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+{"text":"\\section{\\bf{Introduction}}\n\nQED in $(1+1)$ dimension, e.g., Schwinger model \\cite{SCH} is a\nvery interesting field theoretical model. It has been widely\nstudied over the years by several authors in connection with the\nmass generation, confinement aspect of fermion (quark), charge\nshielding etc. \\cite{LO, COL, CAS, AG}. The description of this\nmodel in noncommutative space time has also been found to give\ninteresting result \\cite{APR, APR1}. Effect of non commutativity\nin space time has showed up as an interacting background with the\nmassive boson of the usual vector Schwinger model. In the\nSchwinger model massless fermion interact with the Abelian gauge\nfield. Photon acquires mass via a kind of dynamical symmetry\nbreaking and the quarks disappear from the physical spectra. The\nexactly solvable nature of the model leads to express this model\nin terms of canonical boson field. It is a remarkable feature of\n(1+1) dimensional exactly solvable interacting fermionic field\ntheory. Gauge invariance was there in the fermionic version of the\nmodel. So initially the model in the bosonised version too studied\nmaintaining gauge invariance. However gauge non invariant\nregularized version of this model also found to be an important\nfield theoretical model \\cite{AR}. Here we find that a one\nparameter class of regularization commonly used to study the\nchiral Schwinger model has been introduced in the vector Schwinger\nmodel. For a specific choice,\n i.e., for the vanishing value of the parameter the model reduces to\n the usual vector Schwinger model but for\n the other admissible value of this parameter the phase space structure\n as well as  the the physical spectra\ngets altered remarkably. This new regularization leads to a change\nin the confinement scenario of the quark too. In fact, the quarks\ngets liberated as it was happened in the Chiral Schwinger model\n\\cite{JR, RABIN, GIR, GIR1, GIR2}.\n\nRecently, we find that the Schwinger model is studied adding\nmasslike term for gauge field with the lagrangian at the classical\nlevel \\cite{USHA1, USHA2}. This model is structurally equivalent\nto the model studied in \\cite{AR}. The masslike term for gauge\ngauge field are introduced in the two models with different\nperspective. In \\cite{AR}, masslike term occurred as a one loop\ncorrection in order to remove the divergence of the fermionic\ndeterminant appeared during bosonization whereas in \\cite{USHA1,\nUSHA2}, the author studied the Schwinger model with the masslike\nterm for gauge field at the classical level. In \\cite{AR}, the\nauthors had reasonably fair  motivation to introduce the masslike\nterm since it is known that in QED, a regularization gets involved\nwhen one calculates the effective action by integrating the\nfermions out. The ambiguity in the regularization has been\nexploited by different authors in different times in (1+1)\ndimensional QED and Chiral QED and different interesting scenarios\nhave been resulted in \\cite{JR, RABIN, GIR, GIR1, GIR2, PM, MG,\nKH, ABD}. The most remarkable one is the chiral Schwinger model\nstudied by Jackiw and Rajaraman \\cite{JR}. They saved the long\nsuffering of the chiral generation of the Schwinger model due to\nHagen \\cite{HAG} from the non-unitary problem introducing a one\nparameter class of regularization. In \\cite{CASA}, the authors\nstudied the Schwinger model introducing masslike term for the\ngauge field at the classical level and showed that for a\nparticular value of the ambiguity parameter the lost gauge\ninvariance of the so called nonconfining (anomalous) Schwinger\nmodel \\cite{AR} gets restored. In \\cite{USHA1, USHA2}, however the\nauthors presented a surprising and untrustworthy result adding the\nsame masslike term at the classical level. There we find that the\nmass generated for the boson is $m=\\sqrt{2}e$ and it does not\ncontain the parameter involved within the masslike term of the\ngauge field!\nIf we look into the the work \\cite{AR}, a structurally equivalent\nmodel to \\cite {USHA1}, we find that the theoretical spectrum\ncontains a massive and a massless boson and the mass of the\nmassive boson acquires a generalized expression with the ambiguity\nparameter. However a massless boson is there in \\cite{USHA1,\nUSHA2}. A parameter free mass term may appear if the added term\nworks as gauge fixing. However the added term was not a legitimate\ngauge fixing term in \\cite{USHA1, USHA2}. A question therefore,\nautomatically comes how did the authors get such untrustworthy\nresult in \\cite{USHA1, USHA2}? The model is thus reinvestigated in\nthis note.\n\nThe vector Schwinger model is described by the following\ngenerating functional.