorn}\\xspace has the unique design rationale of deliberately allowing noise\nfrom heuristic choices \n(like coreference resolution, triple extraction, and alignment insertion)\ninto early stages of\nthe answering pipeline and coping with it later.\nThis makes it a practical choice for \nproviding direct answers in Web-based QA, with Steiner Trees\nadding the bonus of interpretable insights into answer derivation.\n\n\\section{Concepts and Notation}\n\\label{sec:concepts}\n\nWe now introduce salient concepts necessary for an accurate understanding of \\textsc{Uniqorn}\\xspace.\nA glossary of concepts and notation is provided in Table~\\ref{tab:notation}.\n\n\\subsection{General Concepts}\n\\label{subsec:gen-con}\n\n\\myparagraph{Knowledge graph} An RDF knowledge graph $K$, like Wikidata,\nconsists of \nentities $E$ (like \\phrase{Leonardo DiCaprio}), predicates $P$\n(like \\phrase{award received}), types $\\mathds{T}$ (like \\phrase{film}), and \nliterals $L$ (like \\phrase{07 January 2016}), organized as a set of\nsubject-predicate-object (SPO) triples $\\{T^K\\}$ where $S \\in E$ and\n$O \\; \\in \\; E \\; \\cup \\; \\mathds{T} \\; \\cup \\; L$. Optionally, a triple $T_i^K$ may be accompanied\nby one or more qualifiers as <qualifier-predicate, qualifier-object> tuples that provide additional context.\nThe following is an example of a triple with\n\\textit{qualifiers}\\footnote{\\url{https:\/\/www.wikidata.org\/wiki\/Help:Qualifiers}}:\n\\struct{<LeonardoDiCaprio, awardReceived, AcademyAward;\nforWork, TheRevenant; pointInTime, 2016>}. Each $T_i^K$ represents a fact in $K$.\nUnder the graph model used in this work, each entity, predicate, type and literal becomes a node in the graph.\nEdges connect different components of a fact.\nThey run from the subject to the predicate and on to the object of a fact.\nFor facts with qualifiers, there are additional edges from the main predicate to the qualifier predicate and on to the qualifier object.\nA common alternative model for KGs represents predicates as edge labels, but this makes it difficult to incorporate qualifiers (a more realistic setting).\n\n\\myparagraph{Text corpus} A text corpus $D$ is a collection of\ndocuments, where each document $D_i$ contains a set of natural \nlanguage snippets $\\mathcal{S}$. Each snippet is defined as a span of text in $D_i$ that contains at least two tokens from the question, within a specified context window.\nSuch documents could come from a static collection\nlike ClueWeb12, Common Crawl,\nWikipedia articles, or the open Web. To induce structure on $D$,\nOpen IE~\\cite{del2013clausie,angeli2015leveraging,mausam2016open}\nis performed on each\n$\\mathcal{S}_j \\in D_i$, for every $D_i \\in D$, to return a set of triples\n$\\{T_k^{D_i}\\}$\nin SPO format, where each such triple represents a fact (evidence) mined from some $D_i$.\nThese triples are augmented with those built from Hearst\npatterns~\\cite{hearst1992automatic} run over $D$ that indicate entity-type\nmemberships.\nThis non-canonicalized (open vocabulary) triple store is referred to as\na quasi knowledge graph $K^D$ built from the document corpus, where\n$K^D = \\{T^D\\} = \\bigcup_i T^{D_i}$.\nThus, each $T^D_i$ is a fact in $K^D$.\n\n\\myparagraph{Evidence} We refer to a fact in a KB or a snippet from text by the unifying term ``evidence''. \n\n\\myparagraph{Heterogeneous source} We refer to the mixture of the knowledge\ngraph $K$ and the quasi-KG $K^D$ as a heterogeneous source $K^H = K \\cup K^D$,\nwhere each triple in this heterogeneous KG can come from either \nthe RDF store or the text corpus.\n\n\\myparagraph{Question} A question $q  = \\; \\langle q_1 q_2 \\ldots q_{|q|} \\rangle$ is posed \neither as a full-fledged interrogative sentence\n(\\utterance{Who is the director of the western film for which Leonardo\nDiCaprio won an Oscar?}) or in \ntelegraphic~\\cite{joshi2014knowledge,sawant2019neural}\n\/ utterance~\\cite{abujabal2018never,yih2015semantic} form\n(\\utterance{director of western with Oscar for Leo}), where\nthe $q_i$'s are the tokens in the question,\nand $|q|$ is the total number of tokens in $q$.\nQuestion tokens are either words, or phrases like entity mentions detected by a named entity recognition (NER) system~\\cite{qi2020stanza}.\nStopwords are not considered as question tokens.\n\n\\myparagraph{Complex question} \\textsc{Uniqorn}\\xspace is motivated\nby the need for a unified approach to\ncomplex questions~\\cite{sun2019pullnet,lu2019answering,vakulenko2019message,dubey2019lc,talmor2018web}, as\nsimple questions are already well-addressed\nby prior works~\\cite{petrochuk2018simplequestions,abujabal2018never,bast2015more,yih2015semantic}.\nWe call a question ``simple'' if\nit translates into a query or a logical form\nwith a single entity and a single predicate\n(like \\utterance{capital of Greece?}).\nQuestions where the simplest proper query requires\nmultiple\n$E$ or $P$, are considered ``complex''\n(like \\utterance{director of the western for which Leo won an Oscar?}).\nThere are other notions of complex\nquestions~\\cite{hoffner2017survey}, \nlike those requiring\ngrouping and aggregation,\n(e.g., \\utterance{which director made the highest \nnumber of movies that won an Oscar in any category?}), \nor when questions involve negations\n(e.g., \\utterance{which director has won multiple Oscars but never a Golden Globe?}). \nThese are not considered in this paper.\n\n\\myparagraph{Answer} An answer $a \\in A$ to question $q$ is an\nentity $e \\in E$ or a literal $l \\in L$ in the KG $K$ (like\nthe entity \\struct{The Revenant}\nin Wikidata), or a span of text (a sequence of words)\nfrom some sentence $S_j \\in D_i$ in the corpus $D$\n(like \\phrase{The Revenant film}). $A$ is the set of all correct answers\nto $q$ ($|A| \\geq 1$).\n\n\\begin{table} [t]\n\t\\centering\n\n\t\t\\setlength{\\tabcolsep}{0.5em}\n\t\t\\begin{tabular}{l l l l}\n\t\t\t\\toprule\n\t\t\t\\textbf{Notation}\t\t\t& \\textbf{Concept}\t\t\t\t& \\textbf{Notation}\t\t\t\t\t\t\t\t\t& \t\\textbf{Concept}\t\t\t\t\t\t\t            \\\\ \\toprule\n\t\t\t$K$\t\t\t\t\t\t\t& Knowledge graph (KG)\t\t\t\t& $\\mathcal{S}$\t\t\t\t\t\t\t\t\t\t&  \tSnippet\t\t\t\t\t\t\t\t\t            \\\\ \n\t\t\t$D$\t\t\t\t\t\t\t& Document corpus\t\t\t\t\t& $q = \\langle q_i\\rangle$\t\t\t\t\t\t\t& \tQuestion, question words\t\t\t\t\t        \\\\ \n\t\t\t$K^D$\t\t\t\t\t\t& Quasi KG from $D$\t\t\t\t\t& $A = \\{a\\}$ \t\t\t\t\t\t\t\t\t\t& \tAnswer to $q$\t\t\t\t\t\t\t\t        \\\\ \n\t\t\t$K^H$\t\t\t\t\t\t& Heterogeneous $KG$\t\t\t\t& $XG^{(\\cdot)}(q)$ \t\t\t\t\t\t\t\t& \tContext graph for $q$ built from $K$\/$K^D$\/$K^H$    \\\\ \n\t\t\t$E = \\{e\\}$\t\t\t\t\t& Entity\t\t\t\t\t\t\t& $N = \\{n\\}$\t\t\t\t\t\t\t\t\t\t& \tNode \t\t\t\t\t\t\t\t\t\t        \\\\\n\t\t\t$P = \\{p\\}$ \t\t\t\t& Predicate \t\t\t\t\t\t& $\\mathcal{E} = \\{\\epsilon\\}$\t\t\t\t\t\t& \tEdge\t\t\t\t\t\t\t\t\t\t        \\\\ \n\t\t\t$\\mathds{T} = \\{t\\}$\t\t& Type\t\t\t\t                & $N^{\\mathcal{T}}, \\mathcal{E}^\\mathcal{T}$\t\t& \tNode types, edge types\t\t\t\t\t\t        \\\\\n\t\t\t$L = \\{l\\}$\t\t\t\t\t& Literal\t\t\t\t\t\t\t& $N^{W}, \\mathcal{E}^W$\t\t\t\t\t\t\t& \tEdge weights, edge weights\t\t\t\t\t        \\\\\n\t\t\t$S$\t\t\t\t\t\t\t& Subject\t\t\t\t\t\t\t& $M_{\\mathcal{T}}^N(\\cdot), M^\\mathcal{E}(\\cdot)$\t& \tMapping function from node\/edge to type\t\t\t    \\\\ \n\t\t\t$O$\t\t\t\t\t\t\t& Object \t\t\t\t\t\t\t& $M_W^N(\\cdot), M_W^\\mathcal{E}(\\cdot)$ \t\t\t& \tMapping function from node\/edge to weight\t\t\t\\\\ \n\t\t\t$T^{(\\cdot)}$\t\t\t\t& Triple in $K$\/$K^D$\/$K^H$\t\t\t& $\\tau_{align}^{(\\cdot)}$\t\t\t\t\t\t\t& \tAlignment threshold for $E\/P$\t\t\t\t        \\\\\n\t\t\t$m$\t\t\t\t\t\t\t& Entity mention\t\t\t\t\t& $\\mathbb{T}_i$\t\t\t\t\t\t            & \tAny tree in $XG^{(\\cdot)}(q)$ connecting anchors\t\\\\ \n\t\t\t$\\mathcal{A}$\t\t\t\t& Anchor node\t\t\t\t\t\t& $\\mathbb{T}$*\t\t\t\t\t\t\t\t\t\t& \tGST\t\t\t\t\t\t\t\t\t\t            \\\\ \\bottomrule\n\t\\end{tabular}\n\t\\caption{Concepts and notation.}\n\t\\label{tab:notation}\n\t\\vspace*{-0.7cm}\n\\end{table}\n\n\\subsection{Graph Concepts}\n\\label{subsec:graph-con}\n\n\\myparagraph{Context graph} A context graph $XG$ for a question $q$ is defined\nas a subgraph of the full \/ quasi \/ heterogeneous knowledge graph, i.e.,\n$XG^K(q) \\subset K$ (KG)\nor $XG^D(q) \\subset K^D$ (text)\nor $XG^H(q) \\subset K^H$ (mixture), such that it contains all\ntriples $T^K_i(q)$ or $T^D_i(q)$ potentially relevant for answering $q$.\nThus, $XG(q)$ is expected to contain every answer entity $a \\in A$.\nAn $XG$ has nodes $N$ and edges $\\mathcal{E}$ with types\n($N^\\mathcal{T}, \\mathcal{E}^\\mathcal{T}$) and weights ($N^W, \\mathcal{E}^W$) as \ndiscussed below. Thus, an $XG$ is always \\textit{question-specific}, and to\nsimplify notation, we often write only $XG$ instead of $XG(q)$.\nFig.~\\ref{fig:xg} shows possible context graphs for our running example question in each of the three setups.\n\n\\myparagraph{Node types} A node $n \\in N$ in an $XG$ is mapped to one of\nfour categories $N^\\mathcal{T}$:\n(i) entity, (ii) predicate, (iii) type, or (iv) literal, via a \nmapping function $M_{\\mathcal{T}}^N(\\cdot)$, where\n$n \\in E \\cup P \\cup \\mathds{T} \\cup L$. Each $n_j$ is produced from\nan $S$, $P$, or $O$, from the triples in $K$ or $K^D$.\nFor\nKG facts,\nwe make no distinction between predicates and qualifier\npredicates, or objects and qualifier objects (no qualifiers\nin\ntext).\n\nLet us use Fig.~\\ref{fig:xg} for reference.\nEntities and literals are shown in rectangles with sharp corners,\nwhile predicates and types are in rectangles with rounded corners.\nEven though it is standard practice to treat predicates as\nedge labels in a KG, we model them as nodes, because this design \nsimplifies the \napplication of our graph algorithms.\n\nNote that predicates originating from different triples are assigned\nunique identifiers in the graph. For instance, for triples\n$T_1 = $ \\struct{<BarackObama, married, MichelleObama>},\nand $T_2 = $ \\struct{<BillGates, married, MelindaGates>}, we will obtain\ntwo \\struct{married} nodes,\nthat will be marked as \\struct{married-1} and \\struct{married-2} in\nthe context graph.\nSuch distinctions prevent false inferences when answering over some XG.\nFor simplicity, we do not show such predicate indexing in Fig.~\\ref{fig:xg}.\n\nFor text-based $XG^D$, we make no\ndistinction between $E$ and $L$, as Open IE markups that produce these\n$T^D$ often lack such ``literal'' annotations. Type nodes $\\mathds{T}$\nin $XG^K$ come from the objects of \\struct{instanceOf} and \\struct{occupation} predicates in $K$\n(e.g., for Wikidata),\nwhile those in $XG^D$ originate from the outputs of Hearst \npatterns~\\cite{hearst1992automatic} in $D$. \nIn Fig.~\\ref{fig:xg-kg}, nodes \\struct{TheRevenant},\n\\struct{director}, \\struct{AcademyAwards}, and \\struct{2016} are\nof type $E$, $P$, $\\mathds{T}$, and $L$, respectively.\nIn Fig.~\\ref{fig:xg-text}, nodes \\phrase{The Revenant},\n\\phrase{directed by}, and \\phrase{2015 American western film}, are\nof type $E$, $P$, and $\\mathds{T}$, respectively.\n\n\\myparagraph{Edge types} An edge $\\epsilon \\in \\mathcal{E}$ in an $XG$ is\nmapped to\none of three categories $\\mathcal{E}^\\mathcal{T}$:\n(i) triple, (ii) type, or (iii) alignment edges, via\na mapping function $M_{\\mathcal{T}}^\\mathcal{E}(\\cdot)$. \nTriples in $\\{T^K\\}$ or $\\{T^D\\}$ or $\\{T^H\\}$, where $M^N(O) = \\{E \\cup L\\}$,\ni.e., the object is of node type entity or literal, contribute triple edges\nto the $XG$. For example, in Fig.~\\ref{fig:xg-kg},\nthe two edges between \\phrase{The Revenant}\nand \\phrase{genre} and between \\phrase{genre} and \\phrase{Western film} are\ntriple edges.\nIn Fig.~\\ref{fig:xg-text},\nthe two edges between \\phrase{Alejandro}\nand \\phrase{director of}, and between \\phrase{director of} and \\phrase{Survival drama Revenant} are\ntriple edges.\n\nTriples where $M_{\\mathcal{T}}^N(O) = \\{\\mathds{T}\\}$ \n(object is a type) are\nreferred to as type triples, and each type triple contributes two\ntype edges to the $XG$. Examples are: edges between \\struct{AcademyAwardForBestDirector} and \\struct{instanceOf}, and \\struct{instanceOf} and \n\\struct{AcademyAwards} in Fig.~\\ref{fig:xg-kg}; \nand \nedges between \\phrase{The Revenant} and \\phrase{type}, and \\phrase{type} and \n\\phrase{2015 American western film} in Fig.~\\ref{fig:xg-text}.\n\nAlignment edges \nrepresent potential synonymy between nodes, and run \\textit{only} between nodes of the same type.\nAlignments are inserted in $XG^D$ or $XG^H$ via\nexternal sources of similarity like aliases or word embeddings.\nThere are no alignment edges in $XG^K$ (or more generally, in $K$) as all items in a KG are canonicalized.\nExamples of alignment edges are\nthe bidirectional dotted edges\nbetween\n\\phrase{The Revenant} and \\phrase{Survival drama Revenant} in \nFig.~\\ref{fig:xg-text},\nand \\struct{director} and \\phrase{directed by} in Fig.~\\ref{fig:xg-hetero}.\nInsertion of alignment edges as opposed to\na naive merging of synonymous nodes is a deliberate choice. This enables\nmore matches with question tokens and a subsequent disambiguation of question concepts. It also precludes\nthe problem of choosing a representative label for merged clusters, and avoids\ntopic drifts arising from the transitive effects of merging nodes at this stage.\n\n\\myparagraph{Node weights} A node $n \\in N$ in an $XG$ is weighted by\na function $M_W^N(\\cdot) \\in [0, 1]$ according to its similarity\nto the question.\nThis is obtained averaging BERT scores of the evidences of which $n$ is a part of.\nNode weights are used as a criterion for answer ranking.\n\n\\myparagraph{Edge weights} Each edge $\\epsilon = (n_i, n_j)$ in an $XG$ is assigned a weight by a function $M_W^\\mathcal{E}(\\cdot) \\in [0, 1]$.\nEdge weights are later converted to edge costs that are vital for computing Group Steiner Trees on the XG.\nIn \\textsc{Uniqorn}\\xspace, edge weights are also the BERT scores of evidences (KB facts or text snippets) from where the edge originates.\nEdge weights are averaged if the same edge is found in multiple evidences. This holds for triple and type edges.\n\nWeights of alignment edges come from similarities between nodes in the XG. For entities, this is computed as the Jaccard overlap of character-level trigrams~\\cite{yih2015semantic} between the node labels that are the endpoints of the edge (for KBs, entity aliases available as part of the KB are appended to entity names before computing the similarity). Character-level $n$-grams make the matching robust to spelling errors. Lexical matching is preferred for entities and literals as we are more interested in \\textit{hard equivalence} rather than a soft relatedness. For predicates and types, lexical matches are not enough and semantic similarity computations are necessary.\nSo alignment scores are computed as pairwise embedding similarities (cosine values) over words in the two node labels (one word from each node), followed by a maximum over these pairs. This is then min-max normalized to $[0, 1]$.\nWikipedia2vec~\\cite{yamada2020wikipedia2vec}, that taps into both corpus statistics and link structure in Wikipedia, was used for computing embeddings.\n\nAn alignment\nedge is inserted into an $XG^D$ or $XG^H$ if the similarity exceeds or equals\nsome threshold $\\tau_{align}$, i.e.,\n$\\textrm{sim}(\\textrm{label}(n_i), \\textrm{label}(n_j)) \\geq \\tau_{align} \\in \n(0,1]$. Zero is not an acceptable value for $\\tau_{align}$ as that would mean\ninserting an edge\nbetween every pair of possibly unrelated nodes. This alignment insertion threshold\n$\\tau_{align}$ could be \npotentially different for entities ($\\tau_{align}^E$) and\npredicates ($\\tau_{align}^P$), due to the use of different similarity functions.\nThese (and other) hyper-parameters are tuned on a \ndevelopment set of question-answer pairs.\n\n\\myparagraph{Anchor nodes} A node $n$ in an $XG$ is an anchor node\n$\\mathcal{A}$ if\nit matches one of the question tokens. Such matches may either be lexical (\\phrase{western} $\\mapsto$ \\struct{2015 American western film} in Fig.~\\ref{fig:xg-text}), or more sophisticated mapping of entity mentions $m$ in questions to KG entities $\\{e\\}$ via named entity recognition and disambiguation (NERD) systems~\\cite{li2020efficient,hoffart2011robust,ferragina2010tagme}\n(\\phrase{Leo} $\\mapsto$ \\struct{Leonardo DiCaprio} in Fig.~\\ref{fig:xg-kg}).\nAnchors are grouped into sets,\nwhere a set\n$\\mathcal{A}^k = \\{\\mathcal{A}^k_1, \\mathcal{A}^k_2, \\ldots \\}$,\ndepending upon which question token $q_k$ the elements of the set match.\nIn other words, more than one XG node can match the same question token, and hence the need for the superscript $k$: the anchor nodes corresponding to $q_2$ would be denoted by $\\{\\mathcal{A}^2_1, \\mathcal{A}^2_2, \\ldots\\}$.\nFor example, \\phrase{director} in the question matches nodes \\phrase{director of}, \\phrase{directed by}, \\phrase{Best Director}, $\\ldots$ in Fig.~\\ref{fig:xg-text}.\nAnchors thus identify the question-relevant nodes of the $XG$,\nin the vicinity of which answers can be expected to be found.\nAny node of category entity, predicate, type, or literal, can qualify\nas an anchor.\n\n\\section{Experimental Setup}\n\\label{sec:exp-setup}\n\n\\subsection{Benchmarks}\n\\label{subsec:data}\n\n\\subsubsection{Knowledge sources}\n\nAs our \\textbf{knowledge graph} we use the NTriples dump of the full Wikidata\\footnote{\\url{https:\/\/dumps.wikimedia.org\/wikidatawiki\/entities\/}} as of 20 April 2021,\nincluding all qualifiers.\nThe original dump consumed about $2$ TB of disk space, and contained about $12$B triples.\nWe use a cleaned version\\footnote{\\url{https:\/\/github.com\/PhilippChr\/wikidata-core-for-QA}}\nthat prunes language tags, external identifiers, additional schema labels and so on (KB cleaning steps in~\\cite{christmann2022beyond}).\nThis left us with about $2$B triples with $40$ GB disk space.\nThe cleaned KB is accessed via the recently proposed \\textsc{Clocq} API\\footnote{https:\/\/clocq.mpi-inf.mpg.de\/}.\nNote that there is nothing in our method specific to Wikidata, and can be easily\nextended to other KGs like YAGO~\\cite{suchanek2007yago} or DBpedia~\\cite{auer2007dbpedia}. Wikidata was chosen as it is one of the popular choices\nin the community today, and\nhas an active community that contributes to growth of the KG, akin to Wikipedia\n(both are supported by the Wikimedia foundation).\nFurther, in 2016, Google ported a large volume of the erstwhile Freebase into\nWikidata~\\cite{pellissier2016freebase}. \n\nFor \\textbf{text}, we create a pool of\n$10$ Web pages per question, obtained from Google Web search\nin June 2021.\nSpecifically, we issue the full question as a query to Google Web search,\nand create a question-specific corpus from these top-$10$ results obtained.\nThis was done using the Web search option inside\nthe Google Custom Search API\\footnote{\\url{https:\/\/developers.google.com\/custom-search\/v1\/overview}}.\nThis design choice of fetching pages from the Web was made to be close\nto the direct-answering-over-the-Web-setting, and not be restricted to specific\nchoices of benchmarks that have associated corpora. This also enables comparing\nbaselines from different QA families on a fair testbed. This was done despite the fact\nthat it entails significant resources, as this has to be done for thousands of questions,\nand \\textit{in addition}, these Web documents have to be entity-linked to Wikidata\nto enable training some of the supervised baselines.\n\nThe \\textbf{heterogeneous} answering setup was created by considering the above two sources together.\nTo be specific, each question from our benchmark is answered over the entire KG and\nthe corresponding text corpus.\n\n\\textit{All baselines were exposed to the same knowledge sources as \\textsc{Uniqorn}\\xspace},\nin all the three setups -- KG, text, or heterogeneous inputs.