\n\\begin{equation} Z[A] = \\int\nd\\psi d\\bar\\psi e^{\\int d^2x{\\cal L}_f}\n\\end{equation}\\b with\n\\begin{equation}\n{\\cal L}_f = \\bar\\psi\\gamma^\\mu[i\\partial_\\mu + e\\sqrt\\pi\nA_\\mu]\\psi \\nonumber\n\\end{equation}\nThe effective bosonized lagrangian density obtained by integrating\nout the  the fermion one by is\n\\begin{equation}\n{\\cal L}_B = \\frac{1}{2}\\partial_\\mu\\phi \\partial^\\mu\\phi -\ne\\epsilon_{\\mu\\nu}\\partial^\\nu\\phi A^\\mu. \\label{BLD}\n\\end{equation}\nIf electromagnetic background is introduced with masslike term for\nthe gauge field the model reads\n\\begin{equation}\n{\\cal L}_B = \\frac{1}{2}\\partial_\\mu\\phi \\partial^\\mu\\phi -\ne\\epsilon_{\\mu\\nu}\\partial^\\nu\\phi A^\\mu -\n\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu} + \\frac{1}{2}m_0 A_{\\mu}A^{\\mu}.\n\\label{BLM}\n\\end{equation}\nHere Lorentz indices runs over the two values $0$ and $1$\ncorresponding to the two space time dimension and the rest of the\nnotation are standard. The antisymmetric tensor is defined with\nthe convention $\\epsilon_{01}=+1$. The coupling constant $e$ has\none mass dimension in this situation. The parameter $m_0$ is\nintroduced to represent the masslike term for the gauge field at\nthe classical level in the same way as it was done in the\nThirring-Wess model \\cite{THIR}. In \\cite{USHA1, USHA2} the\nauthors used $m_0=ae^2$. Here we  would like to mention that the\nauthors in \\cite{USHA1} used a regularization parameter $M$ in the\ngeneralized bosonized lagrangian and set it to zero along with few\nothers parameter to get vector Schwinger mode. Then again the\nauthor added a mass like term for the gauge field ${\\cal L}_m =\n\\frac{1}{2}ae^2A_\\mu A^\\mu$ and termed this $a$ again as standard\nregularization parameter. It is highly confusing at this level.\n\nLet us now proceed to study the phase space structure of the\n model. To this end it is necessary to calculate the\nthe momenta corresponding to the field $A_0$, $A_1$ and $\\phi$.\nFrom the standard definition of the momentum we obtain\n\\begin{equation}\n\\pi_0=0 \\label{M1},\n\\end{equation}\n\\begin{equation}\n\\pi_1 = F_{01}\\label{M2},\n\\end{equation}\n\\begin{equation}\n\\pi_\\phi = \\dot\\phi - eA_1\\label{M3},\n\\end{equation}\nwhere $\\pi_0$, $\\pi_1$ and $\\pi_\\phi$ are the momenta\ncorresponding to the field $A_0$, $A_1$ and $\\phi$. Using the\nequations (\\ref{M1}), (\\ref{M2}) and (\\ref{M3}), the Hamiltonian\ndensity are calculated:\n\\begin{equation}\n{\\cal H} = \\frac{1}{ 2}(\\pi_\\phi +eA_1)^2 + \\frac{1}{2}\\pi_1^2 +\n\\frac{1}{2}\\phi'^2 + \\pi_1A_0' - eA_0\\phi' - {1\\over 2}m_0(A_0^2 -\nA_1^2).\\end{equation} Note that $\\omega = \\pi_0 \\approx 0$, is the\nfamiliar primary constraint of the theory. The preservation of the\n constraint $\\omega$\nrequires $[\\omega(x), H(y)] = 0$, which leads to the Gauss' law as\na secondary constraint:\n\\begin{equation}\n\\tilde\\omega = \\pi_1' + e\\phi' + m_0A_1 \\approx 0. \\label{SCO}\n\\end{equation}\nThe constraints (\\ref{M1}) and (\\ref{SCO}) form a second class\nset. Treating (\\ref{M1}) and (\\ref{SCO}) as strong condition one\ncan eliminate $A_0$ and obtain the reduced Hamiltonian density as\nfollows.\n\\begin{equation}\n{\\cal H}_r = \\frac{1}{2}(\\pi_\\phi + eA_1)^2 +\n\\frac{1}{2m_0}(\\pi'_1 + e\\phi')^2 + \\frac{1}{2}({\\pi_1}^2 +\n\\phi'^2)^2 + \\frac{1}{2}m_0A_1^2. \\label{RHAM}\n\\end{equation}\nAccording to the Dirac's prescription \\cite{DIR} of quantizing a\ntheory with second class constraint the Poisson brackets becomes\ninadequate for this situation. This type of systems however remain\nconsistent with the Dirac brackets \\cite{DIR}. It is\nstraightforward to show that the Dirac brackets between the fields\ndescribing the reduced hamiltonian (\\ref{RHAM}) remain canonical.\nUsing the canonical Dirac brackets the following first order\nequations of motion  are found out from the reduced Hamiltonian\ndensity (\\ref{RHAM}).\n\\begin{equation}\n\\dot A_1= \\pi_1 -\\frac{1}{m_0}(\\pi_1'' + e\\phi''), \\label{EQM1}\n\\end{equation}\n\\begin{equation}\n\\dot\\phi = \\pi_\\phi + eA_1,\\label{EQM2}\\end{equation}\n\\begin{equation}\n\\dot \\pi_\\phi = (1+ \\frac{e^2}{m_0})\\phi'' + \\frac{e}{m_0}\\pi_1''\n, \\label{EQM3}\n\\end{equation}\n\\begin{equation}\n\\dot\\pi_1 = -e\\pi_\\phi - (m_0+e^2)A_1. \\label{EQM4}\n\\end{equation}\nNote that in \\cite{USHA1},  all the equations of motion except the\nequation  of motion corresponding to equation (\\ref{EQM3}) were\nidentical. In \\cite{USHA1}, the calculational mistake started from\nthat erroneous equation of motion. A little algebra converts the\nabove first order equations (\\ref{EQM1}), (\\ref{EQM2}),\n(\\ref{EQM3}) and (\\ref{EQM4}) into the following second order\nequations:\n\\begin{equation}\n(\\Box + (m_0+ e^2)\\pi_1 = 0, \\label{SP1}\n\\end{equation}\n\\begin{equation}\n\\Box[\\pi_1 + \\frac{m_0+e^2}{e}\\phi] = 0. \\label{SP2}\n\\end{equation}\nEquation (\\ref{SP1}) describes a massive boson field with mass $m\n=\\sqrt{m_0+ e^2}$ whereas equation (\\ref{SP2}) describes a\nmassless scalar field. The result clearly shows that the mass\nacquires a generalized expression with the parameter involved in\nthe masslike term at the classical level as it can be expected\nfrom the result of the paper \\cite{AR}. This boson can be\nidentified with the photon that acquired mass via a dynamical\nsymmetry breaking. The massless boson (\\ref{SP2}) is equivalent to\nthe massless fermion in (1+1) dimension. So fermion here does not\nconfine. It remains free as it has been found in chiral Schwinger\nmodel \\cite{JR, RABIN, GIR, GIR1, GIR2} and the so called\nnonconfining Schwinger model \\cite{AR, AR1}. In this context we\nshould mention that there is some confusion in the literature\nregarding the conclusion concerning confinement and de-confinement\nscenario of fermion but the result in this context is considered\nto be more or less standard now \\cite{LO, AR, GIR2, PM, MG, AR1}.