\n\n\\begin{table} [t] \n\t\\centering\n\t\\resizebox*{\\textwidth}{!}{\n\t\t\\setlength{\\tabcolsep}{0.5em}\n\t\t\\begin{tabular}{l r l}\n\t\t\t\\toprule\n\t\t\t\\textbf{Benchmark}\t\t\t\t\t\t\t\t\t\t\t\t\t\t& \\textbf{\\#Questions}\t& \\textbf{Answers}\t                                    \\\\ \\toprule\n\t\t\tComplex questions from LC-QuAD $2.0$~\\cite{dubey2019lc}\t\t\t\t\t& $4921$\t\t\t\t& Wikidata entities\t\t\t\t\t\t\t\t        \\\\ \\midrule\n\t\t\tComplex questions from LC-QuAD $1.0$~\\cite{trivedi2017lc}\t\t\t\t& $1459$\t\t\t\t& DBpedia entities, mapped to Wikidata via Wikipedia\t\\\\\n\t\t\tComplex questions from WikiAnswers (CQ-W)~\\cite{abujabal2017automated}\t& $150$\t\t\t\t\t& Freebase entities, mapped to Wikidata via Wikipedia \t\\\\\n\t\t\tComplex questions from Google Trends (CQ-T)~\\cite{lu2019answering}\t\t& $150$\t\t\t\t\t& Entity mention text, manually mapped to Wikidata      \\\\\n\t\t\tComplex questions from QALD $1-9$~\\cite{ngomo20189th}\t\t\t\t\t& $70$\t\t\t\t\t& DBpedia entities, mapped to Wikidata via Wikipedia \t\\\\\n\t\t\tComplex questions from ComQA~\\cite{abujabal2019comqa}\t\t\t\t\t& $202$ \t\t\t\t& Wikipedia URLs, mapped to Wikidata entities\t\t\t\\\\ \\midrule\n\t\t\tTotal number of complex questions\t\t\t\t\t\t\t\t\t\t& $6952$\t\t\t\t& Wikidata entities\t\t\t\t\t\t\t            \\\\ \\bottomrule\n\t\\end{tabular}}\n\t\\caption{Test sets with sampled questions from each benchmark, totaling about $7k$ complex questions.}\n\t\\label{tab:data}\n\t\\vspace*{-0.7cm}\n\\end{table}\n\n\\begin{table} [t] \n\t\\centering\n\n\t\t\\setlength{\\tabcolsep}{0.5em}\n\t\t\\begin{tabular}{l p{10.5cm}}\n\t\t\t\\toprule\n\t\t\t\\textbf{Benchmark}\t\t\t\t\t\t\t\t&   \\textbf{Complex Questions}\t                                                \\\\ \\toprule\n\t\t\tLC-QuAD $2.0$~\\cite{dubey2019lc}\t\t\t\t&   \\utterance{Which of Danny Elfman's works was nominated for an Academy Award for Best Original Score?}       \\\\\t                                                \\cmidrule{2-2} \n    \t\t\t\t\t\t\t                            &   \\utterance{When did Glen Campbell receive a Grammy Hall of Fame award?}                                        \\\\ \\midrule\n\t\t\tLC-QuAD $1.0$~\\cite{trivedi2017lc}\t\t    \t&   \\utterance{Which award that has been given to James F Obrien, had used Laemmle Theatres for some service?}                                                     \\\\ \\cmidrule{2-2}\n\t\t\t                                                &   \\utterance{Which home stadium of 2011-12 FC Spartak Moscow season is also the location of birth of the Svetlana Gounkina?}\n\t\t\t                                                    \\\\ \\midrule\n\t\t\tCQ-W~\\cite{abujabal2017automated}\t            & \\utterance{Which actor is married to Kaaren Verne and played in Casablanca?} \t        \\\\ \\cmidrule{2-2}\n        \t\t\t\t                                    & \\utterance{Who graduated from Duke University and was the president of US?} \t        \\\\ \\midrule\n\t\t\tCQ-T~\\cite{lu2019answering}\t\t\t\t\t\t& \\utterance{In which event did Taylor Swift and Joe Alwyn appear together?}            \\\\ \\cmidrule{2-2}\n\t\t\t\t\t\t\t\t\t                        & \\utterance{Who played for Barcelona and managed Real Madrid?}                         \\\\ \\midrule\n\t\t\tQALD $1-9$~\\cite{ngomo20189th}\t\t\t\t\t& \\utterance{Which street basketball player was diagnosed with Sarcoidosis?} \t        \\\\ \\cmidrule{2-2}\n\t\t\t\t\t\t\t\t                            & \\utterance{Which recipients of the Victoria Cross fought in the Battle of Arnhem?}    \\\\ \\midrule\n\t\t\tComQA~\\cite{abujabal2019comqa}\t\t\t\t\t& \\utterance{What is the river that borders Mexico and Texas?}\t\t\t                \\\\ \\cmidrule{2-2}\n\t\t\t\t\t\t\t\t                            & \\utterance{Who Killed Dr Martin Luther King Jr's mother?}\t\t\t                    \\\\ \\midrule\n\t\\end{tabular\n\t\\caption{Examples of complex questions from each benchmark.}\n\t\\label{tab:data-eg}\n\t\\vspace*{-1cm}\n\\end{table}\n\n\\subsubsection{Question-answer pairs} \n\nThere is no QA benchmark that is directly suitable for answering complex questions on heterogeneous sources.\nAs a result, we choose six relatively recent benchmarks (Table~\\ref{tab:data}) with realistic questions, proposed for KG-QA, and curate text corpora for questions in these benchmarks\\footnote{Note that the opposite strategy of adapting text-QA benchmarks for KG and heterogeneous QA, is not a practical option. This is because answers in mainstream text-QA may be arbitrary spans of text from given passages, and may not map to crisp entities -- and hence would be out of scope of factoid QA, the focus of this work.}\nOut of these, the most recent one, LC-QuAD $2.0$, served as the main benchmark (owing to its relatively larger size), and the rest were used to test the generalizability of QA systems in open-domain experiments (running pre-trained models on unseen questions).\nBaselines that require supervision were trained on the LC-QuAD 2.0 train set ($18$k questions), and hyperparameter tuning for all methods were done on a random sample of $1000$ questions from the LC-QuAD 2.0 development set ($6$k questions).\n\\textsc{Uniqorn}\\xspace only requires tuning of the hyperparameters inside the BERT model and the alignment thresholds, and for this, used only the $1000$-question sample above.\n\nTo ensure that the benchmark questions pose \nsufficient difficulty to the systems under test, all questions from individual sources\nwere manually examined to ensure that they have at least entity and more than two relations.\nQuestions that do not have a ground truth answer in our our KG Wikidata, and aggregations and existential \nquestions (i.e., questions with counts and yes\/no answers, respectively) were also removed.\nThe number of questions finally contributed by each source is shown in Table~\\ref{tab:data}.\nThese are factoid questions: each question usually has one or a few entities\nas correct answers ($72\\%$ of the total number of questions across all benchmarks have exactly one ground truth answer, and most of the rest have $2-3$ gold answers).\nDetails of specific benchmarks are provided below. Some examples of complex questions from these benchmarks are shown in Table~\\ref{tab:data-eg}.\nAll datasets and code for this project are available at \\url{qa.mpi-inf.mpg.de\/uniqorn}.\n\n(i) \\textbf{LC-QuAD 2.0}~\\cite{dubey2019lc}: This very recent and large QA benchmark from October 2019\nwas compiled using crowdsourcing\nwith human annotators verbalizing KG templates that were shown to them. Answers are Wikidata\/DBpedia SPARQL queries that can be\nexecuted over the corresponding KGs to obtain entities\/literals as answers. \n\n(ii) \\textbf{LC-QuAD 1.0}~\\cite{trivedi2017lc}: This dataset was curated by the same team\nas LC-QuAD 2.0, by a similar process, but where the crowdworkers directly corrected\nan automatically generated natural language question. Answers are DBpedia entities --\nwe linked them to Wikidata for our use via their Wikipedia URLs, that act as\nbridges between popular curated KGs like Wikidata, DBpedia, Freebase, and YAGO.\n\n(iii) \\textbf{CQ-W}~\\cite{abujabal2017automated}: The questions (complex questions from \nWikiAnswers, henceforth CQ-W) were sampled from\nWikiAnswers with multiple entities or relations. Answers are Freebase entities.\nWe mapped them to Wikidata using Wikipedia URLs.\n\n(iv) \\textbf{CQ-T}~\\cite{lu2019answering}: This contains complex questions about emerging\nentities\n(e.g., people dated by celebrities) created from queries in Google Trends\n(complex questions from Google Trends, henceforth CQ-T).\nAnswers are text mentions of entities, that we manually map to Wikidata.\n\n(v) \\textbf{QALD}~\\cite{ngomo20189th}: We collated these questions by going through\nnine years of QA datasets\nfrom the QALD\\footnote{\\url{http:\/\/qald.aksw.org\/}} benchmarking campaigns\n(editions $1-9$).\nAnswers are DBpedia entities, that we map to Wikidata using Wikipedia links.\n\n(vi) \\textbf{ComQA}~\\cite{abujabal2019comqa}: These factoid questions, like~\\cite{abujabal2017automated},\nwere also sampled from the WikiAnswers community QA corpus. Answers are Wikipedia URLs\nor free text, which we map to Wikidata.\n\n\\subsection{Systems under test}\n\\label{subsec:base}\n\n\\begin{table} [t] \n\t\\centering\n\t\\setlength{\\tabcolsep}{0.25em}\n\n\t\\begin{tabular}{l ccc c ccc}\n\t\t\\toprule\n\t\t\\multirow{2}{*}{\\textbf{Setup}} & \\multicolumn{3}{c}{\\textbf{\\#Nodes}}\t&&  \\multicolumn{3}{c}{\\textbf{\\#Edges}} \t\t\\\\\n\t\t\\cmidrule{2-4} \t\t\\cmidrule{6-8}\t\t\t\n\t\t                    & \\textbf{Entity} \t& \\textbf{Predicate}\t& \\textbf{Type} \t&& \\textbf{Triple} \t& \\textbf{Alignment} \t& \\textbf{Type} \\\\ \\midrule\n\t\t\\textbf{KG+Text} \t& $215.2$\t\t\t& $255.6$\t\t\t\t& $12.4$\t\t    && $508.6$\t        & $6151.9$\t            & $2.5$        \\\\ \n\t\t\n\t\t\\textbf{KG} \t\t& $10.3$\t\t\t& $27.6$\t\t\t\t& $11.5$\t\t\t&& $38.7$\t        & N. A.\t\t            & $16.4$\t    \\\\\n\t\t\\textbf{Text} \t\t& $147.9$\t\t\t& $182.3$\t\t\t\t& $3.6$\t\t        && $362.6$\t        & $3.9$\t            & $4.3$        \\\\ \\bottomrule\n\t\\end{tabular\n\t\\caption{Basic properties of \\textsc{Uniqorn}\\xspace's XGs, averaged over LC-QuAD $2.0$ test questions.\n\t\t\tParameters affect these sizes: best values in each setting were used in these measurements.\n\t\tThe graphs typically get bigger and denser as the alignment edge insertion thresholds are lowered.\n\t\tThe KG+Text may not be exact additions of the two settings due the computation of the largest connected component, and slight parameter variations.\n\t\tExpectedly, the heterogeneous setting has the largest number of nodes\n\t\tand edges.\n\t}\n\t\\label{tab:graph-size}\n\t\\vspace*{-0.7cm}\n\\end{table}\n\n\\subsubsection{\\textsc{Uniqorn}\\xspace implementation}\n\\label{subsubsec:init}\n\nAs mentioned in Sec.~\\ref{subsec:evi-retr}, we need NERD systems for the KG-QA pipeline.\nTo improve answer recall, we used two complementary NERD systems on questions -- one biased towards precision (\\textsc{Elq}\\xspace~\\cite{li2020efficient}, with default configuration), and one towards recall (\\textsc{TagMe}\\xspace~\\cite{ferragina2010tagme}, with zero cut-off threshold $\\rho$).\nBoth systems disambiguate to Wikipedia, and we map them to Wikidata using links available in the KG.\nAll baselines were also given the advantage of these same disambiguations.\nFor Text-QA, POS tagging and NER on questions and documents were done using spaCy\\footnote{\\url{https:\/\/spacy.io\/}}.\nFor simplicity, POS tags were mapped to the Google Universal Tagset~\\cite{petrov2012universal}.\nThese POS tags were used for running our open IE pattern extractors, and identifying noun phrases necessary for matching Hearst patterns.\nEntity alignment edges were added using Jaccard coefficients on KG items labels augmented with aliases, and predicate and type alignments using $100$-dimensional Wikipedia2vec~\\cite{yamada2020wikipedia2vec} embedding similarities (Sec~\\ref{subsec:graph-con}).\nBERT fine-tuning needs a train-dev split (the best model is then applied on the test set): the 1000-question LC-QuAD 2.0 development set was split in an $80:20$ ratio for this purpose.\nThe following hyperparameters were found to work best on the development set: a batch size of $50$, a learning rate of $3 \\times 10^{-5}$, and a gradient accumulation of $4$.\nThe maximum token length was $512$.\nHugging Face libraries for the BERT-base-cased model\\footnote{\\url{https:\/\/huggingface.co\/bert-base-cased}} was used, for the sequence pair classification task\\footnote{\\url{https:\/\/huggingface.co\/docs\/transformers\/model_doc\/bert\\#tfbertforsequenceclassification}}.\nThe top-$5$ evidences were returned from the BERT fine-tuning, i.e. $\\varepsilon = 5$.\nThe three hyperparameters for the GST step ($\\tau_{align}^E, \\tau_{align}^P$, and top-$g$ GSTs to consider)\nwere tuned on the $1000$-question development test using grid search in each of the three settings.\nWe obtained:\n(i) for KG+text: $\\tau_{align}^E = 0.8, \\tau_{align}^P = 0.7, g = 10$.\n(ii) for KG: $g = 10$ (no alignment necessary for KGs);\n(iii) for text: $\\tau_{align}^E = 0.5, \\tau_{align}^P = 0.9, g = 10$;\nEffects of these parameters are examined later in Sec.~\\ref{subsec:analysis}.\nUsing these configurations, context graphs in the KG, text, and heterogeneous settings, $XG^K$, $XG^D$ and $XG^H$,\nare constructed in the manner described in text Sec.~\\ref{subsec:xg-build}.\n\\textsc{Uniqorn}\\xspace's XGs in the different setups are characterized in Table~\\ref{tab:graph-size}.\n\n\\subsubsection{Baselines for the heterogeneous setup}\n\nWe use the state-of-the-art systems \\textsc{UniK-QA}\\xspace~\\cite{oguz2021unikqa}, \\textsc{PullNet}\\xspace~\\cite{sun2019pullnet}, and its predecessor \\textsc{GRAFT-Net}\\xspace~\\cite{sun2018open}, as baselines for the KG+text heterogeneous setup.\n\\textsc{UniK-QA}\\xspace is one of the first systems to explore answering over heterogeneous sources by verbalizing all inputs to text sequences (almost opposite to our approach of inducing structure on all sources), and is state-of-the-art in the prominent retrieve-and-read paradigm (to which all our text-QA methods belong, as well) for QA over heterogeneous sources.\n\n(i) \\myparagraph{\\textsc{UniK-QA}\\xspace~\\cite{oguz2021unikqa}} \\textsc{UniK-QA}\\xspace verbalizes all structured inputs (KGs, tables, lists) to sentence form using simple rules, and adds them to the text corputs. This is then queried through a neural retriever (dense passage retrieval, DPR~\\cite{karpukhin2020dense}), and the top-$100$ evidences are fed into a generative reader model (fusion-in-decoder, \\textsc{FiD}\\xspace~\\cite{izacard2021leveraging}). DPR was designed to work over Wikipedia: so we fine-tuned DPR for LC-QuAD 2.0 by using the subset of Wikipedia pages retrieved among the Google results containing a gold answer as positive instances, and randomly choosing a negative instance from the positive instance of a different question (see the original DPR paper~\\cite{karpukhin2020dense} for details). We also fine-tune \\textsc{FiD}\\xspace, originally pre-trained using T5, on LC-QuAD 2.0.\nSome parts of the \\textsc{UniK-QA}\\xspace code is available publicly (\\textsc{DPR}\\xspace, \\textsc{FiD}\\xspace): the rest was reimplemented.\n\n(i) \\myparagraph{\\textsc{PullNet}\\xspace~\\cite{sun2019pullnet}} \\textsc{PullNet}\\xspace uses an iterative process to construct a question-specific subgraph for complex questions, where in each step, a relational graph convolutional neural network (R-GCN) is used to find subgraph nodes that should be expanded (to support multihop conditions) using pull operations on the KG and corpus. After the expansion, another R-GCN is used to predict the answer from the expanded subgraph.\n\\textsc{PullNet}\\xspace code is not public: it was completely reimplemented.\n\n(ii) \\myparagraph{\\textsc{GRAFT-Net}\\xspace~\\cite{sun2018open}} \\textsc{GRAFT-Net}\\xspace (Graphs of Relations Among Facts and Text Networks) also uses R-GCNs specifically designed to operate over heterogeneous graphs of KG facts and entity-linked text sentences. \\textsc{GRAFT-Net}\\xspace uses LSTM-based updates for text nodes and directed propagation using Personalized PageRank (PPR) and tackles the heterogeneous setting with an early fusion philosophy.\n\\textsc{GRAFT-Net}\\xspace code is available for the deprecated KG Freebase, necessary (non-trivial) adaptations were made for answering over Wikidata.\n\\textsc{PullNet}\\xspace and \\textsc{GRAFT-Net}\\xspace need entity-linked text for learning their models, for which we used the same text corpora described earlier ($10$ documents per question from Google Web search) that were tagged with \\textsc{TagMe}\\xspace and \\textsc{Elq}\\xspace\nwith the same $\\rho$ threshold of zero (see Sec.~\\ref{subsubsec:init}).\n\n\\subsubsection{KG-QA baselines}\n\n\\textsc{UniK-QA}\\xspace, \\textsc{PullNet}\\xspace, and \\textsc{GRAFT-Net}\\xspace\ncan be \nrun in KG-only modes as well. So these naturally add to KG-QA baselines\nfor us.\nIn addition, we use the systems\n\\textsc{QAnswer}\\xspace~\\cite{diefenbach2019qanswer,diefenbach2020towards}\\footnote{\\url{https:\/\/qanswer-frontend.univ-st-etienne.fr\/}}\nand\n\\textsc{Platypus}\\xspace~\\cite{tanon2018demoing}\\footnote{\\url{https:\/\/askplatyp.us\/}}\nas baselines for KG-QA.\nAs of March 2021, to the best of our knowledge,\nthese are the only systems \nfor Wikidata with sustained online services and APIs,\nwith \\textsc{QAnswer}\\xspace having state-of-the-art performance.\nWe use public APIs for both systems.\n\n(i) \\myparagraph{\\textsc{QAnswer}\\xspace~\\cite{diefenbach2020towards}} This is an extremely efficient method that relies on an over-generation of SPARQL queries based on simple templates, that are subsequently ranked, and the best query is executed to fetch the answer. The queries are generated by a fast graph breadth-first search (BFS) algorithm relying on HDT indexing\\footnote{\\url{https:\/\/www.rdfhdt.org\/}}.\n\n(ii) \\myparagraph{\\textsc{Platypus}\\xspace~\\cite{tanon2018demoing}} This was designed as a QA system driven by natural language understanding, targeting \\textit{complex questions} using grammar rules and template-based techniques. Questions are translated not exactly to SPARQL, but to a custom logical representation inspired by dependency-based compositional semantics.\n\n\\subsubsection{Text-QA baselines}\n\nThe mainstream manifestation of text-QA in the literature today is in the form machine reading comprehension (MRC), where given a question and a collection of passages, the system derives an answer. There are many, many MRC systems today, along with open retrieval variants where the question-relevant passages are not given but need to be retrieved from a large repository. As for text-QA baselines in this work against which we compare \\textsc{Uniqorn}\\xspace, we focus on distantly supervised methods. These include neural QA models which are pre-trained on large question-answer collections, with additional input from word embeddings. These methods are well-trained for QA in general, but not biased towards specific benchmark collections. We are interested in robust behavior for ad hoc questions, to reflect the rapid evolution and unpredictability of topics in questions on the open Web. Hence this focus on distantly supervised methods. As specific choices, we adopt the well-known \\textsc{PathRetriever}\\xspace~\\cite{asai2020learning}, \\textsc{DocumentQA}\\xspace~\\cite{clark2018simple}, and \\textsc{DrQA}\\xspace~\\cite{chen2017reading} systems as representatives of robust open-source implementations, that can work seamlessly on unseen questions and passage collections. All methods can deal with multi-paragraph documents, and are deep learning-based systems with large-scale training via Wikipedia and reading-comprehension resources like SQuAD~\\cite{rajpurkar2016squad} and TriviaQA~\\cite{joshi2017triviaqa}, and we run pre-trained QA models on our test sets.\n\n(i) \\myparagraph{\\textsc{PathRetriever}\\xspace~\\cite{asai2020learning}} This method is specifically developed for \\textit{complex questions}. The approach used is supervised iterative graph traversal (akin to \\textsc{PullNet}\\xspace~\\cite{sun2019pullnet}) and uses a novel recurrent neural network (RNN) method to learn to sequentially retrieve relevant passages in reasoning paths for complex questions, by conditioning on the previously retrieved documents. It is a very robust method that makes effective use of data augmentation and smart negative sampling to boost its learning, and has a minimal set of hyperparameters. The standard BERT QA model is used as the reader. Code is available at \\url{https:\/\/github.com\/AkariAsai\/learning_to_retrieve_reasoning_paths}. The default configuration was used.\n\n(ii) \\myparagraph{\\textsc{DocumentQA}\\xspace~\\cite{clark2018simple}} The \\textsc{DocumentQA}\\xspace system adapted passage-level reading comprehension to the multi-document setting. It samples multiple paragraphs from documents during training, and uses a shared normalization training objective that encourages the model to produce globally correct output. The \\textsc{DocumentQA}\\xspace model uses a sophisticated neural pipeline consisting of multiple CNN and bidirectional GRU layers, coupled with self-attention. For \\textsc{DocumentQA}\\xspace, we had a number of pre-trained models to choose from, and we use the one trained on TriviaQA (\\texttt{TriviaQA-web-shared-norm}) as it produced the best results on our dev set. Default configuration was used for the remaining parameters\\footnote{\\url{https:\/\/github.com\/allenai\/document-qa}}.\n\n(iii) \\myparagraph{\\textsc{DrQA}\\xspace~\\cite{chen2017reading}} This system combines a search component based on bigram hashing and TF-IDF matching with a multi-layer RNN  model trained to detect answers in paragraphs. Since we do not have passages manually annotated with answer spans, we run the \\textsc{DrQA}\\xspace model pre-trained on SQuAD~\\cite{rajpurkar2016squad} on our test questions with passages from the ten documents as the source for answer extraction. Default configuration settings were used\\footnote{\\url{https:\/\/github.com\/facebookresearch\/DrQA}}.\n\n\\subsubsection{Graph baselines}\n\nWe compare our GST-based method to simpler graph algorithms based on bread-first search (BFS) and \\textsc{ShortestPaths}\\xspace as answering mechanisms:\n\n(i) \\myparagraph{\\textsc{BFS}\\xspace~\\cite{kasneci2009star}} In \\textsc{BFS}\\xspace for graph-based QA, iterators start from each anchor node in a round-robin manner, and whenever at least one iterator from each anchor group meets at some node, it is marked as a candidate answer. At the end of $1000$ iterations, answers are ranked by the number of iterators that met at the nodes concerned.\n\n(ii) \\myparagraph{\\textsc{ShortestPaths}\\xspace} As another intuitive graph baseline, shortest paths are computed between every pair of anchors, and answers are ranked by the number of shortest paths they lie on.\n\nWe perform fine-tuning for the \\textsc{BFS}\\xspace and \\textsc{ShortestPaths}\\xspace baselines by finding the best thresholds for alignment insertion ($\\tau_{align}^E$ and $\\tau_{align}^P$) for these methods, using the development set.\n\n\\subsection{Metrics}\n\\label{subsec:metrics}\n\nMost systems considered return exactly one answer (\\textsc{UniK-QA}\\xspace, \\textsc{PathRetriever}\\xspace, \\textsc{DocumentQA}\\xspace, \\textsc{DrQA}\\xspace, \\textsc{QAnswer}\\xspace, \\textsc{Platypus}\\xspace) and so we use Precision@1 (P@1) as our metric. P@1 is a standard one for factoid QA~\\cite{saharoy2021question}: in several practical use cases like voice-based personal assistants, only one best answer can be returned to the user. The gold answer sets for all our benchmarks are entities and literals grounded (or canonicalized) via the Wikidata KG. Thus, for KG-QA, evaluation is standard: answers are already KG entities and literals, and when the top-$1$ matches any of the ground truth exactly, P@1 = 1, else 0. This is trickier for text, or KG+Text (when the answer source is text) settings, where the system response can be an arbitrary span of text (\\phrase{The Rhine river}, \\phrase{lotr}, \\phrase{two million dollars}, etc.). \nThis also applies for \\textsc{UniK-QA}\\xspace, that uses a generative model for formulating a textual answer, that may not exactly be found in any source evidence.\nWe tackle such cases in a \\textit{KG-grounded evaluation setup} as follows: when the returned answer span matches a Wikidata entity\/literal in the ground truth set exactly, P@1 = 1. If no matches are found, we search for the string in KG entity aliases of the ground truth answers. If an exact match is found, then P@1 = 1, else we proceed to a human evaluation as discussed below.\n\n\\myparagraph{User evaluation} For the remaining cases, we perform a crowdsourced human evaluation via Amazon Mechanical Turk (AMT)\\footnote{\\url{https:\/\/www.mturk.com\/}}. The task is straightforward: given a factoid question and an answer string, verify its correctness by matching against the gold answer set and optionally searching the Web in unsure cases.\nSystem response strings were pooled for every question to reduce annotation overhead (all string responses to a question like \\utterance{Which river flows through Austria and Germany?} were grouped within an AMT HIT, a unit task on the platform).\nResponses from the different systems were shuffled with respect to their ordering in the AMT interface to avoid \\textit{position bias} in annotations.\nTo prevent spam, only Master Turkers\\footnote{\\url{https:\/\/www.mturk.com\/help}} were allowed to participate. Workers were paid at the standard rate of $15$ USD per hour.\nFor being robust to noisy annotations, three labels were collected per QA pair, and the majority vote was taken as the answer.\nFor all questions where a system-generated answer is deemed as correct by human annotation majority vote, P@1 = 1. For all remaining cases, P@1 = 0.\nOn examining anecdotal cases, we found that this human evaluation to be particularly worthwhile: the matching task is non-trivial -- matching abbreviations (St. with Street or Saint), noun phrases (Rhine with The Rhine River), varying date formats, and incomplete KB aliases are some of the particularly tricky cases that would have made a machine evaluation noisy.\n\n\\begin{table} [t] \\small\n\t\\centering\n\t\\caption{Basic statistics for the AMT study.}\n\t \\vspace*{-0.3cm}\n\n\n\t\\begin{tabular}{p{3.5cm} p{9.5cm}}\n\t\t\\toprule\n\t\tTitle\t\t\t            &   Verify correctness of machine-generated answers to fact-based questions\t\\\\\n\t\tDescription\t\t            &   Match machine-generated answers to a fact-based question and its set of correct answers.\n\t\t                                If you consider the pair to match, mark ``yes'', otherwise ``no''.\n\t\t                                When you are unsure, please consult Web Search.\n\t\t                                Cases to look out for are abbreviations, partial matches, and alternative formulations,\n\t\t                                i.e., the machine answer is correct but the exact surface forms vary.       \\\\ \\midrule\n\t\tParticipants\t\t\t    &   $870$ unique Master Turkers                                                  \\\\\n\t\tTime allotted per HIT\t    &   $45$ minutes maximum                                                        \\\\ \n\t\tTime taken per HIT \t        &   $60$ minutes on average                                                     \\\\ \n\t\tAnnotations per QA pair     &   $3$                                                                         \\\\\n\t\tQuestions in one HIT        &   $20$ (About $200$ QA pairs)                                                 \\\\\n\t\tPayment per HIT\t\t\t    &   $5$ USD                                                                     \\\\  \\midrule\n\t\tTotal no. of questions      &   $6,952$ (over all benchmarks, c.f. Table~\\ref{tab:data})                     \\\\\n\t    Total no. of QA pairs       &   $87,387$  (over all benchmarks, systems, and sources)                       \\\\   \n        Central tendency of \\#pairs &   $12.57$ (mean), $11$ (median), $11$ (mode)                                  \\\\  \\bottomrule\n\t\\end{tabular}\n\t\\label{tab:amt}\n\\end{table}\n\n\\section{\\textsc{Uniqorn}\\xspace: Graph Construction}\n\\label{sec:method-1}\n\n\\begin{figure} [t]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{images\/uniqorn-block-diagram.pdf}\n\t\\caption{System architecture of \\textsc{Uniqorn}\\xspace.}\n\t\\label{fig:block}\n\t\\vspace*{-0.3cm}\n\\end{figure}\n\nFig.~\\ref{fig:block} gives an overview of the \\textsc{Uniqorn}\\xspace\narchitecture. The two main stages -- construction of the\nquestion-specific context graph (Sec.~\\ref{sec:method-1}), and \ncomputing answers by Group Steiner Trees (Sec.~\\ref{sec:method-2})\non this context graph --\nare described in this section and the next.\n\nWe describe the XG construction, using our example question\nfor illustration:\n\\utterance{director of the western for which Leo won an Oscar?}\nInstantiations of key factors in the two settings, KG-QA and Text-QA, are \nshown\nin Table~\\ref{tab:kg-vs-text}. The heterogeneous setting can be understood\nas simply the union of the two individual cases.\nThe context graph is built in two stages: (i) identifying question-relevant evidences from the knowledge sources, and (iii) creating a graph from these top evidences.\n\n\\begin{table} [t]\n\t\\centering\n\t\\resizebox*{\\textwidth}{!}{\n\t\t\\setlength{\\tabcolsep}{0.5em}\n\t\t\\begin{tabular}{l l l}\n\t\t\t\\toprule\n\t\t\t\\textbf{Scenario}\t\t        & \\textbf{KG}\t\t\t\t\t\t\t    & \\textbf{Text}\t\t\t\t\t\t\\\\ \\toprule\n\t\t\tTriples in XG\t\t\t        & NERD + KG-lookups + BERT\t\t\t\t    & IR system + BERT + Open IE \\\\ \\midrule\n\t\t\tNode types\t\t\t\t        & Entity, predicate, type, literal\t\t    & Entity, predicate, type\t\t\t\\\\\n\t\t\tType nodes \t\t\t            & \\struct{instance of, occupation} triples  & Hearst patterns \t\t\t\t\t\\\\ \\midrule\n\t\t\tEdge types\t\t\t\t        & Triple, alignment, type\t\t\t\t    & Triple, alignment, type \t\t\t\\\\\n\t\t\tEntity alignments\t\t        & None \t\t\t\t\t\t                & With character trigrams   \t\t\\\\\n\t\t\tPredicate and type alignments\t& None\t\t\t\t\t                    & With word embeddings\t\t\t\t\\\\ \\midrule\n\t\t\tNode weights\t\t\t        & With BERT \t\t\t\t                & With BERT \t\t\t            \\\\\n\t\t\tEdge weights\t\t\t        & With BERT\t\t\t\t\t\t\t        & With BERT \t                    \\\\ \\bottomrule\n\t\t\\end{tabular}\n\t}\n\t\\caption{Instantiations of different factors of XGs from KG and text corpus.}\n\t\\label{tab:kg-vs-text}\n\t\\vspace*{-0.7cm}\n\\end{table}\n\n\\subsection{Retrieving question-relevant evidences}\n\\label{subsec:evi-retr}\n\nOur underlying goal is to reduce the huge knowledge repositories to a reasonably-sized question-relevant subgraphs, over which graph algorithms can be run with interactive response times.\n\n\\myparagraph{From knowledge graph} \nA typical curated KG $K$ contains billions of facts, with millions of entities and thousands of predicates, occupying multiple terabytes of disk space.\nTo reduce this enormous search space in $K$, \nwe first identify entities $E(q)$\nin the question $q$\nby using methods for named entity recognition and disambiguation (NERD)~\\cite{ferragina2010tagme,hoffart2011robust,van2020rel,li2020efficient}.\nThis produces KG entities\n(\\struct{LeonardoDiCaprio}, \\struct{AcademyAwards})\nas output.\nNext, all KG triples are fetched that are in the $1$-hop\nneighborhood of an entity in $E(q)$\\footnote{One entity to another entity is considered one ``hop''. The $2$-hop\nentity neighborhood in a KG can be enormously large,\nespecially for popular entities\nlike countries, with thousands of 1-hop neighbors.\nThis is further exacerbated by proliferations via type nodes\n(all humans are within two hops of each other on Wikidata). See~\\cite{christmann2022beyond} for more on KG neighborhoods.}.\nTo reduce this large and noisy set of facts (equivalently, KG-evidences) to a few question-relevant ones (to obtain $T^K(q)$), we fine-tune BERT~\\cite{devlin2019bert} (Sec.~\\ref{subsec:bert-tune}).\n\n\\myparagraph{From text corpus} The Web is the analogue of large KGs in text-QA.\nSimilar to the case for the KG, we \ncollect a small set of potentially relevant documents for $q$ from $D$\nusing a commercial search engine (for $D$ being the Web) or\nan IR system like ElasticSearch (for $D$ being a fixed corpus\nsuch as Wikipedia full-text).\n\nKG-style entity-centric retrieval is not a practical approach for open-domain text: it requires entity linking on a potentially large set of sentences, greatly limiting the efficiency of an on-the-fly procedure.\nRather, we take a noisy and recall-oriented process:\nwe locate question tokens (stopwords not considered) within the relevant documents, and consider a window of words (window length = $50$ in all our experiments) to each side of a match as a question-relevant snippet $\\mathcal{S}$. In case two snippets overlap, they are merged to form a longer snippet.\nThese snippets form our candidate text-evidences for locating an answer, and analogous to the KG facts obtained via NERD, are passed on to a BERT-based pruning model.\n\n\\subsection{Finding question-relevant evidences with BERT}\n\\label{subsec:bert-tune}\n\n\\myparagraph{Training a classifier for question-relevance} The goal our fine-tuned BERT model is to classify an evidence as being relevant to the question or not. In other words, we want a model that can score each evidence based on its likelihood of containing the answer. To build such a model, we prepare training data as follows. The (KG fact\/text snippet) retrieval step above feeds a set of evidences to the classifier. Out of these evidences, the ones that contain the gold answer(s) are labeled as \\textit{positive instances}. This is a \\textit{distant or noisy supervision signal} as the mere \\textit{presence} of an answer is not necessarily conclusive evidence of relevance to the question: the fact contained in the evidence may not be the one intended in the question. As an example, consider the question \\utterance{Who played Hugh in The Revenant?}: the KG fact \\struct{The Revenant, award received, Academy Award for Best Actor; winner, Leonardo DiCaprio} would be deemed relevant due to the presence of the gold answer \\struct{Leonardo DiCaprio}. Nevertheless, the method works: such false positives are rare in reality. For each positive instance, we randomly sample five \\textit{negative instances} from the evidences that do not contain the answer. Sampling question-specific negative instances helps learn a discriminative classifier, as all negative instances are guaranteed to contain at least one entity\/keyword from the question. Using \\textit{all facts} that do not contain an answer as negative examples would result in severe class imbalance, as this is much higher than the number of positive instances.\n\nWe then pool together the $\\langle$question, evidence$\\rangle$ paired positive and negative instances for all training questions. Text snippets are already in natural language and amenable for use in a BERT model.\nTo bring KG facts close to an NL form, we \\textit{verbalize} them by concatenating their constituents; qualifier statements are joined using \\phrase{and}~\\cite{oguz2021unikqa}. \nFor example, the verbalizaton of: \\struct{<The Revenant, nominated for, Academy Award for Best Director; nominee, Alejandro Gonz\\'{a}lez I\\~{n}\\'{a}rritu>}. This fact has one qualifier, and would be verbalized simply as \\phrase{The Revenant nominated for Academy Award for Best Director and nominee Alejandro Gonz\\'{a}lez I\\~{n}\\'{a}rritu.}.\nThe questions and the verbalized evidences, along with the binary ground-truth labels, are fed as training input to a \\textit{sequence pair classification model} for BERT: one sequence is the question, the other being the evidence.\n\n\\myparagraph{Applying the classifier}\nFollowing~\\cite{devlin2019bert}, the question and the evidence texts are concatenated with the special separator token \\struct{[SEP]} in between, and the special classification token \\struct{[CLS]} is prepended to this sequence. The final hidden vector corresponding to \\struct{[CLS]}, denoted by $C \\in \\mathbb{R}^H$ ($H$ is the size of the hidden state), is considered to be the accumulated representation. Weights $W$ of a final classification layer are the introduced during fine-tuning, where $W \\in \\mathbb{R}^{\\mathcal{L} \\times H}$, where $\\mathcal{L}$ is the number of class labels ($\\mathcal{L} = 2$ here, an evidence is either question-relevant or it is not). The value $\\log(\\text{softmax}(CW^T))$ is used as the classifier loss. Once this classifier is trained, given a new $\\langle$question, evidence$\\rangle$ pair, it outputs the probability (and the label) of the evidence containing an answer to the question. We make this prediction for all candidate evidences retrieved for a question, and sort them in descending order of this question relevance likelihood. We pick the top-$\\varepsilon$ evidences from here as our question-relevant set for constructing the XG. \n\n\\subsection{Materializing the context graph}\n\\label{subsec:xg-build}\n\n\\myparagraph{For KG evidences} Evidences from the KG are facts, and can be trivially cast into a graph that is a much smaller subset of the entire KG. Entities, predicates, types and literals constitute nodes, while edges represent connections between pieces of the same fact (recall Fig.~\\ref{fig:xg-kg}). Each distinct entity, type, and literal form one node, which (possibly repeated) predicate gets its own node (recall Sec.~\\ref{subsec:graph-con}). \nWe then add type information to this graph, useful for QA in several ways~\\cite{saharoy2021question,abujabal2017automated,yavuz2016improving,ziegler2017efficiency}. For this, we look up the most frequent KG types for each entity (an entity can belong to multiple KG types, but usually one is the most notable) in the qualifying set of evidences, and additionally look up occupations for humans (recall Sec.~\\ref{subsec:graph-con}), and add these triples to the graph.\nTo ensure connectivity in this graph, as far as possible, we add shortest paths from the KG between NERD entities detected in this question to this graph (complex questions often have more than one entity). This often helps reintroduce question-relevant connections between entities in the question. Such shortest paths can be obtained via the \\textsc{Clocq}\\footnote{\\url{https:\/\/clocq.mpi-inf.mpg.de\/}} API.\nThe largest connected component (LCC) is extracted from this graph, and the final structure thus obtained constitute $XG^K(q)$, being made up of individual triples $T^K(q)$. The LCC is expected to cover the most pertinent information and concepts relevant to the question.\n\n\\myparagraph{For text evidences} There is no natural graph structure in the NL snippets, so we induce it using a simple version of open information extraction (open IE). The goal of open IE is to extract informative riples from raw text sources. As off-the-shelf tools like Stanford Open IE~\\cite{angeli2015leveraging},\nOpenIE 5.1~\\cite{saha2018open},\nClausIE~\\cite{del2013clausie}\nor MinIE~\\cite{gashteovski2017minie},\nall have limitations regarding either precision or recall or efficiency, we developed our own custom-made open IE extractor. We start by part-of-speech (POS) tagging and named entity recognition (NER) on the original sentences from $D$ (and not on the snippets $\\mathcal{S}$, to preserve necessary sequence information vital for such taggers).\nThis is followed by light-weight coreference resolution by replacing each third person personal and possessive pronoun (\\phrase{he, him, his, she, her, hers})\nby its nearest preceding entity of type \\struct{PERSON}. We then define\na set of POS patterns that may indicate an entity or a predicate. \nEntities are marked by an unbroken sequence of\n\\struct{nouns, adjectives, cardinal numbers},\nor mention phrases from the NER system\n(e.g., \\phrase{Leonardo DiCaprio}, \\phrase{2016 American western film}, \\phrase{Wolf of Wall Street}). \nTo capture both verb- and noun-mediated relations, predicates correspond to\nPOS patterns \\struct{verb}, \\struct{verb+preposition} or \\struct{noun+preposition}\n(e.g.,\n\\phrase{produced}, \\phrase{collaborated with}, \\phrase{director of}).\nSee node labels in Fig.~\\ref{fig:xg-text} for examples. \nThe patterns are \napplied to\neach snippet $\\mathcal{S}_j \\in D$, to produce a markup like\n$\\mathcal{S}_j \\equiv E_1 \\ldots E_2 \\ldots P_1 \\ldots E_3 \\ldots P_2 \\ldots E_4.