\nThe nature of the theoretical spectrum becomes more transparent if\nwe calculate the fermionic propagator to which we now tern.\n\nTo calculate fermion propagator one needs to work with the\noriginal fermionic model. The calculation is analogous to the so\ncalled nonconfining Schwinger model \\cite{AR}. The same\ncalculation for chiral Schwinger model is available in \\cite{GIR,\nGIR1}. The effective action obtained by integrating out $\\phi$\nfrom the bosonized action (\\ref{BLM}) is\n\\begin{equation}\nS_{eff}=\\int d^2x\\frac{1}{2}[A_\\mu(x)\nM^{\\mu\\nu}A_\\nu(x)],\\end{equation} where,\n\\begin{equation}\nM^{\\mu\\nu}= m_0 g^{\\mu\\nu} - \\frac{\\Box\n+e^2}{\\Box}\\tilde\\partial^\\mu\\tilde\\partial^\\nu.\\end{equation}\nHere we have used the standard notation\n$\\tilde\\partial^\\mu=\\epsilon^{\\mu\\nu}\\partial_\\nu.$\n The gauge field propagator is just\nthe inverse of $M^{\\mu\\nu}$ and it is found to be\n\\begin{equation}\n\\Delta_{\\mu\\nu}(x-y)=\\frac{1}{m_0}[g_{\\mu\\nu}+ \\frac{\\Box +e^2}{\n\\Box(\\Box + (m_0+e^2))} \\tilde\\partial_\\mu\\tilde\\partial_\\nu\n]\\delta(x-y).\n\\end{equation}\nNote that the two poles of propagator are found at the expected\npositions. One at zero and another at $m_0 + e^2$ indicating a\nmassive and a massless excitations.\n\nSetting an  Ansatz, for the Green function of the Dirac operator\n$(i\\partial\\!\\!\\!\\!\/ -gA\\!\\!\\!\\!\/)$, enable us to construct the\npropagator of the original fermion $\\psi$. The conventional\nconstruction of the Ansatz is\n\\begin{equation}\nG(x,y;A)=e^{ie(\\Phi(x)-\\Phi(y))}S_F(x-y),\\label{ANST}\n\\end{equation} where $S_F$  is  the  free, massless fermion\npropagator and $\\Phi$ is determined when the Ansatz (\\ref{ANST})\nis plugged into the equation for the Green function. From the\nstandard construction the Green function can be written down as\n\\begin{equation}\nG(x,y;A)=e^{ie\\int d^2zA^\\mu(z)J_\\mu(z)}S_F(x-y),\\end{equation}\nwhere the {\\it current} $J_\\mu$ has the following expression.\n\\begin{equation}\nJ_\\mu=(\\partial^z_\\mu  +  \\gamma_5\\tilde\\partial^z_\\mu)(D_F(z- x)\n-D_F(z-y)).\\end{equation} Here $D_F$ is the propagator of a\nmassless  free  scalar  field. Such propagators have to be\ninfra-red regularized in two dimensions \\cite{LO}:\n\\begin{equation}\nD_F(x)=-{i\\over 4\\pi}ln(-\\mu^2x^2+i0),\\end{equation} where $\\mu$\nis the infra-red regulator mass.\n\nFinally we obtain the  fermion  propagator by functionally\nintegrating $G(x,y;A)$ over the gauge field:\n\\begin{eqnarray}\nS'_F&=&\\int {\\cal D}A e^{\\frac{i}{2}\\int d^2z~(A_\\mu(z)\nM^{\\mu\\nu}A_\\nu(z) + 2eA_\\mu   J^\\mu)}S_F(x-y)\\nonumber\\\\&=&{\\cal\nN} \\exp[\\frac{D_F}{\\frac{m_0^2 + m_0e^2}{e^4}}+\n\\frac{\\Delta_F(m^2=m_0 +\ne^2)}{\\frac{m_0+e^2}{e^2}}]~~S_F.\\label{PROP} \\end{eqnarray} Here\n$\\Delta_F$ is the propagator of a massive free scalar field and\n${\\cal N}$ is a wave function renormalization factor.\n\nBoth the results therefore confirms strongly that the mass\nacquires a generalized expression with bare coupling constant and\nthe parameter involved in the masslike term for the gauge field.\nIt is known that in vector Schwinger model this type of parameter\nfree mass generates. In fact, in the bosonized vector Schwinger\nmodel no such parameter exists if gauge fixing is not introduced\nat the lagrangian level. In this context, note that setting $a=0$,\nin the bosonized lagrangian of the so called nonconfining\nSchwinger model \\cite{AR} one can obtain vector Schwinger model\n\\cite{SCH}.\n\nSo from our result it can be concluded that it is the\ncalculational error that led the authors of \\cite{USHA1, USHA2} to\nland in to this type of untrustworthy result. We have pointed out\nin the body of this paper where the the actual error started.\nActually, the use of that erroneous equation corresponding to the\nequation  (\\ref{EQM3}) led them to reach to the wrong expression\nof the mass term for photon. The fermionic propagator presented in\n\\cite{USHA1, USHA2} also carried the same error. In\nequation(\\ref{PROP}) of this note the correct expression fermionic\noperator is calculated.\n\nAs a concluding remark we would like to mention that the term\n$\\frac{1}{2}ae^2A^\\mu A_\\mu$ is not a legitimate gauge fixing\nterm. With this term the vector Schwinger model turns into the so\ncalled nonconfining Schwinger model \\cite{AR, AR1}. Introduction\nof proper gauge fixing term like $\\frac{\\alpha}{2e^2}(\\partial_\\mu\nA^\\mu)^2$ in the starting lagrangian of vector Schwinger  model\nhowever leads to a result that corresponds to parameter free mass\nterm for the photon though there exists a parameter $\\alpha$ in\nthe starting lagrangian.