$\nThe ellipses ($\\ldots$) denote intervening words in $\\mathcal{S}_j$.\nFrom this markup, \\textsc{Uniqorn}\\xspace finds all $(E_{i_1}, E_{i_2})$ pairs that have\nexactly one predicate $P_k$ between them, this way creating triples.\n$<\\!E_1, P_1, E_3\\!>$ and $<\\!E_2, P_1, E_3\\!>$\n(but excluding\n$<\\!E_1, P_2, E_4\\!>$). \nPatterns from~\\cite{yahya2014renoun}, specially\ndesigned for noun phrases as relations (e.g., \n\\phrase{Oscar winner}), are applied as well.\nSnippets that contain two (or more) entities but no predicate contribute triples with a special \\struct{cooccurs} predicate.\nFor example if a snippet contains three entities $E_1, E_2, E_3$ but zero predicates, we would add triples:\n\\struct{<$E_1$, cooccurs, $E_2$>, <$E_1$, cooccurs, $E_3$>, <$E_2$, cooccurs, $E_3$>} to the open IE triples.\nSuch rules help tap into information in snippets like \\phrase{Leonardo was in Inception}, where the intended relation is implicit (\\phrase{starred}).\n The rationale for this heuristic extractor is to \nachieve high \\textit{recall} towards answers, at the cost of \nintroducing noise. The noise is taken care of in the answering\nstage later.\n\nAs for KG evidences, we would like to to extract entity type information here as well.\nTo this end, we leverage Hearst patterns~\\cite{hearst1992automatic}, like\n\\phrase{$NP_2$ such as $NP_1 \\ldots$}\n(matched by, say, \\phrase{\\underline{western films} such as\n\t\\underline{The Revenant}})\n\\phrase{$NP_1$ is a(n) $NP_2 \\ldots$}\n(matched by, say, \\phrase{\\underline{The Revenant} is a\n\t\\underline{2015 American western film}}),\nor \\phrase{$NP_1$ and other $NP_2 \\ldots$}\n(matched by, e.g., \\phrase{\\underline{Alejandro I\\~{n}\\`{a}rritu} and other\n\\underline{Mexican film directors} $\\ldots$}).\nHere $NP$ denotes a noun phrase, detected by a constituency parser.\nThe resulting triples about type membership\n(of the form: \\struct{<$NP_1$, type, $NP_2$>}\nfor e.g., \\struct{<The Revenant, type, 2015 American western film>})\nare added to the triple collection.\nFinally, as for the KG-QA case,\nall triples are joined by $S$ or $O$ with exact string matches,\nand the LCC is computed, to produce the final $XG^D(q)$. \nTo compensate for \nthe diversity of surface forms where different strings may denote\nthe same entity or predicate, alignment edges\nare inserted into $XG^D(q)$ for node pairs\nas discussed in Sec.~\\ref{subsec:graph-con}.\n$XG^D(q)$ is made up of individual triples $T^D(q)$.\nThus, a large quasi KG $K^D$ is never materialized, and we directly construct $XG^D$, on which our graph algorthims may run.\n\n\\myparagraph{For heterogeneous evidences} The union of $XG^K$ and $XG^D$ form $XG^H$, the heterogeneous context graph for question $q$. Fine-tuned BERT models are run individually on the KG and text repositories for retrieving the source-specific evidences, which are processed as described above.\n\n\n\\section{\\textsc{Uniqorn}\\xspace: Graph Algorithm}\n\\label{sec:method-2}\n\nFor the given context graph $XG(q)$, we find candidate answers $a \\in A$ as follows. First, nodes in the $XG$ are identified\nthat best capture the question; these nodes become anchors.\nFor $XG^K$, question entities detected by NERD systems become entity anchors.\nFor predicates, types and literals, any node with a token in its label that matches any of the question tokens, becomes an anchor (\\struct{cast member} becomes an anchor if the question has \\utterance{Who was \\underline{cast} as ...}). To ensure better semantic coverage, node labels are augmented with aliases from the KB, that are a rich source of paraphrases.\nFor example, Wikidata contains the following aliases for \\struct{cast member}: \n\\phrase{starring}, \\phrase{actor}, \\phrase{actress}, \\phrase{featuring}, and so on.\nThus, \\struct{cast member} will become an anchor node if the question has any of the synonyms above.\nFor $XG^D$, any node with a token in its label that matches any of the question tokens becomes an anchor.\nNode and edge weights in $XG^K$ or $XG^D$ are obtained by using the BERT-based scores that the original evidence (source of the node or edge) received from the fine-tuned model.\nIf a node or edge originates from multiple evidences, their scores are averaged.\nThis helps us harness redundancy of information across evidences.\n\nAnchors are grouped into equivalence classes $\\{\\mathcal{A}^k\\}$ based on\nthe question token that they correspond to.\nAt this stage, we have the directed and weighted\ncontext graph $XG$ as a $6$-tuple:\n$XG^{(\\cdot)}(q) = (N, \\mathcal{E}, N^T, \\mathcal{E}^T, N^W, \n\\mathcal{E}^W)$.\nFor simplification, \nwe disregard the direction of edges, turning $XG$ into an\nundirected graph.\n\n\\myparagraph{Group Steiner Tree} We postulate that the criteria for identifying good answer candidates in the XG are as follows:\n(i) answers lie on \\textit{paths} connecting anchors;\n(ii) \\textit{shorter} paths with \\textit{higher} weights are more likely to contain correct answers;\nand (iii) including at least one instance of an anchor from each \\textit{group} is necessary\n(to satisfy all conditions in a complex question $q$). \nFormalizing these desiderata leads us to the notion of\n\\textit{Steiner Trees}~\\cite{feldmann2016equivalence,bhalotia2002keyword,kacholia2005bidirectional,kasneci2009star}:\nfor an undirected weighted graph, and a subset of nodes \ncalled \\phrase{terminals}, find the tree of least cost that connects \nall terminals.\nIf the number of terminals is two, then this is the weighted\nshortest path problem, and if all nodes of the graph are terminals, this \nbecomes the minimum spanning tree (MST) problem.\nIn our setting, the graph\nis the $XG$, and terminals are the anchor nodes. \nAs anchors are arranged into groups, we pursue the\ngeneralized notion of\n\\textit{Group Steiner Trees (GST)}~\\cite{garg2000polylogarithmic,li2016efficient,ding2007finding,shi2020keyword,chanial2018connectionlens,sun2021finding}:\ncompute a minimum-cost Steiner Tree \nthat connects at least one terminal from each group,\nwhere weights of edges $\\mathcal{E}_k$ are\nconverted into costs\nas $\\textrm{cost}(\\mathcal{E}_k)  = 1 - M_W^\\mathcal{E}(\\mathcal{E}_k)$. \nAt this point, the reader is referred to Fig.~\\ref{fig:xg} again,\nfor illustrations of what GSTs look like (shown\nin orange).\nTo tackle questions with a chain-join~\\cite{saharoy2021question} component (like \\utterance{profession of father of DiCaprio?}), the complete evidence of predicate anchors (resembling predicates dangling from the tree) is admitted into the GST.\n\nFormally, the GST problem in our setting can \nbe defined as follows. Given\na question $q$ with $|q|$ tokens,\nan undirected and weighted graph $XG(q) = (N, \\mathcal{E})$,\nand groups of anchor nodes\n$\\{\\mathcal{A}^1, \\ldots \\mathcal{A}^{|q|}\\}$ with\neach $\\mathcal{A}^k \\subset N$, find the\nminimum-cost tree $\\mathbb{T}^* = (N^*, \\mathcal{E}^*) =\n\\arg \\min_i \\textrm{cost}(\\mathbb{T}_i, XG)$,\nwhere $\\mathbb{T}_i$ is any tree\nthat connects at least one node from each of\n$\\{\\mathcal{A}^1, \\ldots \\mathcal{A}^{|q|}\\}$, such that\n$\\mathcal{A}^k \\cap N^* \\neq \\phi$ for each $\\mathcal{A}_k$,\nand cost of $\\mathbb{T}_i$,\n$\\textrm{cost}(\\mathbb{T}_i) = \\sum_{k}\\textrm{cost}(\\mathcal{E}_k)$,\nwhere $\\mathcal{E}_k \\in \\mathbb{T}_i$.\n\nWhile finding the GST is an NP-hard problem, there are\napproximation algorithms~\\cite{garg2000polylogarithmic}, and also exact methods that are\nfixed-parameter tractable with respect to the number of terminal nodes.\nWe adapted the method\nof Ding et al.~\\cite{ding2007finding} from the latter family, which is exponential in the\nnumber of terminals \nbut $O(n \\; log \\; n)$ in the graph size.\nLuckily for us, the number of anchors\nis indeed typically much less of a bottleneck than the sizes of\nthe $XG$\nin terms of nodes and edges.\nSpecifically, the terminals are the anchor nodes derived\nfrom the question tokens -- so their numbers are not\nprohibitive with respect to computational costs.\n\nThe algorithm is based on \\textit{dynamic programming} and \nworks as follows. It starts from a set of singleton trees, one for\neach terminal group, rooted at one of the corresponding anchor nodes. These\n\\textit{trees are grown} iteratively by exploring immediate \nneighborhoods for\nleast-cost nodes as expansion points. {Trees are merged} \nwhen common nodes are encountered while growing. The trees are stored\nin an efficient implementation of priority queues with \\textit{Fibonacci heaps}.\nThe process terminates when a GST is found that {covers} all terminals \n(i.e., one per group).\nBottom-up dynamic programming guarantees that the\nresult is \\textit{optimal}.\n\n\\myparagraph{Relaxation to top-$k$ GSTs} It is possible that the GST simply\nconnects a terminal from each anchor group, without having any internal \nnodes at all, or with predicates and\/or types as internal nodes. Since we\nneed entities or literals as answers, such\npossibilities necessitate a relaxation of our solution to \ncompute a number of top-$k$\nleast-cost GSTs.\nGST-$k$ ensures that we always get a non-empty\nanswer set, albeit at the cost of making some detours in the graph. \nMoreover, using GST-$k$ provides a natural answer ranking strategy, where\nthe score for an answer can be appropriately reinforced if it appears in\nmultiple such low-cost GSTs. \nThis postulate, and the effect of $k$, is later quantified in our experiments.\nNote that since the tree with the least cost is always kept\nat the top of the priority\nqueue, the $k$ trees can be found in the increasing order of cost, and\nno additional sorting is needed.\nIn other words, the priority-queue-based implementation of the GST algorithm~\\cite{ding2007finding} \neasily supports this top-$k$ computation.\nThe time\nand space complexities for obtaining GST-$k$ is the same as that\n for GST-$1$.\nFig.~\\ref{fig:gst-k} gives an example of GST-$k$ ($k = 3$).\n\n\\begin{figure} [t]\n  \\centering\n    \\includegraphics[width=0.5\\columnwidth]{images\/gst-k.pdf}\n      \\caption{Illustrating GST-$k$, showing edge costs and node weights. Anchors (terminals) and answer candidates (non-terminals A1, A2, A3) are shown in black and white circles respectively. \\{(C11, C12), (C21, C22, C23), (C31)\\} represent anchor groups. Edge costs are used in finding GST-$k$, while node weights are used in answer ranking. A1 is likely to be a better answer due to its presence in two GSTs in the top-3.}\n      \\label{fig:gst-k}\n      \\vspace*{-0.5cm}\n\\end{figure}\n\n\\myparagraph{Answer ranking} Non-terminal entities in GSTs are candidates for final answers. However, this mandates ranking. \nWe use the number of top-$k$ GSTs that an answer candidate lies on, as the ranking criterion. Alternative choices, like weighting these trees by their total node weight, tree cost, or the answers' proximity to anchor nodes, are investigated in Sec.~\\ref{subsec:analysis}. The top-ranking answer is presented to the end user.\n\n\\section{Related Work}\n\\label{sec:related}\n\n\\subsection{QA over Knowledge Graphs}\n\\label{subsec:kg-qa}\n\nThe inception of large KGs like Freebase~\\cite{bollacker2008freebase}, YAGO~\\cite{suchanek2007yago}, DBpedia~\\cite{auer2007dbpedia}, and Wikidata~\\cite{vrandevcic2014wikidata}, gave rise to question answering over knowledge graphs (KG-QA) that typically provides answers as \\textit{single entities} or \\textit{entity lists} from the KG. KG-QA is now an increasingly active research avenue, where the goal is to translate an NL question into a structured query, usually in SPARQL syntax or an equivalent logical form, that is directly executable over the KG triple store containing entities, predicates, classes and literals~\\cite{wu2020perq,qiu2020stepwise,vakulenko2019message,bhutani2019learning,christmann2019look}.\nChallenges in KG-QA include entity disambiguation, bridging the vocabulary gap between phrases in questions and the terminology of the KG, and finally, constructing the SPARQL(-like) query. Early work on KG-QA built on paraphrase-based mappings and question-query templates that typically had a single entity or a single predicate as slots~\\cite{berant2013semantic,unger2012template,yahya2013robust}. This direction was advanced  by~\\cite{bast2015more,bao2016constraint,abujabal2017automated,hu2017answering}, including templates that were automatically learnt from graph patterns in the KG. Unfortunately, this dependence on templates prevents such approaches from coping with arbitrary syntactic formulations in a robust manner. This has led to the development of deep learning-based methods with sequence models, and especially\nkey-value memory networks~\\cite{xu2019enhancing,xu2016question,tan2017context,huang2019knowledge,chen2019bidirectional,jain2016question}.\nThese have been most successful on\nbenchmarks like WebQuestions~\\cite{berant2013semantic} and QALD~\\cite{ngomo20189th}. However, all\nthese methods critically build on sufficient amounts of training data\nin the form of question-answer pairs. In contrast, \\textsc{Uniqorn}\\xspace is fully\nunsupervised and needs neither templates nor training data.\n\n\\textit{Complex question answering} is an area of intense focus in KG-QA \nnow~\\cite{lu2019answering,bhutani2019learning,qiu2020stepwise,ding2019leveraging,vakulenko2019message,hu2018state,jia18tequila}, where the general approach is often \nguided by the existence and detection of substructures for the executable query. \\textsc{Uniqorn}\\xspace treats this as a potential drawback and adopts a joint disambiguation of \nquestion concepts using algorithms for Group Steiner Trees, instead of looking for nested question units that can be mapped to simpler queries. \nApproaches based on question decomposition (explicit or implicit) are brittle due to the huge variety of question formulation patterns (especially for complex \nquestions), and are particularly vulnerable when questions are posed in telegraphic form (\\utterance{oscar-winnning nolan films?}, has to be interpreted as \n\\utterance{Which films were directed by Nolan and won an Oscar?}: this is highly non-trivial).\nAnother branch of complex KG-QA rose from the task of\nknowledge graph reasoning (KGR)~\\cite{das2018go,qiu2020stepwise,dhingra2020differentiable,cohen2020scalable,zhang2018variational}, where the key idea is given a KG entity (\\struct{Albert Einstein}) and a textual relation (\\phrase{nephew}), the best KG path from the input entity to the target entity (answer) is sought. This can be generalized into a so-called QA task where topic (question) entity is known and the question is assumed to be a paraphrase of the multi-hop KG relation (there is an assumption that \\phrase{nephew} is not directly a KG predicate). Nevertheless, this is a restricted view of complex KG-QA, and only deals with such indirection or chain questions (\\phrase{nephew} has to be matched with \\struct{sibling} followed by \\struct{child} in the KG),\nevaluated on truncated subsets of the full KG that typically lack the complexity of qualifier triples.\n\n\n\\subsection{QA over Text}\n\\label{subsec:text-qa}\n\nOriginally, in the late 1990s and early 2000s, question answering considered textual document collections as its underlying source. Classical approaches based on statistical scoring~\\cite{ravichandran2002learning,voorhees1999trec} extracted answers as short text units from passages or sentences that matched most cue words from the question. Such models made intensive use of IR techniques for scoring of sentences or passages and aggregation of evidence for answer candidates. IBM Watson~\\cite{ferrucci2010building}, a thoroughly engineered system that won the Jeopardy! quiz show, extended this paradigm by combining it with learned models for special question types. TREC ran a QA benchmarking series from 1999 to 2007, and more recently revived it as the LiveQA~\\cite{agichtein2015overview} and Complex Answer Retrieval (CAR) tracks~\\cite{dietz2017trec}. \n\n\\textit{Machine reading comprehension (MRC)} was originally motivated by the goal of whether algorithms actually \\textit{understood textual content}. This eventually became a QA variation where a question needs to be answered as a short \\textit{span of words} from a given text paragraph~\\cite{rajpurkar2016squad,yang2018hotpotqa}, and is different from the typical fact-centric answer-finding task in IR. Exemplary approaches in MRC that extended the original single-passage setting to a multi-document one can be found in DrQA~\\cite{chen2017reading} and DocumentQA~\\cite{clark2018simple} (among many, many others). \nIn the scope of such text-QA, we compared with, and outperformed both the aforementioned models, which can both select relevant documents and extract\nanswers from them. Traditional fact-centric QA over text, and multi-document MRC are recently emerging as a joint topic referred to as\n\\textit{open-domain question answering}~\\cite{lin2018denoising,dehghani2019learning,wang2019document}. Open-domain QA tries to combine an IR-based retrieval pipeline and NLP-style reading comprehension, to produce crisp answers extracted from passages retrieved on-the-fly from large corpora.\n\n\\subsection{QA over Heterogeneous Sources}\n\\label{subsec:hybrid-qa}\n\nLimitations of QA over KGs has led to a revival of considering textual sources, in combination with KGs~\\cite{savenkov2016knowledge,xu2016question,sun2018open,sun2019pullnet}. Early methods like PARALEX~\\cite{fader2013paraphrase} and OQA~\\cite{fader2014open} supported noisy KGs in the form of triple spaces compiled via Open IE~\\cite{mausam2016open} on Wikipedia articles or Web corpora. TupleInf~\\cite{khot2017answering} extended and generalized the Open-IE-based PARALEX approach to complex questions, and is geared specifically for \\textit{multiple-choice answer options},\nand is thus inapplicable for our task. TAQA~\\cite{yin2015answering} is another generalization of Open-IE-based QA, by constructing a KG of $n$-tuples from Wikipedia full-text and question-specific search results. \nUnfortunately this method is restricted to questions with prepositional and adverbial constraints only.\nWebAsKB~\\cite{talmor2018web} is an MRC-inspired method that addressed complex questions by decomposing them into a sequence of simple questions, but relies on crowdsourced large-scale training data. Some methods for such hybrid QA start with KGs as a source for candidate answers and use text corpora like Wikipedia or ClueWeb as additional evidence~\\cite{xu2016question,das2017question,sun2018open,sun2019pullnet,sydorova2019interpretable,xiong2019improving}, or start with answer sentences from text corpora and combine these with KGs for giving crisp entity answers~\\cite{sun2015open,savenkov2016knowledge}. Most of these are based on neural networks, and\nare only designed for simple questions like those in the WebQuestions~\\cite{berant2013semantic}, SimpleQuestions~\\cite{bordes2015large},\nor WikiMovies~\\cite{miller2016key} benchmarks. In contrast,\n\\textsc{Uniqorn}\\xspace, through its anchor graphs and GSTs, can handle arbitrary kinds of complex questions and can construct explanatory evidence for its answers -- an unsolved concern for neural methods.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{Intro} Introduction}\n\nThe successful development of atomic laser-cooling techniques has facilitated the study of atomic collisions in the ultracold regime and the macroscopic quantum degeneracy of the Bose--Einstein condensate \\cite{Cornell2002,Ketterle2002}. This progress has inspired interest in the generation of ultracold molecules in prescribed low-energy internal states, opening prospects from ultrahigh precision spectroscopy to the coherent control of chemical reactions and new techniques for quantum computing \\cite{DeMille2002,Tesch2002}. Atomic cooling techniques, however, cannot be generalized even to simple diatomic molecules due to the lack of availability of a closed-loop cooling cycle within the rich molecular internal energy level structure; instead, alternative approaches must be pursued.