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper we develop an algorithm to invert the $j$--function in quasilinear time, and give an application to testing whether an elliptic curve has complex multiplication.\n\nIt is known that the inverse of $j$ is equal to a ratio of two Gaussian hypergeometric functions, though we do not believe any analysis has been made of the running time or required precision to invert $j$ by this method.\nOur method is similar to that of \\cite{fasttheta}, which makes use of addition formulae between Jacobi's theta functions in order to reduce the argument to a fixed compact set -- here we repeatedly make use of the modular polynomial \n\\begin{align*}\n \\Phi_2(X,Y)=X^3&+Y^3-X^2Y^2+1488X^2Y+1488XY^2-162000X^2-162000Y^2\\\\+&40773375XY+8748000000X+8748000000Y-157464000000000,\n\\end{align*}\nwhich has the property that the roots of $\\Phi_2(j(\\tau),z)$ are $j(2\\tau)$, $j\\left(\\frac{\\tau}{2}\\right)$, and $j\\left(\\frac{\\tau+1}{2}\\right)$,\n to either compute $j(2^k\\tau)$, the logarithm of which is a close approximation to $-2^{k+1}\\pi\\tau$, or to manipulate the argument of $j$ into a compact set to which Newton's method may be applied.\n\n We define the \\emph{regulated precision} of an approximation $\\tilde{\\alpha}$ to $\\alpha$ to be\n\\begin{equation*}\n \\frac{\\lvert\\alpha-\\tilde{\\alpha}\\rvert}{\\max\\{1,\\lvert\\alpha\\rvert\\}},\n\\end{equation*}\nand denote by $M(P)$ the computational complexity of multiplication of two $P$--bit integers, which by a recent result of Harvey and Hoeven \\cite{fastmult} may be taken to be $O(P\\log P)$. We obtain the following,\n\\begin{theorem}\n Suppose that $\\tilde{j}$ is an approximation to $j(\\tau)$, $\\tau\\in\\mathcal{F}$, of regulated precision $2^{-P}$, with $P\\geq400$. Let $Q=P\/6$ if $\\lvert \\tau-i\\rvert\\leq2^{-30}$ or $\\left\\lvert \\tau-\\frac{\\pm1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-30}$, and $Q=P-\\max\\{11\\log P,100\\}$ otherwise. Then we may obtain an approximation to $\\tau$ of relative precision $2^{-Q}$ in time\n \\begin{equation*}\n  O(M(P)\\log (P)^2).\n \\end{equation*}\n\\end{theorem}\nThe $j$--function has two ramification points in its fundamental domain, which entails the loss of precision in its inversion when $j$ is close to $0$ or $1728$.\nWe apply this algorithm to test for complex multiplication of elliptic curves -- given an approximation to the $j$--invariant of an elliptic curve and a bound upon its height and degree, we may invert it and determine if the inverse is a quadratic irrational, determining also the discriminant,\n\\begin{theorem}\n Suppose that $j(\\tau)$ is the $j$--invariant of an elliptic curve $E$, with $j(\\tau)$ of degree bounded by $d$ and height bounded by $H\\geq e^e$. Then it may be determined from $d$, $H$, and an approximation to $j$ of regulated precision $2^{-300d^2\\log H(\\log d+\\log\\log H)^2-200}$ whether $E$ has complex multiplication, and if so the associated discriminant, in time, letting $T=d^2\\log H(\\log d+\\log\\log H)^2$, \n\\begin{equation*}\n O(M(T)(\\log T)^2).\n\\end{equation*}\n\\end{theorem}\nPrevious methods include that of \\cite{achter}, based on reduction of elliptic curves at primes, which has an unconditional running time of $O(H^{cd})$, and assuming the Generalized Riemann Hypothesis a running time of $O(d^2(\\log H)^2)$, and two tests of \\cite{xavier}, comprising a deterministic algorithm based on Galois representations associated to torsion points, with running time $O(d^{c_1}(\\log H)^{c_2})$, with an ineffective implicit constant, and a probabilistic algorithm, also of polynomial running time.\n\nWe note that one may also apply our algorithm for the inversion of $j$ to detecting isogenies between two elliptic curves of running time $O(N\\log N\\log\\log N)$, where the degree of the isogeny is bounded by $N$, though our implicit constant is ineffective as explicit bounds on the coefficients of modular polynomials $\\Phi_N(X,Y)$ for composite indices are not currently available.\n\\section{Preliminaries}\nWe will denote by $\\mathcal{F}$ the usual fundamental domain of $j(z)$, $\\left\\{z|-\\frac{1}{2}<\\text{\\normalfont Re}(z)\\leq\\frac{1}{2},\\lvert z\\rvert>1\\right\\}\\cup\\{z|\\lvert z\\rvert=1,\\text{\\normalfont Re}(z)\\geq0\\}$.\nThroughout we will make use of the following results,\n\\begin{lemma}[Lemma 1 of \\cite{j2079}]\\label{j_q}\nIf $\\tau\\in\\mathcal{F}$\n \\begin{equation*}\n  \\left\\lvert j(\\tau)-e^{-2\\pi i\\tau}\\right\\rvert\\leq2079.