\n\nVarious alternative direct cooling techniques have been developed that stabilize molecules efficiently into the lowest vibrational levels, but are unable to cool their translational motion significantly below the millikelvin regime \\cite{Weinstein1998,Bethlem2000}. In order to reach microkelvin or nanokelvin temperatures, it is necessary instead to associate ultracold atoms via approaches such as photoassociation \\cite{Thorsheim1987,Fioretti1998}, the manipulation of Feshbach resonances \\cite{Greiner2003,Zwierlein2003}, or three-body collisions \\cite{Jochim2003}. These processes, however, favour the formation of ultracold but vibrationally excited molecules.\n\nRecently, an incoherent optical cycling scheme was demonstrated that accumulates population in the Cs$_2$ singlet ground vibrational state via spontaneous decay \\cite{Viteau2008}. Precise knowledge of the system spectroscopy is not required; however it relies upon the serendipitous availability within Cs$_2$ of a specific continuous-wave (c.w.) pump-decay photoassociative pathway that pre-populates deeply bound vibrational states. Such a pathway is not, in general, available in other dimers, and an optical repumping scheme would require many more spontaneous decay cycles (and hence greater heating of the centre-of-mass motion) before significant enhancement of the ground vibrational state population is obtained. In certain heteronuclear systems, high-power, c.w.\\ photoassociation has been used to produce an incoherent mixture that includes $v=0$ ground-state singlet dimers \\cite{Sage2005,Deiglmayr2008}. In another approach, stimulated Raman adiabatic passage (STIRAP) has been used to transfer molecules to the lowest vibrational ground state in Rb$_2$ triplets \\cite{Lang2008} and Cs$_2$ singlets \\cite{Danzl2009,Mark2009}, as well as heteronuclear KRb molecules to both triplet and singlet $v=0$ states \\cite{Ni2008}. The transfer is coherent but requires full spectroscopic knowledge of the system.\n\nA complementary quantum control strategy to those outlined above is the coherent control \\cite{Warren1993,Rabitz2000} of the photoassociation process using broadband tailored optical fields. The broad coherent bandwidth of a Ti:sapphire ultrafast pulse is well matched to typical diatomic molecule potential binding energies, while the carrier frequency is appropriate for ground- to excited-state transitions. An ultracold scattering event may therefore be steered towards a target state through a variety of vibronic pathways, with phase and amplitude shaping of the excitation pulse offering control over the system Hamiltonian via the dipole interaction. Furthermore, closed-loop control may be applied to the choice of pulse shape in order to identify an empirical optimum. The generality of this approach is thus limited merely by the frequency and bandwidth of the laser system rather than any requirement for precise spectroscopic knowledge.\n\nThe use of ultrafast `pump' and `dump' pulses would permit the fully coherent transfer of population from free atom pairs to deeply bound vibrational states \\cite{Koch2006a}. An excited-state molecular wavepacket is formed by photoassociation, and evolves freely before being optically de-excited at a time that yields optimal Franck--Condon overlap with the target state.\nThrough the exploitation of a non-stationary coherent superposition in the excited state, this wavepacket approach thus offers the advantage of favourable Franck--Condon factors for both `pump' and `dump' stages without relying upon the overlap between individual wavefunctions of specific vibrational states. Previous ultrafast `pump-decay' photoassociation experiments have revealed a coherent quenching effect on pre-formed triplet molecules \\cite{Brown2006,Staanum2006}; a pump-dump approach may enable the coherent enhancement of bound molecular population before the onset of spontaneous decay.\n\nLearning about the dynamics present in the excited state after photoassociation is therefore an important step towards efficient stabilization into the ground state. A useful tool for resolving ultrafast molecular dynamics is the pump-probe experiment \\cite{Machholm1994,Fatemi2001}. A pump pulse generates an excitation which evolves during a controllable time delay before being detected by a probe pulse through a reaction-coordinate--selective mechanism such as ionization or bond fragmentation. Recently, a pump-probe study of the short-delay dynamics of rubidium atoms and molecules after photoassociation revealed coherent transient oscillations during the photoassociation pulse \\cite{Salzmann2008}. In this paper we report on pump-probe experiments investigating the coherent control of bound excited-state dimers over longer timescales as the first step towards the development of a pump-dump scheme to produce deeply bound ground-state Rb$_2$ singlet molecules. We demonstrate the production of bound excited-state molecules and compare the experimental observations to numerical simulations. We characterize the initial state of a background molecular population within the cold atom cloud and identify their association mechanism.\nWe show evidence that the pump pulse addresses free atom pairs with internuclear separations of \\unit[30-60]{a$_0$}, rather than the closer range background molecules. This experimental determination of the initial conditions permits improved comparison between theory and experiment, offering further insight into the reaction process.\n\n\n\\section{Methods and apparatus}\n\\label{sec:method}\n\n\\subsection{Overview}\n\\label{sec:overview}\n\n  \\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figure1.pdf}\n    \\caption{(Colour online). A schematic of the pump-probe experiment. A femtosecond pump pulse photoassociates atom pairs from the ground state, forming a wavepacket spanning a range of vibrational states within the \\mbox{5S + 5P} potential manifold. A typical simulated population distribution is shown (shaded). After a variable time delay $\\tau$, a probe pulse ionizes the excited-state population in preparation for detection via TOF mass spectrometry.} \n    \\label{fig:schematic}\n  \\end{figure}\n\nA schematic of the pump-probe experiment is given in Fig.\\ \\ref{fig:schematic}. A broadband pump pulse photoassociates ultracold ground-state 5S atom pairs within a rubidium magneto-optical trap (MOT), forming long-range molecular wavepackets within the \\mbox{5S + 5P} potential manifold. The excited dimer is allowed to evolve before being ionized by a broadband probe pulse. In order to study the excited-state dynamics over the oscillation timescales of long-range molecules, this pump-probe delay is varied over up to \\unit[250]{ps}. Atomic ions are also produced via nonresonant multi-photon processes. The nascent atomic and molecular ions are detected and distinguished via time-of-flight (TOF) mass spectrometry. In order to ensure that the atom pairs commence the experiment in their ground state, the MOT lasers (which, like the pump pulse, operate on the 5S $\\rightarrow$ 5P transition) are shuttered for a \\unit[2]{$\\mu$s} window spanning the arrival of the pump and probe pulses (see Fig.\\ \\ref{fig:timing}(a)).\n \n\\subsection{Magneto-optical trap}\n\\label{sec:MOT}\n\n  \\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figure2.pdf}\n    \\caption{(Colour online). (a) A schematic of the experimental timing. The MOT lasers are shut off for a \\unit[2]{$\\mu$s} window spanning the arrival of the ultrafast pulses. The TOF pushing electrode is switched to a positive voltage \\unit[3.3]{$\\mu$s} after ionization. (b) Experimental layout. Pulses from a CPA laser pass through a beamsplitter, with the majority of the pulse energy pumping the NOPA to form the probe pulses whilst the remainder is spectrally filtered in a 4f line and acts as the pump. The pulses are combined with a variable delay $\\tau$ and focussed onto the MOT. Ionized rubidium atoms and molecules are accelerated towards an MCP detector and distinguished via TOF mass spectrometry.}\n    \\label{fig:timing}\n  \\end{figure}\n\nThe choice of rubidium for this study was motivated by the suitability of its atomic and molecular spectroscopy to Ti:sapphire wavelengths, as well as the extent to which its collisional properties and photoassociation spectroscopy have been studied and catalogued. It therefore provides a convenient test bed for the control scenarios, though the concepts are expected to be of much broader utility.\n\n$^{85}$Rb atoms were cooled and trapped in a MOT generated in an ultrahigh vacuum stainless steel chamber with multiple windows for optical access. Three orthogonal pairs of counter-propagating beams of opposite circular polarizations intersected at the minimum of a quadrupole magnetic field generated by a pair of anti-Helmholtz current coils. Stray magnetic fields in the chamber were nulled using three orthogonal pairs of compensation coils. Rb was loaded into the MOT from a background vapour maintained by the application of a constant current to a series of standard commercial getters sources. \n\nThe master trapping light was generated by a home-built external-cavity diode laser (ECDL). A pick-off of the trapping light was passed through a rubidium vapour cell, and a feedback loop was used to lock its frequency to a feature in the resultant Doppler-free saturated absorption spectrum. The master laser was in turn injected into a higher power, free-running slave diode laser. The output of the slave laser was detuned \\unit[15]{MHz} to the red of the $\\textrm{F}=3 \\rightarrow \\textrm{F}'=4$ trapping transition using an acousto-optic modulator (AOM) before passing through a single-mode polarization-maintaining fibre to the MOT. A total of \\unit[30]{mW} trapping light was split equally over three retro-reflected beams of \\unit[4.0]{mm} half-maximum diameter in intensity. A second ECDL was similarly locked to the $\\textrm{F}=2 \\rightarrow \\textrm{F}'=3$ transition to repump population that accumulated in the dark $\\textrm{F}=2$ lower hyperfine state, with a power of \\unit[3]{mW} at the MOT. The AOMs were additionally employed as fast optical shutters for the trapping and repumping light.\n\nThe total fluorescence and density profile of the MOT cloud were monitored by a photodiode and pair of CCD cameras respectively. $4\\times10^6$ atoms were trapped in the MOT with a maximum density of \\unit[$1.1\\times10^9$]{cm$^{-3}$} and a half-maximum diameter of \\unit[0.8]{mm}. The MOT temperature was measured by the release-and-recapture method \\cite{Lett1988} to be \\unit[110]{$\\mu$}K. \n\n\\subsection{Ultrafast system}\n\\label{sec:ultrafast}\n\n  \\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\columnwidth]{Figure3.pdf}\n    \\caption{(Colour online). (a) Spectral intensity and phase of the CPA pulse used to generate the pump and probe (\\unit[53]{fs} FWHM). (b) Typical pump spectrum with cut at Rb D1 line.(\\unit[85]{fs} FWHM). (c) Typical NOPA probe spectrum, filtered at \\unit[480]{nm} (\\unit[390]{fs} FWHM).}\n    \\label{fig:pulses}\n  \\end{figure}\n\nThe ultrafast pulses for this experiment were derived from a regenerative chirped-pulse amplified Ti:sapphire laser (CPA) \\cite{Backus1998} producing \\unit[250]{$\\mu$J}, \\unit[800]{nm} pulses with a \\unit[20]{nm} FWHM at a repetition rate of \\unit[2]{kHz}. Pulse characterization using spectral phase interferometry for direct electric field reconstruction (SPIDER) \\cite{Iaconis1998} revealed near--transform-limited pulses of \\unit[53]{fs} FWHM (see Fig.\\ \\ref{fig:pulses}(a)). The CPA pulses were split into two components: a few microjoules were used to generate the pump pulse while the remainder pumped a non-collinear optical parametric amplifier (NOPA) which acted as the probe \\cite{Cerullo2003}.\n\nThe pump pulse was spectrally filtered by a translatable razor blade in the Fourier plane of a zero-dispersion 4f line (\\unit[1200]{lines\/mm} diffraction gratings separated by \\unit[75]{cm}) with a resolution of \\unit[0.1]{nm} (as determined by the beam waist in the Fourier plane and the 4f line dispersion). The position of the spectral cut was set to eliminate excitation of unbound excited-state population above the \\mbox{5S + 5P$_{1\/2}$} `D1 line' atomic asymptote (see Fig.\\ \\ref{fig:pulses}(b)). Interferometric autocorrelation indicated pump pulses of \\unit[85]{fs} FWHM duration. The NOPA spectrum was tuned to \\unit[500]{nm} (\\unit[20]{nm} FWHM) to correspond to the transitions between the \\mbox{5S + 5P$_{1\/2}$} and molecular ion states (see Fig.\\ \\ref{fig:pulses}(c)).  A long-pass filter with a \\unit[480]{nm} cut-off prevented the resonant ionization of atoms. Intensity autocorrelation indicated probe pulses of \\unit[390]{fs} FWHM duration. A typical pulse energy was \\unit[4]{$\\mu$J} with a fluctuation of less than \\unit[5]{\\%}.\n\nThe pump pulse was passed through a variable delay line, combined with the probe using a dichroic mirror (see Fig.\\ \\ref{fig:timing}(b)) and focussed onto the MOT.\n\n\\subsection{Detection scheme}\n\\label{sec:detection}\n\nThe efficiency of the photoassociation process is limited by the density of the MOT; hence the Rb$_2^+$ ionization rate is exceeded by Rb$^+$ ionization despite the off-resonant nature of the latter process. The challenge is therefore to resolve the molecular ions from the atomic ions with a sensitive detection mechanism. This was achieved using TOF mass spectrometry with a multi-channel plate (MCP) detector. The ions were accelerated towards the MCP by a gated pushing electrode situated opposite the MCP inside the vacuum chamber. The electrode was switched from \\unit[0]{V} to a positive voltage \\unit[3.3]{$\\mu$s} after the arrival of the pump pulse. This provided for an expansion of the ion cloud that reduced the impact of saturation within the central MCP ion channels by distributing the ions over a greater detector surface area \\cite{Kraft2007}.\n\nThe molecular ions arrived at the MCP within a time window narrower than \\unit[100]{ns}. The sensitivity of the detection was changed by varying the pushing electrode voltage from \\unit[100]{V} to \\unit[1100]{V}: a high voltage ensured sensitive detection of  Rb$_2^+$ whilst a low voltage ensured a linear response to  Rb$^+$ without MCP saturation. For each pump-probe delay measurement the Rb$^+$ or Rb$_2^+$ TOF signal was averaged 1000 times by a digital oscilloscope and integrated numerically. This ensured that the duration of the single-point averaging was greater than that of the characteristic NOPA power fluctuations. The signal-to-noise ratio of the pump-probe signal was further improved without compromising immunity to long-term drifts by averaging several successive pump-probe scans.\n\n\\subsection{Bound initial state characterization}\n\\label{sec:REMPI}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Figure4.pdf}\n  \\caption{(Colour online). A schematic of the initial-state characterization experiment. The MOT trapping light photoassociates free atom pairs to loosely bound states within the \\mbox{5S + 5P}$_{3\/2}$ $0_{\\textrm{g}}^-$ potential, from where they spontaneously decay to the $a^3\\Sigma_u^+$ ground triplet potential. These `background' ground triplet molecules may be detected using a two-photon resonant ionization pathway via deeply bound states within the $(2)^3\\Sigma_g^+$ potential. The high-lying populated states within the $a^3\\Sigma_u^+$ ground state are more closely spaced than the deeply bound intermediates within the $(2)^3\\Sigma_g^+$ potential, allowing the ground-state population distribution to be deduced from the resultant spectrum.}\n  \\label{fig:init_schematic}\n\\end{figure}\n\nThe intuitive picture of this pump-probe experiment concerns the photoassociation of a scattering pair of unbound atoms. There are, however, two mechanisms that produce a `background' population of loosely bound ground-state molecules in an $^{85}$Rb MOT in the absence of dedicated photoassociation light: three-body recombination \\cite{Takekoshi1999}, and photoassociation by the MOT trapping light \\cite{Fioretti1998}. Experimental studies in $^{85}$Rb have concluded that the former mechanism dominates at low trapping light intensities \\cite{Gabbanini2000} but that the latter plays a more prominent role at higher intensities \\cite{Caires2005}. \n\nIt is possible that these preformed bound ground-state molecules play an important role in the pump-probe dynamics despite their paucity in comparison to unbound atom pairs, since the low atomic density within the MOT implies a low probability for two neighbouring atoms to interact. A precise characterization of the initial state of the interacting atom pairs is therefore essential for meaningful subsequent numerical simulation of the pump-probe process. The formation mechanism effected by the trapping light involves photoassociation predominantly to the \\mbox{5S + 5P$_{3\/2}$ $0_{\\textrm{g}}^-$} potential followed by spontaneous decay to high-lying states within the $a^3\\Sigma_u^+$ ground triplet potential (see Fig.\\ \\ref{fig:init_schematic}). This photoassociative mechanism is expected to dominate three-body recombination within our setup, as our experimental parameters more closely match those of Ref.\\ \\cite{Caires2005}. A comparison of the measured ground-state occupancy to the calculated population distribution resulting from this decay channel enables this hypothesis to be confirmed. \n\nThe initial distribution of bound ground-state triplet molecules was studied via resonant multi-photon ionization (REMPI). A two-photon process preferentially ionized the Rb$_2$ via a resonant intermediate potential, while atomic ionization followed an off-resonant pathway and required an extra photon. A common REMPI pathway for detection of ground triplet Rb$_2$ employs loosely bound vibrational states within the $(2)^3\\Pi_g$ potential as an intermediate \\cite{Gabbanini2000}. Since the vibrational level spacings of these initial and intermediate states are comparable, it is not easy to unravel the initial distribution from a spectroscopic measurement via this pathway. Instead, this study uses an alternative REMPI route that passes through more deeply bound levels within the $(2)^3\\Sigma_g^+$ potential (see Fig.\\ \\ref{fig:init_schematic}) \\cite{Lozeille2006}. The resultant spectroscopic scan enables distinction between the large energy spacing of the intermediate states and the close spacing of the ground-state triplet Rb$_2$ whose occupancy it is desired to study.\n\nAn estimation of the population distribution of molecules photoassociated by the MOT lasers was made by considering the Franck--Condon overlap of high-lying $0_{\\textrm{g}}^-$ states with each $a^3\\Sigma_u^+$ vibrational state. The expected resultant spectrum for REMPI ionization via the $(2)^3\\Sigma_g^+$ was calculated using the excited-state vibrational level energies given in Ref.\\ \\cite{Lozeille2006}, together with the corresponding transition Franck--Condon factors. A realistic bandwidth for the REMPI laser was assumed.\n\nThe light for the REMPI pathway outlined above was generated with a dye laser (using Pyridine 1 dye) that was pumped by the second harmonic of a Q-switched Nd:YAG laser. Pulses of \\unit[10]{ns} duration and up to \\unit[400]{$\\mu$J} energy were produced at a repetition rate of \\unit[50]{Hz}. The pulse wavelength was tunable between \\unit[685]{nm} and \\unit[705]{nm}, and was calibrated with reference to two-photon atomic transitions to the 4p$^6$6d configuration.\n\n\n\\section{Experimental results}\n\\label{sec:results}\n\n\\subsection{Bound initial state study}\n\\label{sec:initial-state}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Figure5.pdf}\n  \\caption{(Colour online). Spectroscopy of the initial state. High-lying vibrational states within the $a^3\\Sigma_u^+$ ground triplet formed through MOT laser photoassociation are ionized using a REMPI pathway through the $(2)^3\\Sigma_g^+$ potential. The Rb$_2^+$ signal is shown as a function of ionization energy. (a) The full experimental spectrum reveals the large vibrational level spacing within the $(2)^3\\Sigma_g^+$ potential. The atomic resonance used for calibration is indicated. The overall spectrum is modulated by the tuning range of the ionization laser. (b) A close-up plot of the spectral feature at \\unit[14500]{cm$^{-1}$} reveals the initial population distribution within the $a^3\\Sigma_u^+$. A close agreement is found with simulations of the population distribution obtained by MOT photoassociation to the \\mbox{5S + 5P$_{3\/2}$ $0_{\\textrm{g}}^-$} potential. The locations of the highest-lying $a^3\\Sigma_u^+$ vibrational levels are shown and enumerated according to increasing binding energy.}\n  \\label{fig:init-state}\n\\end{figure}\n\nFigure \\ref{fig:init-state}(a) shows the result of a spectroscopic scan of the REMPI ionization signal from the high-lying ground vibrational states populated via background mechanisms. The full spectrum reveals a repeating structure at intervals of \\unit[50]{cm$^{-1}$} that corresponds to the spacing of the vibrational levels within the $(2)^3\\Sigma_g^+$ excited state (as recorded in Fig.\\ 6 of Lozeille et al.\\ \\cite{Lozeille2006}). The detected Rb$_2^+$ signal strength is modulated by the available ionization laser pulse energy as the wavelength is tuned.\n\nAs discussed in Section \\ref{sec:REMPI}, the repeated structure within the spectrum reveals the initial population distribution within the $a^3\\Sigma_u^+$ ground-state occupancy. A close-up of the spectral feature at \\unit[14500]{cm$^{-1}$} is shown in Fig.\\ \\ref{fig:init-state}(b), together with the calculated population distribution formed via MOT laser photoassociation to the $0_{\\textrm{g}}^-$ potential followed by spontaneous decay. The greatest population is found to be contained within the ninth-highest vibrational level, which corresponds to the level with the greatest Franck--Condon overlap from the excited state populated by the MOT light. Around ten bound states are found to be populated in total.\n\nThe experimental spectrum is in close agreement with the calculated population distribution formed by MOT laser photoassociation via the \\mbox{5S + 5P$_{3\/2}$ $0_{\\textrm{g}}^-$} potential. This agreement confirms that this, rather than three-body recombination, is the dominant mechanism for background molecule formation within this experimental setup. The detection of a strong background triplet molecule signal implies that this bound initial population, additionally to unbound scattering atom pairs, is a candidate for contribution to the photoassociation dynamics. This is analyzed further in Section \\ref{sec:bound-dimers}.\n\n\\subsection{Numerical pump-probe simulations}\n\\label{sec:simulations}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics [width=\\columnwidth]{Figure6.pdf}\n  \\caption{(Colour online). Simulations of the time- and position-dependent excited-state population density induced by the pump pulse. Coherent dynamics occur at a range of internuclear separations (represented by the dark shaded areas) despite the incoherence of the initial states. Results are shown for transitions from an unbound initial population to both gerade (a) and ungerade (b) excited states, as well as from a bound initial population to gerade excited states only (c) (single photon transitions to ungerade states are prohibited from the  $a^3\\Sigma_u^+$ potential). }\n  \\label{fig:dynamics}\n\\end{figure}\n\nNumerical simulations were conducted of the excited-state dynamics induced by the pump pulse, with consideration given to both the initial-state scenarios discussed above. The time-dependent Schr\\\"{o}dinger equation was solved for two different initial density matrices: a thermal distribution of scattering states, and the incoherent mixture of bound \\mbox{5S + 5P$_{3\/2}$ $0_{\\textrm{g}}^-$} decay products. The details of the coupled-channel model and the numerical implementation are as discussed in Ref.\\ \\cite{Martay2009}, but with the pump pulse selected to match Fig.\\ \\ref{fig:pulses}(b). The time- and position-dependent population density for both initial density matrices is shown in Fig.\\ \\ref{fig:dynamics}. The dynamics due to transitions from the unbound initial state to excited states of both gerade and ungerade symmetries are shown separately; however dipole selection rules prohibit the population of ungerade excited states from the bound $a^3\\Sigma_u^+$ initial state. The excited-state population exhibits a time-varying internuclear separation distribution, and so a position-sensitive measurement should observe a time-varying signal. Close-range, few-picosecond timescale dynamics are predicted for the bound initial state, compared to slower and longer-range dynamics from the unbound initial state.\n\nThe time-variation is caused by two effects. Firstly, the population density matrix that describes a time-independent distribution in the ground potential, no longer describes a time-independent distribution in the excited states since different potential energy curves govern its motion. The thermal distribution has a nodal structure that, when transferred to the excited state, forms distinct pockets of population that fall inwards. The anharmonicity of the excited potential at large internuclear separation precludes harmonic motion, but coherent interference effects still cause time-varying oscillations in the population at any internuclear separation. The second effect is that the pump-pulse bandwidth and transition Franck--Condon factors only transfer population in a narrow window in internuclear separation. This further improves the coherence of the excited-state population.\n\nThe simulations show that the incoherence of the initial state, whether from thermally distributed scattering states or from preassociated molecules, is not a barrier to the formation of coherently oscillating excited-state population by the pump pulse. The subsequent action of the ionizing probe pulse is less well theoretically understood due to the scope for the ejected electron to carry surplus energy and momentum. Due to the absence of a suitable experimentally verified model for molecular ionization, the probe pulse has been omitted from these calculations. The experimental probe pulse described in Section \\ref{sec:ultrafast} is sufficiently short to resolve these dynamics provided that it permits the requisite position-selective measurement.\n\n\n\n\n\\subsection{Rubidium pump-probe dynamics}\n\\label{sec:pump-probe}\n\n\\begin{figure}\n  \\centering\n   \\includegraphics[width=0.9\\columnwidth]{Figure7.pdf}  \n  \\caption{(Colour online). Typical atomic and molecular ion signals as a function of pump-probe delay (main figure), together with a typical molecular pump-probe signal at larger delays (inset). The atomic ion signal has been scaled for comparison. Both atomic and molecular signals exhibit a step at zero delay and a peak at small positive delays due to transient effects induced by the pump pulse. Fourier analysis of the subsequent signal reveals no dominant oscillatory timescales.}\n  \\label{fig:pu-pr}\n\\end{figure}\n\nFigure \\ref{fig:pu-pr} shows typical atomic and molecular pump-probe signals as a function of pump-probe delay. A low pump-pulse energy of \\unit[85]{nJ} was used in order to allow atomic and molecular dynamics to be monitored simultaneously without excessive atomic ionization. The inset also shows a typical molecular pump-probe signal at longer delays for the higher pulse energy of \\unit[400]{nJ}. The absolute Rb$^+$ and Rb$_2^+$ signal scalings employed in the plots are arbitrary and have been chosen to take a similar magnitude. Both the Rb$^+$ and Rb$_2^+$ experimental signals show a distinct step at zero delay: the signals are greater for a pump-probe sequential pulse order, though significant off-resonant atomic ionization is still evident with probe-pump timings (Fig.\\ \\ref{fig:pu-pr}(a)). In the Rb$_2$ pump-probe experiment of Salzmann et al.\\ \\cite{Salzmann2008}, the sharp spectral cut applied to the pump pulse introduced a long tail into the pump-pulse envelope in the corresponding temporal domain. In exciting an atom pair, the pump pulse induces an electronic dipole which interacts coherently with this long temporal tail, causing oscillations in the excited-state population and hence the Rb$_2^+$ signal. This behaviour is similar to the observations of Zamith et al.\\ in a Rb vapour cell \\cite{Zamith2001}. These dipole transient effects are not apparent in the long-timescale pump-probe signals of this work due to the employment of a less sharp spectral cut of the pump pulse.\n\n\nFourier analysis of the positive-delay signal that follows the coherent transient peak does not reveal any dominant characteristic oscillatory periods. This is in contrast with the picosecond-timescale coherent excited-state dynamics predicted by the simulations (which the \\unit[400]{fs} probe pulse was short enough to resolve), particularly for the case of the bound initial state. The calculations of Section \\ref{sec:simulations} revealed a wide range of conditions under which coherent oscillations may be formed. It therefore seems likely that the absence of such oscillations in the pump-probe signal can be attributed to an insufficiently position-dependent measurement being effected by the probe, for which there are two likely causes. Firstly, the ionization mechanism of the probe pulse is not as well understood as for neutral-neutral transitions (as outlined in Section \\ref{sec:simulations}). In order to attain a sufficiently short pulse to resolve the dynamics, a large bandwidth is required which may therefore compromise selectivity with respect to internuclear separation. Other indirect mechanisms, such as probe-pulse excitation to neutral Rydberg states that couple to the Rb$_2^+$ potential, may also complicate the process. Secondly, the detection mechanism may not be sensitive to very long-range excited-state population. Further information is thus required concerning the nature of the initial population addressed by the pump pulse. These issues are further discussed below in Section \\ref{sec:bound-dimers}.\n\n\\subsection{Bound excited-state dimers}\n\\label{sec:bound-dimers}\n\n\\begin{figure}\n  \\centering\n   \\includegraphics[width=0.9\\columnwidth]{Figure8.pdf}\n  \\caption{(Colour online). (a) Normalized atomic and molecular signals measured at a fixed pump-probe delay for different detunings of the pump-pulse spectral cut from the D1 line. The shaded area indicates evidence for the formation of bound excited-state dimers. (b) Normalized theoretical calculations of the excited-state population transfer induced by a pump pulse as a function of spectral cut detuning (red dashed lines). The different series represent the behaviour of pairs of atoms of different initial separations within the $a^3\\Sigma_u^+$ ground state (labelled in atomic units). The experimental molecular ion signal from (a) is plotted together with an optimized theoretical fit for a distribution of initial separations around \\unit[50]{a$_0$}.}\n  \\label{fig:bound_mols}\n\\end{figure}\n\nThe Rb$_2^+$ ion signal did not appreciably decay over the delays of hundreds of picoseconds that these experiments considered. In order to demonstrate that this large-delay signal is attributable to bound molecular population within the excited-state manifold (rather than further coherent transient effects or photoassociation of an unbound 5S and 5P atom to bound molecular ions), a further study was performed. A fixed, positive pump-probe delay of \\unit[40]{ps} was selected and the atomic and molecular ion signals were recorded as a function of the position of the pump-pulse spectral cut.\n\nAs the pump-pulse spectral cut was further detuned from the \\mbox{5S + 5P$_{1\/2}$} atomic asymptote (increasing the binding energy of the energetically accessible excited states), the Rb$^+$ signal was observed to fall off more quickly than the Rb$_2^+$ signal (\\mbox{Fig.\\ \\ref{fig:bound_mols}(a)}). Indeed, atomic ion formation became negligible at detunings beyond \\unit[10]{cm$^{-1}$}, while a substantial number of molecular ions continued to be formed. The fast decline in Rb$^+$ signal is caused by the loss of spectral intensity resonant with the atomic transition. Hence less atomic population is excited and subsequently ionized. The Rb$_2^+$ signal persists, however, due to excitation into bound vibrational levels below the atomic asymptote. The difference between the Rb$^+$ and Rb$_2^+$ data series (shaded) therefore provides strong evidence for the formation of bound molecular wavepackets by the pump pulse.\n\nIn order to study the initial state that contributed to this pump-probe signal, theoretical calculations were performed of the excited-state population transfer induced by a realistic experimental pump pulse as a function of spectral cut detuning (\\mbox{Fig.\\ \\ref{fig:bound_mols}(b)}), and repeated for pairs of atoms with a range of initial separations. The asymptotic long-range form of the excited-state potentials scales as $R^{-3}$ due to dipole-dipole interactions, whereas the ground-state potentials scale as $R^{-6}$ due to van der Waals interactions. The pump-pulse transition energy therefore becomes more red-detuned from the atomic asymptote at closer range. As a consequence, these simulated population transfers fall off more slowly with spectral cut detuning for closer range initial populations.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.9\\columnwidth]{Figure9.pdf}\n  \\caption{(Colour online). The population distributions of the bound background molecules characterized in Section \\ref{sec:initial-state} and a \\unit[100]{$\\mu$K} scattering state, together with an analytical best fit of the population that contributes to the pump-probe signal. The contributing population is of longer range than the background molecules, but is consistent with the association of unbound scattering atoms by the pump pulse. Dissociation of molecular ions by the TOF electric field is calculated to account for the lack of contribution of very long-range scattering atom pairs to the Rb$_2^+$ signal.}\n  \\label{Pu-pr_init}\n\\end{figure}\n\nThe atom pairs contributing to this pump-probe process will realistically be distributed over a range of internuclear separations rather than located at a single point; indeed, the fact that the experimental data span a number of these theoretical series corroborates this picture. Excellent agreement was found when the initial population distribution was fitted to the experimental data (\\mbox{Fig.\\ \\ref{fig:bound_mols}(b)}). Several analytical forms of the initial distributions were found to fit the data closely; all shared the common characteristic that population was concentrated at internuclear separations of \\unit[30-60]{a$_0$} (see \\mbox{Fig.\\ \\ref{Pu-pr_init}}). By contrast, the outer turning point of the most populated triplet vibrational state according to the study of Section \\ref{sec:initial-state} is located at \\unit[25]{a$_0$}. This provides evidence that the population addressed by the pump and probe pulses is different from the background triplet distribution photoassociated by the MOT light. Instead, the inferred distribution is consistent with the association of unbound atoms, coinciding with several antinodes of the scattering wavefunction. In this case, the detuning of the probe pulse from the Rb$^+$ asymptote might account for the lack of a contribution from the very long-range portion of the scattering wavefunction. As an alternative explanation, our calculations indicate that the TOF electric field is sufficient to dissociate molecular ions (causing \\mbox{$\\textrm{Rb}_2^+ \\rightarrow \\textrm{Rb} ^+ + \\textrm{Rb}$}) of longer range than around \\unit[90]{a$_0$}.\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nWe have performed a spectroscopic measurement of the initial state of the preformed molecules within our ultracold $^{85}$Rb$_2$ photoassociation experiments. The twin mechanisms of three-body recombination and trapping light photoassociation are known to contribute to a background population of loosely bound ground-state triplets within the MOT with relative impacts that are determined by the particular experimental conditions. Our measurement of the initial ground triplet-state occupancy reveals a distribution that closely matches the photoassociative mechanism, confirming this to be the dominant contribution in this case.\n\nWe have carried out a pump-probe study of the excited-state dynamics of Rb$_2$ dimers as the first step towards a fully coherent pump-dump transfer to the ground state. An ultrafast pump pulse was spectrally filtered to populate bound excited states only, and the resultant wavepacket was allowed to evolve for a controllable delay before being ionized with a probe pulse. By observing the large-delay behaviour of the pump-probe signal after the influence of initial transient effects, we have shown the formation of excited-state population that is demonstrably bound in character.\n\nWe have characterized the population that contributes to the pump-probe signal and have found that its behaviour with respect to spectral cut location is consistent with the association of unbound scattering atom pairs, rather than the excitation of background triplet molecules. Calculations indicated that the bulk of the excited-state dynamics induced from such an unbound initial state would occur at larger internuclear separations than are resonantly addressed by the probe pulse. This may therefore be a factor in the lack of experimental observation of pump-probe dynamics.\n\nThe ultimate goal, however, remains the coherent control of this population so as to maximize the Franck-Condon overlap to the dump-pulse target state after a suitably chosen delay; a signal presenting evidence of wavepacket oscillations is therefore required. Simulations reveal that pre-associated molecules in an incoherent mixture of states can be made to oscillate coherently with \\unit[5]{ps} timescales at relatively close range using a single ultrafast pump pulse \\cite{Martay2009}. By contrast, the classical oscillation period of molecules photoassociated from scattering atom pairs is greater by an order of magnitude or more, and occur predominantly at long range where detection is particularly difficult.\nIncreasing the signal-to-noise size at the larger pump-pulse detunings that address the relatively close-range pre-formed molecules should therefore allow this coherent behaviour to be revealed and controlled.\n\nSince the photoassociation rate scales in proportion to the product of the density and the number of trapped atoms, we identify these two parameters as targets for an improvement to our experimental apparatus.  \\mbox{Figure \\ref{fig:bound_mols}(b)} indicates that an order-of-magnitude improvement in photoassociation rate would provide a sufficient signal-to-noise ratio to probe the background molecular population located around \\unit[30]{a$_0$}. To this end we have trialled a dark spontaneous-force optical trap (SPOT) MOT \\cite{Ketterle1993} in order to improve density. This protocol is difficult to implement effectively in our existing Rb MOT, however, as the low branching ratio in Rb to the dark state requires high extinction of the repumper, and we consequently saw no improvement in signal. Instead, we are constructing a new MOT apparatus with which we expect to obtain this necessary improvement in atom number and density.\n\n\\section{Conclusion}\n\\label{sec:conclusions}\n\nThere is much current research interest devoted to the preparation of translationally ultracold molecules in their absolute ground electronic and rovibrational states. A broad range of strategies is at the disposal of the experimentalist in the pursuit of this goal. Some significant recent advances have used the approach of associating ultracold atoms, with the experimental realization of $v=0$ dimers in their absolute ground electronic states via techniques such as STIRAP or incoherent optical cycling. These breakthroughs for certain specific species are complemented by coherent control strategies, which are expected to be broadly applicable to a range of systems without the need for fortuitous initial bound-state preparation pathways, a detailed knowledge of the system spectroscopy or incoherent transfer processes.