\n \\end{equation*}\n\\end{lemma}\n\n\\begin{theorem}[Kantorovich, \\cite{kantorovich}]\\label{kantorovich}\n Let $F:S(x_0,R)\\subset X\\to Y$ have a continuous Fr\\'{e}chet derivative in $\\overline{S(x_0,r)}$. Moreover, let \\emph{(i)} the linear operation $\\Gamma_0=[F'(x_0)]^{-1}$ exist; \\emph{(ii)} $\\lVert\\Gamma_0 F(x_0)\\rVert\\leq\\eta$; \\emph{(iii)} $\\lVert\\Gamma_0F''(x)\\rVert\\leq K$ $(x\\in\\overline{S(x_0,r)})$. Now, if\n \\begin{equation*}\n  h=K\\eta\\leq\\frac{1}{2}\n \\end{equation*}\nand \n\\begin{equation*}\n r\\geq\\frac{1-\\sqrt{1-2h}}{h}\\eta,\n\\end{equation*}\nthen $F(x)=0$ will have a solution $x^*$ to which the Newton method is convergent. Here,\n\\begin{equation*}\n \\lVert x^*-x_0\\rVert\\leq r_0.\n\\end{equation*}\nFurthermore, if for $h<\\frac{1}{2}$,\n\\begin{equation*}\n r<r_1=\\frac{1+\\sqrt{1-2h}}{h}\\eta,\n\\end{equation*}\nor for $h=\\frac{1}{2}$\n\\begin{equation*}\n r\\leq r_1,\n\\end{equation*}\nthe solution $x^*$ will be unique in the sphere $\\overline{S(x_0,r)}$. The speed of convergence is characterized by the inequality\n\\begin{equation*}\n \\lVert x^*-x_k\\rVert\\leq\\frac{1}{2^k}(2h)^{2^k}\\frac{\\eta}{h}\n\\end{equation*}\nfor $k=0,1,2,\\ldots$.\n\\end{theorem}\nWe note that the condition \n\\begin{equation*}\n r\\geq\\frac{1-\\sqrt{1-2h}}{h}\\eta\n\\end{equation*}\nmay be replaced with\n\\begin{equation*}\n r\\geq 2\\eta.\n\\end{equation*}\n\n\n\\section{Inversion of $j(z)$}\nFirstly, if $\\lvert j \\rvert\\leq 2^{-P\/2}$ or $\\lvert j-1728\\rvert\\leq 2^{-P\/3}$, we return $\\tau = \\frac{1+i\\sqrt{3}}{2}$ or $\\tau = i$ respectively, and otherwise, we split the fundamental domain of $j$ into $4$ sections -- $\\text{\\normalfont Im}(\\tau)\\geq 3$, $\\lvert\\tau-i\\rvert\\leq 2^{-{31}}$, $\\left\\lvert\\tau-\\frac{1+\\sqrt{3}}{2}\\right\\rvert\\leq2^{-31}$, $\\left\\lvert\\tau-\\frac{-1+\\sqrt{3}}{2}\\right\\rvert\\leq2^{-31}$ and the remaining compact subset of the fundamental domain. We may determine $\\tau$ with sufficient precision by taking a low precision inverse via the following expression for $j^{-1}$ in terms of Gaussian hypergeometric functions -- with $\\alpha$ a solution to \n\\begin{equation*}\n j(\\tau)=\\frac{1728}{4\\alpha(1-\\alpha)},\n\\end{equation*}\n$\\tau$ is equal to either\n\\begin{align*}\n i\\frac{_2F_1\\left(\\frac{1}{6},\\frac{5}{6},1,1-\\alpha\\right)}{_2F_1\\left(\\frac{1}{6},\\frac{5}{6},1,\\alpha\\right)}.\n\\end{align*}\nor the negative of its inverse.\nIf $j$ is sufficiently large, or close to $0$ or $1728$, then we do not need to evaluate this in order to determine which section of the fundamental domain $\\tau$ lies in, so we need only compute the above to a fixed precision in a compact set, at points which it is easily shown are bounded away from zeros of $_2F_1\\left(\\frac{1}{6},\\frac{5}{6},1,z\\right)$, so this takes only constant time.\n\\iffalse\nFirstly, we note that in the case of real $j$, one may take the elliptic curve\n\\begin{equation*}\n E:y^2=x^3-\\frac{27j}{j-1728}x-\\frac{27j}{j-1728},\n\\end{equation*}\nwhich has $j$--invariant $j$, and use the algorithm of (???, p.391) to compute a basis $\\omega_1,\\omega_2$ of the lattice of $E$, taking $\\tau = \\frac{\\omega_2}{\\omega_1}$, and then transforming this to the fundamental domain to obtain the inverse of $j$, which would give a faster algorithm, at least for $j$ away from $0$ and $1728$.\n\\fi\n\\subsection{Large $j$}\n\nThroughout this section we assume $\\text{\\normalfont Im}(\\tau)\\geq 3$, and will repeatedly make use of the consequent fact that $\\lvert j(\\tau)\\rvert\\geq 10^8$. We will make use of the modular polynomial $\\Phi_2(X,Y)$ to obtain an approximation to $j(2\\tau)$, and repeat the process until we have an approximation to $j(2^k\\tau)$, where $k$ is sufficiently large, at which point taking the logarithm of $j(2^k\\tau)$ gives a good approximation to $-2^{k+1}\\pi\\tau$, owing to the $q$--series of $j$.\n\n\\begin{proposition}\\label{phi2discrepancy}\n Let $j(\\tau)=j$, and suppose that $\\text{\\normalfont Im}(\\tau)\\geq3$ and $\\tilde{j}$ is an approximation to $j$ of relative precision at least $2^{-P}$, with $P\\geq300$. Then the largest root, in absolute value, of $\\Phi_2(\\tilde{j},z)$ is an approximation to $j(2\\tau)$ of relative precision at least $2^{-P+2}$.\n\\end{proposition}\n\\begin{proof}\n Let $f(z) = \\Phi_2(j,z)$ and $g(z) =\\Phi_2(\\tilde{j},z)$. We first bound the coefficients of $f(z)-g(z)$. For the coefficient of $z^2$, we have, with $\\tilde{j}=j+\\delta$, \n \\begin{align*}\n  \\left\\lvert -j^2+1488j-162000-(-(j+\\delta)^2+1488(j+\\delta)-162000)\\right\\rvert &= \\lvert 2\\delta j+\\delta^2+1488\\delta\\rvert\\\\\n  &\\leq 2.1\\lvert\\delta\\rvert\\lvert j\\rvert,\n \\end{align*}\nas $\\lvert \\delta\\rvert\\leq2^{-300}\\lvert j\\rvert$ and $\\lvert j\\rvert\\geq 10^8$.\nFor the coefficient of $z$, we have\n\\begin{align*}\n  &\\left\\lvert 1488j^2 +40773375j+8748000000-(1488(j+\\delta)^2+40773375(j+\\delta)+8748000000)\\right\\rvert\\\\\n  &\\quad\\quad\\quad= \\lvert 2976\\delta j+1488\\delta^2+40773375\\delta\\rvert\\\\\n  &\\quad\\quad\\quad\\leq 3210\\lvert\\delta\\rvert\\lvert j\\rvert.\n \\end{align*}\n For the constant coefficient, we have\n \\begin{align*}\n  &\\left\\lvert 3\\delta j^2+3\\delta^2j+\\delta^3-16200\\delta j-16200\\delta^2+8748000000\\delta\\right\\rvert\\\\\n  &\\quad\\quad\\quad\\leq 3.4\\lvert\\delta\\rvert\\lvert j\\rvert^2.