\n\nIn this work, we have demonstrated the formation of bound excited-state dimers by an ultrafast photoassociation pulse. This represents an important milestone in the design of a coherent pump-dump transfer strategy. We have determined the initial state addressed by the pump pulse and demonstrated that it predominantly comprises longer-range initial population than those background molecules pre-associated by the trapping light. We have identified improvements in MOT number and density as being critical to allow the detection and control of the coherent excited-state dynamics of closer-range population.\n\n\\begin{acknowledgments}\nThe authors are grateful to Jordi Mur-Petit for valuable discussions and suggestions. \\mbox{H.\\ E.\\ L.\\ M.\\ }acknowledges help from Thorsten K\\\"{o}hler. This work was supported by EPSRC grant number EP\/D002842\/1, Marie Curie Initial Training Network CA-ITN-214962-FASTQUAST, and Bulgarian NSF grant number WU-301\/07.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe {\\em Olrov spectrum} of a triangulated category is introduced by Orlov \\cite{O} based on the works of Bondal and Van den Bergh \\cite{BV} and Rouquier \\cite{R}.\nThis categorical invariant is the set of finite generation times of objects of the triangulated category.\nThe generation time of an object is the number of exact triangles necessary to build the category out of the object, up to finite direct sums, direct summands and shifts.\nIn \\cite{R}, Rouquier studies the infimum of the Orlov spectrum, which is called the {\\em Rouquier dimension} of the triangulated category.\nFor more information on Orlov spectra and Rouquier dimensions, we refer the reader to \\cite{BFK, O, R} for instance.\n\nThe Orlov spectrum and Rouquier dimension measure how big a triangulated category is, but it is basically rather hard to calculate them.\nThus, the notions of a level \\cite{ABIM} and a relative Rouquier dimension \\cite{AAITY} are introduced to measure how far one given object\/subcategory from another given object\/subcategory.\nThe main purpose of this paper is to report on these two invariants for the singularity category of a hypersurface of countable (Cohen--Macaulay) representation type.\n\nLet $k$ be an uncountable algebraically closed field of characteristic not two, and let $R$ be a complete local hypersurface over $k$ with countable representation type.\nDenote by $\\operatorname{\\mathsf{D_{sg}}}(R)$ the singularity category of $R$, that is, the Verdier quotient of the bounded derived category of finitely generated $R$-modules by the perfect complexes.\nLet $\\operatorname{\\mathsf{D_{sg}^o}}(R)$ be the full subcategory of $\\operatorname{\\mathsf{D_{sg}}}(R)$ consisting of objects that are zero on the punctured spectrum of $R$.\nThe main results of this paper are the following two theorems; we should mention that (1) and (2a) of Theorem \\ref{b} are essentially shown in the previous papers \\cite{AIT,T3} of some of the authors of the present paper.\n\n\\begin{thm}\\label{a}\nFor all nonzero objects $M$ of $\\operatorname{\\mathsf{D_{sg}}}(R)$ one has\n$$\n\\operatorname{level}_{\\operatorname{\\mathsf{D_{sg}}}(R)}^M(k)\\le1,\n$$\nthat is, $k$ belongs to ${\\langle M\\rangle}_2^{\\operatorname{\\mathsf{D_{sg}}}(R)}$.\n\\end{thm}\n\n\n\\begin{thm}\\label{b}\nLet $\\mathcal{T}$ be a nonzero thick subcategory of $\\operatorname{\\mathsf{D_{sg}}}(R)$, and let $\\mathcal{X}$ be a full subcategory of $\\mathcal{T}$ closed under finite direct sums, direct summands and shifts.\nThen the following statements hold.\n\\begin{enumerate}[\\rm(1)]\n\\item\n$\\mathcal{T}$ coincides with either $\\operatorname{\\mathsf{D_{sg}}}(R)$ or $\\operatorname{\\mathsf{D_{sg}^o}}(R)$.\n\\item\n\\begin{enumerate}[\\rm(a)]\n\\item\nIf $\\mathcal{T}=\\operatorname{\\mathsf{D_{sg}}}(R)$, then\n$$\n\\dim_\\mathcal{X}\\mathcal{T}=\n\\begin{cases}\n0 & (\\text{if } \\mathcal{X} =\\mathcal{T}),\\\\\n1 & (\\text{if } \\mathcal{X} \\neq \\mathcal{T},\\, \\mathcal{X} \\nsubseteq \\operatorname{\\mathsf{D_{sg}^o}}(R)),\\\\\n\\infty & (\\text{if }\\mathcal{X} \\subseteq \\operatorname{\\mathsf{D_{sg}^o}}(R)).\n\\end{cases}\n$$\n\\item\nIf $\\mathcal{T}=\\operatorname{\\mathsf{D_{sg}^o}}(R)$, then\n$$\n\\dim_\\mathcal{X}\\mathcal{T}=\n\\begin{cases}\n0 & (\\text{if } \\mathcal{X}  =\\mathcal{T}),\\\\\n1 & (\\text{if } \\mathcal{X} \\neq \\mathcal{T},\\,\\#\\operatorname{\\mathsf{ind}} \\mathcal{X}=\\infty),\\\\\n\\infty & (\\text{if }\\#\\operatorname{\\mathsf{ind}} \\mathcal{X} <\\infty).\n\\end{cases}\n$$\n\\end{enumerate}\n\\end{enumerate}\n\\end{thm}\n\nAs a consequence of Theorem \\ref{b}, we get the Orlov spectra and Rouquier dimensions of $\\operatorname{\\mathsf{D_{sg}}}(R)$ and $\\operatorname{\\mathsf{D_{sg}^o}}(R)$.\nNote that the equalities of Rouquier dimensions are already known to hold \\cite{AIT,DT}.\n\n\\begin{cor}\\label{c}\nIt holds that\n$$\n\\operatorname{OSpec}\\operatorname{\\mathsf{D_{sg}}}(R)=\\{1\\},\\qquad\n\\operatorname{OSpec}\\operatorname{\\mathsf{D_{sg}^o}}(R)=\\emptyset.\n$$\nIn particular, $\\dim \\operatorname{\\mathsf{D_{sg}}}(R)=1$ and $\\dim\\operatorname{\\mathsf{D_{sg}^o}}(R)=\\infty$.\n\\end{cor}\n\nThe organization of this paper is as follows.\nIn Section 2, we give the definitions of the Orlov spectrum and the (relative) Rouquier dimension of a triangulated category.\nWe also recall the theorem of Buchweitz providing a triangle equivalence for a Gorenstein ring $R$ between the singularity category of $R$ and the stable category of maximal Cohen--Macaulay $R$-modules.\nIn Section 3, we investigate the Orlov spectra and Rouquier dimensions of the singularity categories of hypersurfaces and their double branched covers to reduce to the case of Krull dimension one.\nIn Section 4, using the results obtained in the previous sections together with the classification theorem of maximal Cohen--Macaulay modules over a hypersurface of countable representation type, we prove our main theorems stated above.\n\n\\begin{conv}\nThroughout this paper, all subcategories are assumed to be full.\nWe often omit subscripts and superscripts if there is no risk of confusion.\n\\end{conv}\n\\section{Preliminaries}\n\nWe recall the definitions of several basic notions which are used in the later sections.\n\n\\begin{nota}\nLet $\\mathcal{T}$ be a triangulated category.\n\\begin{enumerate}[(1)]\n\\item\nFor a subcategory $\\mathcal{X}$ of $\\mathcal{T}$ we denote by $\\langle \\mathcal{X} \\rangle$ the smallest subcategory of $\\mathcal{T}$ containing $\\mathcal{X}$ which is closed under isomorphisms, shifts, finite direct sums and direct summands.\n\\item\nFor subcategories $\\mathcal{X}, \\mathcal{Y}$ of $\\mathcal{T}$ we denote by $\\mathcal{X} * \\mathcal{Y}$ the subcategory consisting of objects $M \\in \\mathcal{T}$ such that there is an exact triangle $X\\to M\\to Y \\to X[1]$ with $X\\in \\mathcal{X}$ and $Y\\in \\mathcal{Y}$.\nSet $\\mathcal{X} \\diamond \\mathcal{Y} := \\langle\\langle\\mathcal{X}\\rangle * \\langle\\mathcal{Y}\\rangle\\rangle$.\n\\item\nFor a subcategory $\\mathcal{X}$ of $\\mathcal{T}$ we put ${\\langle \\mathcal{X}\\rangle}_0:=0$, ${\\langle \\mathcal{X}\\rangle}_1:=\\langle \\mathcal{X}\\rangle$, and inductively define ${\\langle \\mathcal{X}\\rangle}_n:= \\mathcal{X} \\diamond {\\langle \\mathcal{X}\\rangle}_{n-1}$ for $n\\geq 2$.\nWe set ${\\langle M \\rangle}_n := {\\langle \\{ M \\}\\rangle}_n$ for an object $M\\in \\mathcal{T}$.\n\\end{enumerate}\n\\end{nota}\n\n\\begin{dfn}\nLet $\\mathcal{T}$ be a triangulated category.\n\\begin{enumerate}[(1)]\n\\item\nThe {\\em generation time} of an object $M\\in\\mathcal{T}$ is defined by\n$$\n\\operatorname{gt}_{\\mathcal{T}}(M):=\\inf\\{ n\\geq 0\\mid\\mathcal{T}={\\langle M \\rangle}_{n+1} \\}.\n$$\nIf $\\operatorname{gt}_{\\mathcal{T}}(M)$ is finite, $M$ is called a {\\it strong generator} of $\\mathcal{T}$.\n\\item\nThe {\\it Orlov spectrum} and {\\em (Rouquier) dimension} of $\\mathcal{T}$ are defined as follows.\n\\begin{align*}\n\\operatorname{OSpec}(\\mathcal{T})&:=\\{ \\operatorname{gt}_{\\mathcal{T}}(M)\\mid M\\text{ is a strong generator of }\\mathcal{T}\\},\\\\\n\\dim\\mathcal{T}&:=\\inf\\operatorname{OSpec}(\\mathcal{T})=\\inf\\{n\\ge0\\mid\\mathcal{T}={\\langle M\\rangle}_{n+1}\\text{ for some }M\\in\\mathcal{T}\\}.\n\\end{align*}\n\\item\nLet $\\mathcal{X}$ be a subcategory of $\\mathcal{T}$.\nThe {\\em dimension} of $\\mathcal{T}$ {\\em with respect to} $\\mathcal{X}$ is defined by\n$$\n\\dim_\\mathcal{X}\\mathcal{T}:=\\inf\\{n\\ge0\\mid \\mathcal{T}={\\langle\\mathcal{X}\\rangle}_{n+1}\\}.\n$$\n\\item\nLet $M, N$ be objects of $\\mathcal{T}$.\nThen the {\\em level} of $N$ with respect to $M$ is defined by\n$$\n\\operatorname{level}_{\\mathcal{T}}^M(N):=\\inf\\{ n\\geq 0\\mid N \\in {\\langle M \\rangle}_{n+1} \\}.\n$$\n\\end{enumerate}\n\\end{dfn}\n\n\\begin{dfn}\nLet $R$ be a Noetherian ring.\n\\begin{enumerate}[(1)]\n\\item\nWe denote by $\\operatorname{\\mathsf{D^b}}(R)$ the bounded derived category of finitely generated $R$-modules.\n\\item\nA {\\em perfect} complex is by definition a bounded complex of finitely generated projective modules.\n\\item\nWe denote by $\\operatorname{\\mathsf{perf}}(R)$ the subcategory of $\\operatorname{\\mathsf{D^b}}(R)$ consisting of complexes quasi-isomorphic to perfect complexes.\n\\item\nThe {\\em singularity category} of $R$ is defined by\n$$\n\\operatorname{\\mathsf{D_{sg}}}(R):=\\operatorname{\\mathsf{D^b}}(R)\/\\operatorname{\\mathsf{perf}}(R),\n$$\nthat is, the Verdier quotient of $\\operatorname{\\mathsf{D^b}}(R)$ by $\\operatorname{\\mathsf{perf}}(R)$.\n\\end{enumerate}\n\\end{dfn}\n\nNote that every object of the singularity category $\\operatorname{\\mathsf{D_{sg}}}(R)$ is isomorphic to a shift of some $R$-module; see \\cite[Lemma 2.4]{sing}.\n\nLet $R$ be a Cohen--Macaulay local ring.\nLet $\\operatorname{\\mathsf{CM}}(R)$ be the category of maximal Cohen-Macaulay $R$-modules, and $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ the stable category of $\\operatorname{\\mathsf{CM}}(R)$.\nThe following theorem is celebrated and fundamental; see \\cite[Theorem 4.4.1]{B}.\n\n\\begin{thm}[Buchweitz]\\label{ragnar}\nLet $R$ be a Gorenstein local ring of Krull dimension $d$.\nThen $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ has the structure of a triangulated category with shift functor $\\Omega^{-1}$, and there exist mutually inverse triangle equivalence functors\n$$\nF:\\operatorname{\\mathsf{D_{sg}}}(R)\\rightleftarrows\\operatorname{\\mathsf{\\underline{CM}}}(R):G,\n$$\nsuch that $GM=M$ for each maximal Cohen--Macaulay $R$-module $M$ and $FN=\\Omega^dN[d]$ for each finitely generated $R$-module $N$.\n\\end{thm}\n\nBy virtue of Theorem \\ref{ragnar}, for a Gorenstein local ring, the study of generation in the singularity category reduces to the stable category of maximal Cohen--Macaulay modules.\n\n\\section{The relationship between the singularity categories of $R$ and ${R^\\sharp}$}\n\nLet $(R,\\mathfrak{m},k)$ be a complete equicharacteristic local hypersurface of (Krull) dimension $d$.\nThen thanks to Cohen's structure theorem we can identify $R$ with a quotient of a formal power series ring over $k$:\n$$\nR=k[\\![x_0,x_1,\\dots,x_d]\\!]\/(f)\n$$\nwith $0\\neq f \\in (x_0,x_1,\\dots,x_d)^2$. \nWe define a hypersurface of dimension $d+1$:\n$$\n{R^\\sharp}=k[\\![x_0,x_1,\\ldots,x_d, y]\\!]\/(f+y^2).\n$$\nNote that the element $y$ is ${R^\\sharp}$-regular and there is an isomorphism ${R^\\sharp}\/y{R^\\sharp}\\cong R$.\nThe main purpose of this section is to compare generation in the singularity categories $\\operatorname{\\mathsf{D_{sg}}}(R)$ and $\\operatorname{\\mathsf{D_{sg}}}({R^\\sharp})$.\nAs both $R$ and ${R^\\sharp}$ are Gorenstein, in view of Theorem \\ref{ragnar} and the remark following the theorem, it suffices to investigate the stable categories of maximal Cohen--Macaulay modules $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ and $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$.\n\nThe following result is a consequence of \\cite[Proposition 12.4]{Y}, which plays a key role to compare generation in $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ and $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$.\n\n\\begin{lem}\\label{dense}\nThe assignments $M\\mapsto\\Omega_{{R^\\sharp}}M$ and $N\\mapsto N\/yN$ define triangle functors $\\Phi :\\operatorname{\\mathsf{\\underline{CM}}}(R) \\to \\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$ and $\\Psi :\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp}) \\to \\operatorname{\\mathsf{\\underline{CM}}}(R)$ satisfying\n$$\n\\Psi\\Phi(M)\\cong M\\oplus M[1],\\qquad\n\\Phi\\Psi(N) \\cong N\\oplus N[1].\n$$\nIn particular, $\\Phi$ and $\\Psi$ are both equivalences up to direct summands.\n\\end{lem}\n\nApplying this lemma, we deduce relationships of levels in $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ and $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$.\n\n\\begin{prop}\\label{lev}\nOne has the following equalities.\n\\begin{enumerate}[\\rm(1)]\n\\item\n$\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^M(\\Omega_R^dk)=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Omega_{{R^\\sharp}} M}(\\Omega_{{R^\\sharp}}^{d+1}k)$ for each $M\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\n\\item\n$\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^N(\\Omega_{{R^\\sharp}}^{d+1}k)=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{N\/yN}(\\Omega_{R}^{d}k)$ for each $N\\in\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nWe use Lemma \\ref{dense} and adopt its notation.\nThere are (in)equalities\n\\begin{align*}\n\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^M(\\Omega_R^dk)\n\\ge\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Phi M}(\\Phi(\\Omega_R^dk))\n&\\ge\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{\\Psi\\Phi M}(\\Psi\\Phi(\\Omega_R^dk))\\\\\n&=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{M\\oplus M[1]}(\\Omega_R^dk\\oplus\\Omega_R^dk[1])\n=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^M(\\Omega_R^dk),\n\\end{align*}\nwhich show $\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^M(\\Omega_R^dk)=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Phi M}(\\Phi(\\Omega_R^dk))$.\nA similar argument gives rise to $\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^N(\\Omega_{{R^\\sharp}}^{d+1}k)=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{\\Psi N}(\\Psi(\\Omega_{R^\\sharp}^{d+1}k))$.\nThere are isomorphisms in $\\operatorname{\\mathsf{\\underline{CM}}}(R)$:\n\\begin{align}\\label{1e}\n\\Psi(\\Omega_{R^\\sharp}^{d+1}k)\n=\\Omega_{R^\\sharp}^{d+1}k\/y\\Omega_{R^\\sharp}^{d+1}k\n&\\cong\\Omega_{{R^\\sharp}\/y{R^\\sharp}}^{d+1}k\\oplus\\Omega_{{R^\\sharp}\/y{R^\\sharp}}^dk\\\\\n\\notag&\\cong\\Omega_R^{d+1}k\\oplus\\Omega_R^dk\n\\cong\\Omega_R^dk[1]\\oplus\\Omega_R^dk,\n\\end{align}\nwhere the first isomorphism follows from \\cite[Corollary 5.3]{T0}.\nApplying $\\Phi$, we obtain\n\\begin{equation}\\label{2e}\n\\Omega_{R^\\sharp}^{d+1}k\\oplus\\Omega_{R^\\sharp}^{d+1}k[1]\\cong\\Phi(\\Omega_R^dk)[1]\\oplus\\Phi(\\Omega_R^dk).\n\\end{equation}\nIt is observed from \\eqref{1e} and \\eqref{2e} that $\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{\\Psi N}(\\Psi(\\Omega_{R^\\sharp}^{d+1}k))=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{\\Psi N}(\\Omega_R^dk)$ and $\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Phi M}(\\Phi(\\Omega_R^dk))=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Phi M}(\\Omega_{R^\\sharp}^{d+1}k)$, respectively.\nConsequently we obtain\n\\begin{align*}\n\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^M(\\Omega_R^dk)\n&=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Phi M}(\\Phi(\\Omega_R^dk))\n=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Phi M}(\\Omega_{R^\\sharp}^{d+1}k)\n=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Omega_{R^\\sharp} M}(\\Omega_{R^\\sharp}^{d+1}k),\\\\\n\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^N(\\Omega_{{R^\\sharp}}^{d+1}k)\n&=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{\\Psi N}(\\Psi(\\Omega_{R^\\sharp}^{d+1}k))\n=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{\\Psi N}(\\Omega_R^dk)\n=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{N\/yN}(\\Omega_R^dk),\n\\end{align*}\nwhich completes the proof of the proposition.\n\\end{proof}\n\nUsing Lemma \\ref{dense} again, we get relationships of generation times in $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ and $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$.\n\n\\begin{prop}\\label{prop1}\nThe following statements hold true.\n\\begin{enumerate}[\\rm(1)]\n\\item\nIf $M \\in \\operatorname{\\mathsf{\\underline{CM}}}(R)$ is a strong generator, then so is $\\Omega_{R^\\sharp} M\\in\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$, and $\\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}(M) = \\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}(\\Omega_{R^\\sharp} M)$.\n\\item\nIf $N \\in \\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$ is a strong generator, then so is $N\/yN\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)$, and $\\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}(N) = \\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}(N\/yN)$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\n(1) We use Lemma \\ref{dense} and adopt its notation.\nPut $n=\\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}(M)$.\nBy definition, it holds that ${\\langle M\\rangle}_{n+1}=\\operatorname{\\mathsf{\\underline{CM}}}(R)\\ne{\\langle M\\rangle}_n$.\nWhat we need to prove is that ${\\langle\\Phi M\\rangle}_{n+1}=\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})\\ne{\\langle\\Phi M\\rangle}_n$.\nFor each $X\\in\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$ we have $\\Psi X\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)={\\langle M\\rangle}_{n+1}$, and $\\Phi\\Psi X\\in{\\langle\\Phi M\\rangle}_{n+1}$.\nSince $X$ is a direct summand of $\\Phi\\Psi X$, it belongs to ${\\langle\\Phi M\\rangle}_{n+1}$.\nTherefore, we get $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})={\\langle\\Phi M\\rangle}_{n+1}$.\nSuppose that the equality $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})={\\langle\\Phi M\\rangle}_n$ holds.\nTaking any $Y\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)$, we have $\\Phi Y\\in\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})={\\langle\\Phi M\\rangle}_n$, and $\\Psi\\Phi Y\\in{\\langle\\Psi\\Phi M\\rangle}_n$.\nAs $Y$ is a direct summand of $\\Psi\\Phi Y$, it is in ${\\langle\\Psi\\Phi M\\rangle}_n$.\nHence $\\operatorname{\\mathsf{\\underline{CM}}}(R)={\\langle\\Psi\\Phi M\\rangle}_n={\\langle M\\oplus M[1]\\rangle}_n={\\langle M\\rangle}_n$, which is a contradiction.\nThus $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})\\ne{\\langle\\Phi M\\rangle}_n$.\n\n(2) An analogous argument to the proof of (1) applies.\n\\end{proof}\n\nThe Orlov spectra and Rouquier dimensions of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ and $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$ coincide:\n\n\\begin{cor}\\label{cor}\nOne has the following equalities.\n$$\n\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}(R)=\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp}),\\qquad\n\\dim\\operatorname{\\mathsf{\\underline{CM}}}(R)=\\dim\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp}).\n$$\n\\end{cor}\n\n\\begin{proof}\nIt suffices to show the first equality, since the second equality follows by taking the infimums of the both sides of the first equality.