\n \\end{align*}\n We now bound the values of $g(j(\\tau'))$ for $\\tau'\\in\\left\\{2\\tau,\\frac{\\tau}{2},\\frac{\\tau+1}{2}\\right\\}$.\n For $j(2\\tau)$, as $\\lvert j(2\\tau)\\rvert\\leq 1.02\\lvert j\\rvert^2$,\n \\begin{align*}\n  \\lvert g(j(2\\tau))\\rvert &= \\lvert g(j(2\\tau))-f(j(2\\tau))\\rvert\\\\\n  &\\leq 2.1\\lvert\\delta\\rvert\\lvert j\\rvert(1.02\\lvert j\\rvert^2)^2+3210\\lvert\\delta\\rvert\\lvert j\\rvert\\cdot1.02\\lvert j\\rvert^2+3.4\\lvert\\delta\\rvert\\lvert j\\rvert^2\\\\\n  &\\leq 2.2\\lvert\\delta\\rvert\\lvert j\\rvert^5.\n \\end{align*}\n For $g(j(\\tau'))$, $\\tau'\\in\\left\\{\\frac{\\tau}{2},\\frac{\\tau+1}{2}\\right\\}$,  firstly, the absolute values of $j\\left(\\frac{\\tau}{2}\\right)$ and $j\\left(\\frac{\\tau+1}{2}\\right)$ are bounded above and below by $e^{2\\pi\\text{\\normalfont Im}(\\tau)\/2}\\pm2079$, and the absolute value of $j$ is bounded above and below by $e^{2\\pi\\text{\\normalfont Im}(\\tau)}\\pm2079$. As $\\text{\\normalfont Im}(\\tau)\\geq 3$, these yield \n\\begin{equation}\n0.83\\lvert j\\rvert^{1\/2}\\leq\\left\\lvert j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert, \\left\\lvert j\\left(\\frac{\\tau}{2}\\right)\\right\\rvert\\leq 1.17\\lvert j\\rvert^{1\/2},\n\\end{equation}\nso \n\\begin{align*}\n  \\lvert g(j(\\tau'))\\rvert &= \\lvert g(j(\\tau'))-f(j(\\tau'))\\rvert\\\\\n  &\\leq 2.1\\lvert\\delta\\rvert\\lvert j\\rvert(1.17\\lvert j\\rvert^{1\/2})^2+3210\\lvert\\delta\\rvert\\lvert j\\rvert(1.17\\lvert j\\rvert^{1\/2})+3.4\\lvert\\delta\\rvert\\lvert j\\rvert^2\\\\\n  &\\leq 20\\lvert\\delta\\rvert\\lvert j\\rvert^2.\n \\end{align*}\nLet $\\beta_0,\\beta_1,\\beta_2$ be the roots of $g(z)$. Then for $\\beta_i$ the closest root of $g$ to $j(2\\tau)$,\n\\begin{equation*}\n \\lvert j(2\\tau)-\\beta_i\\rvert \\leq \\left( 2.2\\lvert\\delta\\rvert\\lvert j\\rvert^5\\right)^{1\/3}\\leq \\left(2.2\\cdot2^{-100}\\lvert j\\rvert^6\\right)^{1\/3}\\leq10^{-9}\\lvert j\\rvert^2,\n\\end{equation*}\nand we let $\\beta_0$ be the closest root of $g$ to $j(2\\tau)$. As $0.98\\lvert j\\rvert^2\\leq j(2\\tau)\\leq1.02\\lvert j\\rvert^2$, $0.97\\lvert j\\rvert^2\\leq \\lvert\\beta_0\\rvert\\leq1.03\\lvert j\\rvert^2$. \nFor $\\tau'=\\frac{\\tau}{2}$, with $\\beta_i$ the closest root of $g$ to $j(\\tau')$,\n\\begin{equation*}\n \\lvert j(\\tau')-\\beta_i\\rvert \\leq \\left( 20\\lvert\\delta\\rvert\\lvert j\\rvert^2\\right)^{1\/3}\\leq \\left(20\\cdot2^{-300}\\lvert j\\rvert^3\\right)^{1\/3}\\leq10^{-29}\\lvert j\\rvert,\n\\end{equation*}\nand in particular, \n\\begin{equation}\n \\lvert\\beta_i\\rvert\\leq 10^{-29}\\lvert j\\rvert+1.17\\lvert j\\rvert^{1\/2}<0.97\\lvert j\\rvert^2\\leq\\lvert\\beta_0\\rvert,\n\\end{equation}\nso $\\beta_i\\neq \\beta_0$. Let this root of $g$ be $\\beta_1$. Now we may improve the bound on $\\lvert j(\\tau')-\\beta_1\\rvert$,\n\\begin{equation*}\n \\lvert j(\\tau')-\\beta_1\\rvert^2\\leq \\frac{20\\lvert\\delta\\rvert\\lvert j\\rvert^2}{\\lvert j(\\tau')-\\beta_0\\rvert}\\leq\\frac{20\\lvert\\delta\\rvert\\lvert j\\rvert^2}{0.98\\lvert j\\rvert^2-1.17\\lvert j\\rvert^{1\/2}}\\leq30\\cdot2^{-300}\\lvert j\\rvert,\n\\end{equation*}\nso\n\\begin{equation*}\n \\lvert j(\\tau')-\\beta_1\\rvert\\leq 10^{-88}\\lvert j\\rvert^{1\/2},\n\\end{equation*}\nand consequently \n\\begin{equation*}\n 0.82\\lvert j\\rvert^{1\/2}\\leq \\lvert\\beta_1\\rvert\\leq1.18\\lvert j\\rvert^{1\/2}.\n\\end{equation*}\nWe now bound $\\beta_2$. By our bound on the difference between the constant coefficients of $f$ and $g$, the constant coefficient $-\\beta_0\\beta_1\\beta_2$ of $g$ is bounded in absolute value by\n\\begin{equation*}\n \\left\\lvert j(2\\tau)j\\left(\\frac{\\tau}{2}\\right)j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert + 3.4\\delta\\lvert j\\rvert^2\\leq 1.4\\lvert j\\rvert^3+0.01\\lvert j\\rvert^3\\leq 1.5\\lvert j\\rvert^3,\n\\end{equation*}\nso that\n\\begin{equation*}\n \\lvert\\beta_2\\rvert\\leq \\frac{1.5\\lvert j\\rvert^{3}}{0.97\\lvert j\\rvert^2\\cdot0.82\\lvert j\\rvert^{1\/2}}\\leq 2\\lvert j\\rvert^{1\/2}.\n\\end{equation*}\nReturning to $g(j(2\\tau))$, we now bound below the terms $\\lvert j(2\\tau)-\\beta_i\\rvert$ for $i=1,2$ in order to improve our inequality for $\\lvert j(2\\tau)-\\beta_0\\rvert$. We now have, for $i=1,2$,\n\\begin{align*}\n \\lvert j(2\\tau)-\\beta_{i}\\rvert &\\geq \\lvert j(2\\tau)\\rvert - 2\\lvert j\\rvert^{1\/2}\\\\\n &\\geq0.97\\lvert j\\rvert^2.\n\\end{align*}\nThis now improves our bound on the difference of $\\beta_0$ to $j(2\\tau)$,\n\\begin{align*}\n \\lvert j(2\\tau)-\\beta_0\\rvert &\\leq \\frac{2.2\\delta\\lvert j\\rvert^5}{\\lvert j(2\\tau)-\\beta_1\\rvert\\lvert j(2\\tau)-\\beta_2\\rvert} \\\\\n &\\leq \\frac{2.2\\delta\\lvert j\\rvert^5}{(0.97\\lvert j\\rvert^2)^2}\\\\\n &\\leq 2.4\\delta\\lvert j\\rvert.\n\\end{align*}\nSo the relative precision of $\\beta_0$ as an approximation to $j(2\\tau)$ is bounded by\n\\begin{equation*}\n \\frac{\\lvert j(2\\tau)-\\beta_0\\rvert}{\\lvert j(2\\tau)\\rvert}\\leq \\frac{2.4\\lvert \\delta\\rvert\\lvert j\\rvert}{0.97\\lvert j\\rvert^2}\\leq2.5\\frac{\\lvert\\delta\\rvert}{\\lvert j\\rvert}\\leq 2^{-P+2}.\n\\end{equation*}\n\\end{proof}\n\n\\begin{proposition}\n If $\\text{\\normalfont Im}(\\tau)\\geq 3$ and $\\tilde{j}$ is an approximation to $j(\\tau)$ of relative precision $2^{-P}$, where $P$ is at least $300$, applying Newton's method to $\\Phi_2(\\tilde{j},z)$, with starting point\n \\begin{equation*}\n  \\tilde{j}^2-2\\cdot744\\tilde{j}-2\\cdot196884+744^2+744,\n \\end{equation*}\nwill obtain an approximation to $j(2\\tau)$ of relative precision $2^{-P+3}$ after at most $[2\\log P+\\log\\log\\lvert j\\rvert]$ steps of Newton iteration.\n\\end{proposition}\n\\begin{proof}\nWe first give a rough approximation to $\\beta_0$. Writing\n\\begin{equation*}\n j(\\tau) = e^{-2\\pi i\\tau}+744+196884e^{2\\pi i\\tau} + f(\\tau),\n\\end{equation*}\nwe have\n\\begin{align*}\n j(\\tau)^2-2\\cdot744j(\\tau)&+2\\cdot19688+2\\cdot744^2+744 \\\\\n &= e^{-4\\pi i\\tau}+744+196884^2e^{4\\pi i\\tau}+2(196884e^{2\\pi i\\tau}+e^{-2\\pi i\\tau})f(\\tau)+f(\\tau)^2\n\\end{align*}\nso that, as $\\text{\\normalfont Im}(\\tau)\\geq 3$, $f$ is maximized when $\\text{\\normalfont Re}(\\tau)=0$, and $f(\\text{\\normalfont Im}(\\tau))$ is decreasing in $\\text{\\normalfont Im}(\\tau)$,\n\\begin{align*}\n \\lvert j(2\\tau) &- (j(\\tau)^2-744j(\\tau)+19688-744^2+744)\\rvert\\\\\n &\\leq (196884+196884^2)e^{-4\\pi\\text{\\normalfont Im}(\\tau)}+\\lvert f(\\text{\\normalfont Im}(2\\tau))\\rvert\\\\\n &\\quad\\quad+2(196884e^{2\\pi \\text{\\normalfont Im}(\\tau)}+e^{2\\pi \\text{\\normalfont Im}(\\tau)})\\lvert f(\\text{\\normalfont Im}(\\tau))\\rvert+\\lvert f(\\text{\\normalfont Im}(\\tau))\\rvert^2\\\\\n &\\leq 10^{-7}+\\lvert f(6)\\rvert+10^6\\lvert f(3)\\rvert+\\lvert f(3)\\rvert^2\\\\\n &\\leq 0.4\n\\end{align*}\nWe now let \n\\begin{equation*}\n \\gamma_0 = \\tilde{j}^2-744\\tilde{j}+19688-744^2+744.\n\\end{equation*}\nThis will be our starting point for Newton iteration to find $\\beta_0$. We now bound the terms appearing in Kantorovich's theorem. Firstly,\n\\begin{align*}\n\\Phi_2(\\tilde{j},z)=z^3&+(-\\tilde{j}^2+1488\\tilde{j}-162000)z^2\\\\\n  &+(1488\\tilde{j}^2+40773375\\tilde{j}+8748000000)z\\\\\n  &+\\tilde{j}^3-162000\\tilde{j}^2+8748000000\\tilde{j}-157464000000000.\n \\end{align*}\nNow it is clear, as $\\lvert j-\\tilde{j}\\rvert\\leq2^{-300}\\lvert j\\rvert$, that $\\lvert \\gamma_0-j(2\\tau)\\rvert\\leq 0.4+\\frac{\\lvert j\\rvert^2}{2^{90}}$, and so as $0.98\\lvert j\\rvert^2\\leq\\lvert j(2\\tau)\\rvert\\leq1.02\\lvert j\\rvert^2$, we have $0.979\\lvert j\\rvert^2\\leq\\lvert \\gamma_0\\rvert\\leq1.021\\lvert j\\rvert^2$. we will take the $r$ of Kantorovich's theorem to be $0.009\\lvert j\\rvert^2$, and give bounds for the disc $\\lvert\\gamma-\\gamma_0\\rvert\\leq0.009\\lvert j\\rvert^2$. For $\\gamma$ in this disc, $0.97\\lvert j\\rvert^2\\leq\\lvert \\gamma\\rvert\\leq1.03\\lvert j\\rvert^2$ and in addition $\\lvert\\tilde{j}\\rvert\\leq1.01\\lvert j\\rvert$, so for an upper bound on the first derivative, we have\n\\begin{align*}\n \\lvert\\Phi_2'(\\tilde{j},\\gamma)\\rvert&\\leq 3\\lvert\\gamma\\rvert^2+2(\\lvert\\tilde{j}\\rvert^2+1488\\lvert\\tilde{j}\\rvert+162000)\\lvert\\gamma_0\\rvert\\\\\n  &\\quad+(1488\\lvert\\tilde{j}\\rvert^2+40773375\\lvert\\tilde{j}\\rvert+8748000000)\\\\\n  &\\leq 3.2\\lvert j\\rvert^4+2.2\\lvert j\\rvert^4+10^{-7}\\lvert j\\rvert^4\\\\\n  &\\leq 5.5\\lvert j\\rvert^4,\n\\end{align*}\nand for a lower bound, we have\n\\begin{align*}\n \\lvert\\Phi_2'(\\tilde{j},\\gamma)\\rvert&\\geq 3\\lvert\\gamma\\rvert^2-2(\\lvert\\tilde{j}\\rvert^2+1488\\lvert\\tilde{j}\\rvert+162000)\\lvert\\gamma_0\\rvert\\\\\n  &\\quad-(1488\\lvert\\tilde{j}\\rvert^2+40773375\\lvert\\tilde{j}\\rvert+8748000000)\\\\\n  &\\geq 2.82\\lvert j\\rvert^4-2.1\\lvert j\\rvert^4-10^{-7}\\lvert j\\rvert^4\\\\\n  &\\geq 0.71\\lvert j\\rvert^4.\n\\end{align*}\nFor our upper bound on the function at $\\gamma_0$, as\n $\\lvert\\beta_1\\rvert, \\lvert\\beta_2\\rvert\\leq2\\lvert j\\rvert^{1\/2}$, and as $\\lvert\\beta_0-j(2\\tau)\\rvert\\leq 2^{-298}\\lvert j\\rvert^2$ we have $\\lvert\\gamma_0-\\beta_0\\rvert\\leq0.4+2^{-298}\\lvert j\\rvert^2$, so\n\\begin{align*}\n \\lvert\\Phi_2(\\tilde{j},\\gamma_0)\\rvert&=\\lvert\\gamma_0-\\beta_0\\rvert\\lvert\\gamma_0-\\beta_1\\rvert\\lvert\\gamma_0-\\beta_2\\rvert\\\\\n &\\leq(0.4+2^{-289}\\lvert j\\rvert^2)(1.03\\lvert j\\rvert^2+2\\lvert j\\rvert^{1\/2})^2\\\\\n &\\leq 2^{-54}\\lvert j\\rvert^6.\n\\end{align*}\nFor the second derivative, we have for an upper bound\n\\begin{align*}\n \\lvert\\Phi_2''(\\tilde{j},\\gamma)\\rvert&\\leq6\\lvert\\gamma\\rvert+2\\lvert \\tilde{j}\\rvert^2+1488\\lvert \\tilde{j}\\rvert+162000\\\\\n &\\leq 8.3\\lvert j\\rvert^2,\n\\end{align*}\nand for a lower bound,\n\\begin{align*}\n \\lvert\\Phi_2''(\\tilde{j},\\gamma_0)\\rvert&\\geq6\\lvert\\gamma\\rvert-2\\lvert \\tilde{j}\\rvert^2-1488\\lvert \\tilde{j}\\rvert-162000\\\\\n &\\geq 3.8\\lvert j\\rvert^2.