\nUsing Proposition \\ref{prop1}(1), we obtain\n\\begin{align*}\n\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}(R)\n&=\\{\\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}(M)\\mid\\text{$M$ is a strong generator of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$}\\}\\\\\n&=\\{\\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}(\\Omega_{{R^\\sharp}}M)\\mid\\text{$\\Omega_{{R^\\sharp}}M$ is a strong generator of $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$}\\}\\\\\n&\\subseteq\\{\\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}(N)\\mid\\text{$N$ is a strong generator of $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$}\\}\n=\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp}).\n\\end{align*}\nA similar argument using Proposition \\ref{prop1}(2) shows the opposite inclusion $\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})\\subseteq\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\nWe thus conclude that $\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}(R)=\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$.\n\\end{proof}\n\n\\section{The singularity category of a hypersurface of countable representation type}\n\nIn this section, we prove our main results, that is, Theorems \\ref{B} and \\ref{A} from the Introduction.\nWe start by the following lemma on exact triangles in a triangulated category (an exact triangle $A\\to B\\to C\\to A[1]$ is simply denoted by $A\\to B\\to C\\rightsquigarrow$).\n\n\\begin{lem}\\label{ladder}\nLet $\\mathcal{T}$ be a triangulated category.\nLet\n$$\nX\\xrightarrow{{\\left(\\begin{smallmatrix}f_1\\\\f_2\\end{smallmatrix}\\right)}}Y_1\\oplus M\\xrightarrow{{\\left(\\begin{smallmatrix}g_1&\\alpha\\end{smallmatrix}\\right)}}N\\overset{p}\\rightsquigarrow,\\qquad\nM\\xrightarrow{{\\left(\\begin{smallmatrix}\\alpha\\\\g_2\\end{smallmatrix}\\right)}}N\\oplus Y_2\\xrightarrow{\\left(\\begin{smallmatrix}h_1&h_2\\end{smallmatrix}\\right)}Z\\overset{q}\\rightsquigarrow\n$$\nbe exact triangles in $\\mathcal{T}$.\nThen there exists an exact triangle in $\\mathcal{T}$ of the form\n$$\nX\\xrightarrow{\\left(\\begin{smallmatrix}f_1\\\\g_2f_2\\end{smallmatrix}\\right)}Y_1\\oplus Y_2\\xrightarrow{\\left(\\begin{smallmatrix}h_1g_1&-h_2\\end{smallmatrix}\\right)}Z\\rightsquigarrow.\n$$\n\\end{lem}\n\n\\begin{proof}\nThere is an isomorphism\n$$\n\\xymatrix@R=3pc@C=5pc{\nM\\ar[r]^-{\\left(\\begin{smallmatrix}\n0\\\\\n1\\\\\n0\n\\end{smallmatrix}\\right)}\\ar@{=}[d] & Y_1\\oplus M\\oplus Y_2\\ar[r]^-{\\left(\\begin{smallmatrix}\n1 & 0 & 0\\\\\n0 & 0 & 1\n\\end{smallmatrix}\\right)}\\ar[d]_{\\cong}^-{\\left(\\begin{smallmatrix}1&0&0\\\\0&1&0\\\\0&g_2&1\\end{smallmatrix}\\right)} & Y_1\\oplus Y_2\\ar[r]^-{\\left(\\begin{smallmatrix}\n0 & 0\n\\end{smallmatrix}\\right)}\\ar[d]_{\\cong}^-{\\left(\\begin{smallmatrix}1&0\\\\0&-1\\end{smallmatrix}\\right)} & M[1]\\ar@{=}[d] \\\\\nM\\ar[r]^-{\\left(\\begin{smallmatrix}\n0\\\\\n1\\\\\ng_2\n\\end{smallmatrix}\\right)} & Y_1\\oplus M\\oplus Y_2\\ar[r]^-{\\left(\\begin{smallmatrix}\n1 & 0 & 0\\\\\n0 & g_2 & -1\n\\end{smallmatrix}\\right)} & Y_1\\oplus Y_2\\ar[r]^-{\\left(\\begin{smallmatrix}0&0\\end{smallmatrix}\\right)} & M[1] \n}\n$$\nof sequences.\nThe first row is an exact triangle in $\\mathcal{T}$ since it is the direct sum of exact triangles arising from the identity maps of $M$ and $Y_1\\oplus Y_2$ (see \\cite[Proof of Corollary 1.2.7]{N}).\nHence the second row is an exact triangle in $\\mathcal{T}$ as well.\nWe have a commutative diagram\n$$\n\\xymatrix@R=3pc@C=5pc{\n& X\\ar@{=}[r]\\ar[d]^-{\\left(\\begin{smallmatrix}\nf_1\\\\\nf_2\\\\\n0\n\\end{smallmatrix}\\right)} & X\\ar@{..>}[d]^-{\\left(\\begin{smallmatrix}\ns\\\\\nt\n\\end{smallmatrix}\\right)}\\\\\nM\\ar[r]^-{\\left(\\begin{smallmatrix}\n0\\\\\n1\\\\\ng_2\n\\end{smallmatrix}\\right)}\\ar@{=}[d] & Y_1\\oplus M \\oplus Y_2\\ar[r]^-{\\left(\\begin{smallmatrix}\n1 & 0 & 0\\\\\n0 & g_2 & -1\n\\end{smallmatrix}\\right)}\\ar[d]^-{\\left(\\begin{smallmatrix}g_1&\\alpha&0\\\\0&0&1\\end{smallmatrix}\\right)} & Y_1\\oplus Y_2\\ar[r]^{\\left(\\begin{smallmatrix}0&0\\end{smallmatrix}\\right)}\\ar@{..>}[d]^-{\\left(\\begin{smallmatrix}u&v\\end{smallmatrix}\\right)} & M[1]\\ar@{=}[d]\\\\\nM\\ar[r]^-{\\left(\\begin{smallmatrix}\n\\alpha\\\\\ng_2\n\\end{smallmatrix}\\right)} & N \\oplus Y_2\\ar[r]^-{\\left(\\begin{smallmatrix}\nh_1 & h_2 \n\\end{smallmatrix}\\right)}\\ar[d]^-{\\left(\\begin{smallmatrix}p&0\\end{smallmatrix}\\right)} & Z\\ar[r]^-q\\ar@{..>}[d] & M[1]\\\\\n& X[1]\\ar@{=}[r] & X[1]\n}\n$$\nwhere the rows are exact triangles in $\\mathcal{T}$, and so is the left column since it is a direct sum of exact triangles (see \\cite[Proposition 1.2.1]{N}).\nUsing the octahedral axiom, we obtain the right column which is an exact triangle in $\\mathcal{T}$.\nThe diagram chasing shows that $\\left(\\begin{smallmatrix}s\\\\t\\end{smallmatrix}\\right)=\\left(\\begin{smallmatrix}f_1\\\\g_2f_2\\end{smallmatrix}\\right)$ and $\\left(\\begin{smallmatrix}u&v\\end{smallmatrix}\\right)=\\left(\\begin{smallmatrix}h_1g_1&-h_2\\end{smallmatrix}\\right)$.\nThus it is an exact triangle we want.\n\\end{proof}\n\nLet $(R,\\mathfrak{m},k)$ be a complete equicharacteristic local hypersurface of dimension $d$.\nAssume that $k$ is uncountable and has characteristic different from two, and that $R$ has {\\em countable (Cohen--Macaulay) representation type}, namely, there exist infinitely but only countably many isomorphism classes of indecomposable maximal Cohen--Macaulay $R$-modules. \nThen $f$ is either of the following; see \\cite[Theorem 14.16]{LW}.\n\\begin{align*}\n(\\mathrm{A}_\\infty) &:\\ x_0^2+x_2^2+\\cdots +x_d^2,\\\\\n(\\mathrm{D}_\\infty) &:\\ x_0^2x_1+x_2^2+\\cdots +x_d^2.\n\\end{align*}\nIn this case, all objects in $\\operatorname{\\mathsf{CM}}(R)$ are completely classified \\cite{BGS,BD,K}.\n\nNow we can state and prove the following result regarding levels in $\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\n\n\\begin{thm}\\label{B}\nLet $k$ be an uncountable algebraically closed field of characteristic not two.\nLet $R$ be a $d$-dimensional complete local hypersurface over $k$ of countable representation type.\nThen\n$$\n\\Omega_R^dk\\in{\\langle M\\rangle}_2^{\\operatorname{\\mathsf{\\underline{CM}}}(R)}\n$$\nfor all nonzero objects $M\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\nIn other words, $\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^M(\\Omega_R^dk)\\le1$.\n\\end{thm}\n\n\\begin{proof}\nProposition \\ref{lev}(2) reduces to the case $d=1$.\nThus we have the two cases:\n$$\n\\text{(1) }R=k[\\![x,y]\\!]\/(x^2),\\qquad\n\\text{(2) }R=k[\\![x,y]\\!]\/(x^2y).\n$$\n\n(1):\nThanks to \\cite[4.1]{BGS}, the indecomposable objects of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ are the ideals $I_n=(x,y^n)$ with $n\\in\\mathbb{Z}_{>0}\\cup\\{\\infty\\}$, where $I_{\\infty}:=(x)$.\nBy \\cite[6.1]{Sc} there exist exact triangles\n$$\nI_{n}\\to I_{n-1}\\oplus I_{n+1}\\to I_{n}\\rightsquigarrow\\quad(n\\in\\mathbb{Z}_{>0}),\n$$\nwhere $I_0:=0$.\nApplying Lemma \\ref{ladder}, we obtain exact triangles\n$$\nI_{n}\\to I_{1}\\oplus I_{2n-1}\\to I_{n}\\rightsquigarrow\\quad(n\\in\\mathbb{Z}_{>0}),\n$$\nand from \\cite[Proposition 2.1]{AIT} we obtain an exact triangle $I_{\\infty}\\to I_{1}\\to I_{\\infty}\\rightsquigarrow$.\nIt is observed from these triangles that $\\Omega k=I_1$ is in ${\\langle M\\rangle}_2$ for each nonzero object $M\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\n\n(2):\nUsing \\cite[4.2]{BGS}, we get a complete list of the indecomposable objects of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$:\n\\begin{align*}\n&X=R\/(x),\n\\ \\Omega_RX,\n\\ M_0^+=R\/(x^2),\n\\ M_0^-=\\Omega M_0^+,\\\\\n&M_n^+=\\mathrm{Coker}\\left(\\begin{smallmatrix}\nx&y^n\\\\\n0&-x\n\\end{smallmatrix}\\right),\n\\ M_n^-=\\Omega M_n^+,\n\\ N_n^+=\\mathrm{Coker}\\left(\\begin{smallmatrix}\nx&y^n\\\\\n0&-xy\n\\end{smallmatrix}\\right),\n\\ N_n^{-}=\\Omega N_n^{+} \\quad(n\\in\\mathbb{Z}_{>0}).\n\\end{align*}\nAccording to \\cite[(6.1)]{Sc}, for each $n\\in\\mathbb{Z}_{>0}$ there are exact triangles\n$$\nM_n^{\\pm}\\to N_{n+1}^{\\pm}\\oplus N_{n}^{\\mp}\\to M_n^{\\mp}\\rightsquigarrow,\\qquad\nN_n^{\\pm}\\to M_{n}^{\\pm}\\oplus M_{n-1}^{\\mp}\\to N_n^{\\mp}\\rightsquigarrow,\n$$\nwhere $N_0^{\\pm}:=0$.\nLemma \\ref{ladder} gives rise to exact triangles\n$$\nM_n\\to N_1 \\oplus N_{2n} \\to M_n\\rightsquigarrow,\\qquad\nN_n\\to N_1 \\oplus N_{2n-1} \\to N_n\\rightsquigarrow,\n$$\nwhere $M_n$ stands for either $M_n^+$ or $M_n^-$, and so on.\nAlso, by \\cite[Proposition 2.1]{AIT} we get an exact triangle $\\Omega X\\to N_{1}^-\\to X\\rightsquigarrow$.\nThus $\\Omega k=N_1^-$ is in ${\\langle C\\rangle}_2$ for any $0\\ne C\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\n\\end{proof}\n\n\\begin{proof}[\\bf Proof of Theorem \\ref{a}]\nThe assertion is immediate from Theorems \\ref{B} and \\ref{ragnar}.\n\\end{proof}\n\nThe following example shows that Theorem \\ref{a} does not necessarily hold if one replaces $k$ with another nonzero object of the singularity category.\n\n\\begin{ex}\\label{3}\nLet $R=k[\\![x,y]\\!]\/(x^2)$ be a hypersurface over a field $k$.\nThen\n$$\n(x,y^a)R\\notin{\\left\\langle(x,y^b)R\\right\\rangle}_2^{\\operatorname{\\mathsf{D_{sg}}}(R)}\n$$\nfor all positive integers $a,b$ with $a>2b$.\nThus $\\operatorname{level}_{\\operatorname{\\mathsf{D_{sg}}}(R)}^{(x,y^b)R}((x,y^a)R)\\ge2$.\n\\end{ex}\n\n\\begin{proof}\nIn view of Theorem \\ref{ragnar} we replace $\\operatorname{\\mathsf{D_{sg}}}(R)$ with $\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\nLet $I=(x,y^a)R$ and $J=(x,y^b)R$ be ideals of $R$.\nSuppose that $I$ belongs to ${\\langle J\\rangle}_2^{\\operatorname{\\mathsf{\\underline{CM}}}(R)}$.\nSince $\\Omega J\\cong J$, we see that there exists an exact sequence $0 \\to J^{\\oplus m} \\to I\\oplus M \\to J^{\\oplus n} \\to 0$ of $R$-modules with $m,n\\ge0$.\nThis induces an exact sequence\n$$\n\\operatorname{Tor}_1^R(R\/I,J^{\\oplus m}) \\to \\operatorname{Tor}_1^R(R\/I,I)\\oplus\\operatorname{Tor}_1^R(R\/I,M) \\to \\operatorname{Tor}_1^R(R\/I,J^{\\oplus n}).\n$$\nSince $\\operatorname{Tor}_1^R(R\/I,J)\\cong\\operatorname{Tor}_2^R(R\/I,R\/J)$, the first and third Tor modules in the above exact sequence are annihilated by $J$, and hence\n\\begin{equation}\\label{1}\nJ^2\\operatorname{Tor}_1^R(R\/I,I)=0.\n\\end{equation}\nThe minimal free resolution of $R\/I$ is\n$$\nF=(\\cdots\\xrightarrow{\\left(\\begin{smallmatrix}\ny^a&x\\\\\n-x&0\n\\end{smallmatrix}\\right)}R^{\\oplus2}\\xrightarrow{\\left(\\begin{smallmatrix}\n0&-x\\\\\nx&y^a\n\\end{smallmatrix}\\right)}R^{\\oplus2}\\xrightarrow{\\left(\\begin{smallmatrix}\ny^a&x\\\\\n-x&0\n\\end{smallmatrix}\\right)}R^{\\oplus2}\\xrightarrow{(x,y^a)}R\\to0),\n$$\nwhich induces a complex\n$$\nF\\otimes_RR\/I=(\\cdots\\xrightarrow{0}(R\/I)^{\\oplus2}\\xrightarrow{0}(R\/I)^{\\oplus2}\\xrightarrow{0}(R\/I)^{\\oplus2}\\xrightarrow{0}R\/I\\to0).\n$$\nHence $\\operatorname{Tor}_1^R(R\/I,I)\\cong\\operatorname{Tor}_2^R(R\/I,R\/I)=(R\/I)^{\\oplus2}$.\nBy \\eqref{1} we have $J^2(R\/I)^{\\oplus2}=0$, which implies that $J^2$ is contained in $I$.\nTherefore the element $y^{2b}$ is in the ideal $(x,y^a)R$, but this cannot happen since $a>2b$.\n\\end{proof}\n\nRecall that a subcategory of a triangulated category is called {\\em thick} if it is a triangulated subcategory closed under direct summands.\nWe denote by $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ the subcategory of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ consisting of maximal Cohen--Macaulay $R$-modules that are free on the punctured spectrum of $R$.\nThe category $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ is a thick subcategory of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$, and in particular it is a triangulated category.\nFor a subcategory $\\mathcal{X}$ of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ we denote by $\\operatorname{\\mathsf{ind}}\\mathcal{X}$ the set of nonisomorphic indecomposable objects of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ that belong to $\\mathcal{X}$.\nWe can now state and prove the following result concerning relative Rouquier dimensions in $\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\n\n\n\\begin{thm}\\label{A}\nLet $k$ be an uncountable algebraically closed field of characteristic not two.\nLet $R$ be a $d$-dimensional complete local hypersurface over $k$ of countable representation type.\nLet $\\mathcal{T}\\ne0$ be a thick subcategory of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$, and let $\\mathcal{X}$ be a subcategory of $\\mathcal{T}$ closed under finite direct sums, direct summands and shifts.\nThen:\n\\begin{enumerate}[\\rm(1)]\n\\item\n$\\mathcal{T}$ coincides with either $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ or $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$.\n\\item\n\\begin{enumerate}[\\rm(a)]\n\\item\nWhen $\\mathcal{T}=\\operatorname{\\mathsf{\\underline{CM}}}(R)$, one has\n$$\n\\dim_\\mathcal{X}\\mathcal{T}=\n\\begin{cases}\n0 & (\\text{if } \\mathcal{X}=\\mathcal{T}),\\\\\n1 & (\\text{if } \\mathcal{X} \\neq \\mathcal{T}\\text{ and } \\mathcal{X} \\nsubseteq \\operatorname{\\mathsf{\\underline{CM}^o}}(R)),\\\\\n\\infty & (\\text{if } \\mathcal{X} \\subseteq \\operatorname{\\mathsf{\\underline{CM}^o}}(R)).\n\\end{cases}\n$$\n\\item\nWhen $\\mathcal{T}=\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$, one has\n$$\n\\dim_\\mathcal{X}\\mathcal{T}=\n\\begin{cases}\n0 & (\\text{if } \\mathcal{X}=\\mathcal{T}),\\\\\n1 & (\\text{if } \\mathcal{X}\\neq \\mathcal{T}\\text{ and }\\#\\operatorname{\\mathsf{ind}} \\mathcal{X} =\\infty),\\\\\n\\infty & (\\text{if }\\#\\operatorname{\\mathsf{ind}} \\mathcal{X}<\\infty).\n\\end{cases}\n$$\n\\end{enumerate}\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\n(1) We combine \\cite[Theorem 6.8]{T3} and \\cite[Theorem 1.1]{AIT}.\nThe singular locus of $R$ consists of two points $\\mathfrak{p}$ and $\\mathfrak{m}$, and its specialization-closed subsets are $\\operatorname{V}(\\mathfrak{p})$, $\\operatorname{V}(\\mathfrak{m})$ and $\\emptyset$.\nThese correspond to the thick subcategories $\\operatorname{\\mathsf{\\underline{CM}}}(R)$, $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ and $0$.\n\n(2) Part (a) follows from \\cite[Theorem 1.1]{AIT}.\nLet us show part (b).\nWhen $\\#\\operatorname{\\mathsf{ind}} \\mathcal{X}<\\infty$, let $X_1,\\dots,X_n$ be all the indecomposable objects in $\\mathcal{X}$.\nSuppose that $\\dim_\\mathcal{X}\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ is finite, say $m$.\nThen it follows that $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)={\\langle\\mathcal{X}\\rangle}_{m+1}={\\langle X\\rangle}_{m+1}$, where $X:=X_1\\oplus\\cdots\\oplus X_n\\in\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$.\nHence $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ has finite Rouquier dimension.\nBy \\cite[Theorem 1.1(2)]{DT}, the local ring $R$ has to have at most an isolated singularity.\nHowever, in either case of the types $(\\mathrm{A}_\\infty)$ and $(\\mathrm{D}_\\infty)$ we see that the nonmaximal prime ideal $(x_0,x_2,\\dots,x_d)R$ belongs to the singular locus of $R$, which is a contradiction.\nConsequently, we obtain $\\dim_\\mathcal{X}\\operatorname{\\mathsf{\\underline{CM}^o}}(R)=\\infty$.\n\nFrom now on we consider the case where $\\mathcal{X}\\ne\\mathcal{T}=\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ and $\\#\\operatorname{\\mathsf{ind}}\\mathcal{X}=\\infty$.\nWe adopt the same notation as in the proof of Theorem \\ref{B}.\n\nAssume that $R$ has type $(A_\\infty)$.\nAs $\\mathcal{X}$ is a proper subcategory, we can find a positive integer $n$ such that $I_n\\notin\\mathcal{X}$.\nSince there are infinitely many indecomposable objects in $\\mathcal{X}$, we can also find an integer $m>n$ such that $I_m\\in\\mathcal{X}$.\nThere exists an exact triangle\n$$\nI_{m}\\to I_{n}\\oplus I_{2m-n}\\to I_{m}\\rightsquigarrow\n$$\nin $\\mathcal{T}=\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$, which shows that $I_n$ belongs to ${\\langle \\mathcal{X}\\rangle}_2$.\nTherefore, we get $\\dim_\\mathcal{X}\\mathcal{T}\\le1$.\nSince $\\langle\\mathcal{X}\\rangle=\\mathcal{X}\\neq \\mathcal{T}$, we have $\\dim_\\mathcal{X}\\mathcal{T}\\ne 0$.\nConsequently, we obtain $\\dim_\\mathcal{X}\\mathcal{T}=1$.\n\nSuppose that $R$ is of type $(D_\\infty)$.\nSimilarly as above, we find two integers $m>n>0$ such that neither $M_n$ nor $N_n$ belongs to $\\mathcal{X}$ and either $M_m$ or $N_m$ is in $\\mathcal{X}$.\nWhen $M_m$ belongs to $\\mathcal{X}$, there are exact triangles\n$$\nM_{m}\\to M_{n}\\oplus M_{2m-n}\\to M_{m}\\rightsquigarrow,\\qquad\nM_{m}\\to N_{n}\\oplus N_{2m-n+1}\\to M_{m}\\rightsquigarrow.\n$$\nWhen $N_m$ is in $\\mathcal{X}$, we have exact triangles\n$$\nN_{m}\\to M_{n}\\oplus M_{2m-n-1}\\to N_{m}\\rightsquigarrow,\\qquad\nN_{m}\\to N_{n}\\oplus N_{2m-n}\\to N_{m}\\rightsquigarrow.\n$$\nIn either case, both $M_n$ and $N_n$ belong to ${\\langle \\mathcal{X}\\rangle}_2$.\nIt follows that $\\dim_\\mathcal{X}\\mathcal{T}\\le1$.\nAs $\\langle\\mathcal{X}\\rangle=\\mathcal{X}\\neq \\mathcal{T}$, we have $\\dim_\\mathcal{X}\\mathcal{T}\\ne 0$.\nNow we conclude $\\dim_\\mathcal{X}\\mathcal{T}=1$.\n\\end{proof}\n\n\\begin{proof}[\\bf Proof of Theorem \\ref{b}]\nTheorems \\ref{A} and \\ref{ragnar} immediately deduce the assertion.\n\\end{proof}\n\nAs a corollary of Theorem \\ref{A}, we calculate the Orlov spectra and Rouquier dimensions of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ and $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ for a hypersurface $R$ of countable representation type.\n\n\\begin{cor}\\label{x}\nLet $k$ be an algebraically closed uncountable field of characteristic not two. \nLet $R$ be a $d$-dimensional complete local hypersurface over $k$ having countable representation type.\nThen one has the following equalities.\n\\begin{enumerate}[\\rm(1)]\n\\item\n$\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}(R)=\\{1\\}$.\n\\item\n$\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}^o}}(R)=\\emptyset$.\n\\end{enumerate}\nIn particular, $\\dim\\operatorname{\\mathsf{\\underline{CM}}}(R)=1$ and $\\dim\\operatorname{\\mathsf{\\underline{CM}^o}}(R)=\\infty$.\n\\end{cor}\n\n\\begin{proof}\n(1) Applying Theorem \\ref{A}(2a) for $\\mathcal{X}=\\langle M\\rangle$, we see that it suffices to verify that\n\\begin{enumerate}[(a)]\n\\item\n$\\langle M\\rangle\\ne\\operatorname{\\mathsf{\\underline{CM}}}(R)$ for all $M\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)$, and\n\\item\n$\\langle M\\rangle\\ne\\operatorname{\\mathsf{\\underline{CM}}}(R)$ for some $M\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)\\setminus\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$.\n\\end{enumerate}\nStatement (a) is equivalent to saying that $\\dim\\operatorname{\\mathsf{\\underline{CM}}}(R)>0$, which follows from \\cite[Propositions 2.4 and 2.5]{radius}.\nThere exists an object $M\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)\\setminus\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ since $R$ does not have an isolated singularity, and hence (b) follows from (a).\n\n(2) Let $M\\in\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ be an object, and let $\\mathcal{X}=\\langle M\\rangle$ be a subcategory of $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$.\nThen it is observed that $\\#\\operatorname{\\mathsf{ind}}\\mathcal{X}<\\infty$, so applying Theorem \\ref{A}(2b), we obtain $\\dim_\\mathcal{X}\\operatorname{\\mathsf{\\underline{CM}^o}}(R)=\\infty$, which means $\\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}^o}}(R)}(M)=\\infty$.\nThus the assertion follows.\n\\end{proof}\n\n\\begin{proof}[\\bf Proof of Corollary \\ref{c}]\nCombining Corollary \\ref{x} with Theorem \\ref{ragnar} yields the assertion.\n\\end{proof}\n\n\\begin{ac}\nThe authors thank the referee for reading the paper carefully and giving useful suggestions.\n\\end{ac}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}