\n\\end{align*}\nNow, by Theorem \\ref{kantorovich}, as\n\\begin{equation*}\n \\frac{\\lvert\\Phi_2(\\tilde{j},\\gamma_0)\\rvert\\lvert\\Phi_2''(\\tilde{j},\\gamma)\\rvert}{\\lvert\\Phi_2'(\\tilde{j},\\gamma)\\rvert^2}\\leq\\frac{2^{-54}\\lvert j\\rvert^6\\cdot8.3\\lvert j\\rvert^2}{(0.71\\lvert j\\rvert^4)^2}\\leq2^{-49}<\\frac{1}{2},\n\\end{equation*}\nand\n\\begin{align*}\n2\\eta &= 2\\frac{\\lvert\\Phi_2(\\tilde{j},\\gamma_0)\\rvert}{\\lvert\\Phi_2'(\\tilde{j},\\gamma_0)\\rvert}\\leq \\frac{2^{-54}\\lvert j\\rvert^6}{0.71\\lvert j\\rvert^4}\\leq0.001\\lvert j\\rvert^2,\\\\\n r&=0.009\\lvert j\\rvert^2,\n\\end{align*}\nNewton's method will converge to the root $\\beta_0$, with a rate of convergence\n\\begin{equation*}\n \\lvert \\gamma_k - \\beta_0\\rvert\\leq\\frac{1}{2^k}2^{-48\\cdot2^{k}}\\frac{\\lvert\\Phi_2'(\\tilde{j},\\gamma_0)\\rvert}{\\lvert\\Phi_2''(\\tilde{j},\\gamma_0)\\rvert}\\leq\\frac{1}{2^k}2^{-48\\cdot2^k}\\frac{5.5\\lvert j\\rvert^4}{3.8\\lvert j\\rvert^2}\\leq2^{-48\\cdot2^k}\\lvert j\\rvert^2\n\\end{equation*}\nfor $k\\geq 1$. In particular, when $k\\geq [2\\log P+2\\log\\log\\lvert j\\rvert]$, $\\lvert \\gamma_k-\\beta_0\\rvert\\leq 2^{-P}$. Now as, by Proposition \\ref{phi2discrepancy}, $\\beta_0$ is an approximation to $j(2\\tau)$ of relative precision $2^{-P+2}$, it may be easily shown that after $[2\\log P+2\\log\\log\\lvert j\\rvert]$ steps, $\\gamma_k$ will be an approximation to $j(2\\tau)$ of relative precision $2^{-P+3}$.\n\\end{proof}\n\n\\begin{lemma}\n Suppose that $\\lvert j(\\tau)\\rvert\\geq 2^{P+12}$, and $\\tilde{j}$ is an approximation to $j(\\tau)$ of relative precision at least $2^{-P}$. Then\n \\begin{equation*}\n  -\\frac{\\log\\tilde{j}}{2\\pi}\n \\end{equation*}\nis an approximation to $\\tau$ of absolute precision $2^{-P}$.\n\\end{lemma}\n\\begin{proof}\n Let $j(\\tau)+\\delta = \\tilde{j}$. As\n\\begin{equation*}\n \\lvert j(\\tau)-e^{-2\\pi\\tau}\\rvert \\leq 2079,\n\\end{equation*}\nwe have\n\\begin{equation*}\n \\log(\\tilde{j}) = -2\\pi\\tau+ \\log\\left(1+\\frac{\\delta+2079\\theta}{j(\\tau)}\\right),\n\\end{equation*}\nwhere $\\lvert\\theta\\rvert\\leq1$. So as $\\lvert\\log(1+z)\\rvert\\leq1.1\\lvert z\\rvert$ when $\\lvert z\\rvert\\leq 0.05$,\n\\begin{equation*}\n \\lvert\\log\\tilde{j}+2\\pi \\tau\\rvert \\leq 1.1\\cdot2^{-P}+0.6\\cdot2^{-P}\\leq 1.7\\cdot2^{-P}.\n\\end{equation*}\nSo that \n\\begin{equation*}\n \\left\\lvert\\tau-\\left(-\\frac{\\log\\tilde{j}}{2\\pi}\\right)\\right\\rvert\\leq \\frac{1.7}{2\\pi}2^{-P}\\leq 2^{-P}.\n\\end{equation*}\n\\end{proof}\n\nNow the algorithm to invert $j$ when $\\text{\\normalfont Im}(\\tau)\\geq 3$ proceeds as follows -- if $\\lvert j\\rvert\\leq 2^{P+12}$, first iteratively compute approximations to $j(2^k\\tau)$ by Newton's method applied to $\\Phi_2(\\tilde{j},z)$, up to $k = \\left[2\\log\\left(\\frac{P+12}{\\text{\\normalfont Im}(\\tau)}\\right)\\right]+1$. Then calculate the logarithm of $j(2^k\\tau)$ to relative precision $2^{-P-2}$ (note that $\\lvert \\tau\\rvert\\geq 1$), and divide by $-2\\pi$, where we have calculated $2\\pi$ to relative precision $2^{-P-2}$. This process entails a loss of precision of at most $5\\left[2\\log\\left(\\frac{P+12}{\\text{\\normalfont Im}(\\tau)}\\right)\\right]+2$, which is bounded by $11\\log P$ when $P\\geq 400$, and the precision at all applications of Newton's method is at least $2^{-300}$, so our assumptions on the precision of our approximations in the propositions of this section are satisfied at each application. As the computational complexity of Newton's method is $O(M(P)\\log P)$, and of the complex logarithm and computing $\\pi$ are $O(M(P)(\\log P)^2)$, if $\\lvert j\\rvert\\leq2^{P+12}$, the algorithm has time complexity\n\\begin{equation*}\n O\\left(M(P)\\left[2\\log\\left(\\frac{P+12}{\\text{\\normalfont Im}(\\tau)}\\right)+1\\right][2\\log P+2\\log\\log\\lvert j\\rvert]\\right) = O\\left(M(P)(\\log P)^2\\right),\n\\end{equation*}\nand if $\\lvert j\\rvert\\geq 2^{P+12}$, time complexity\n\\begin{equation*}\n O\\left(M(P)(\\log P)^2\\right),\n\\end{equation*}\nwhere the implicit constants are not too large and could be made effective. \n\n\\subsection{Near 1728 and 0}\nWhen $\\tau$ is close to either $i$ or $\\frac{\\pm1+i\\sqrt{3}}{2}$, we will make use of the modular polynomial $\\Phi_2(X,Y)$ to obtain an approximation to one of $j\\left(2\\tau\\right)$, $j\\left( \\frac{\\tau}{2}\\right)$, or $j\\left(\\frac{\\tau+1}{2}\\right)$, for which the SL$_2(\\mathbb{Z})$--equivalent elements of $\\mathcal{F}$ to either $2\\tau$, $\\frac{\\tau}{2}$, or $\\frac{\\tau+1}{2}$ will lie in the aforementioned compact set, to which we may apply Newton's method. We carry out the analysis only for $i$ and $\\frac{1+i\\sqrt{3}}{2}$, as when $\\left\\lvert\\tau-\\frac{-1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-31}$, the SL$_2(\\mathbb{Z})$--equivalent $\\tau+1$ satisfies $\\left\\lvert(\\tau+1)-\\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-31}$.\n\n\n\\begin{lemma}\\label{taylor_series_at_i}\n If $\\lvert\\delta\\rvert\\leq 2^{-30}$ then \n  \\begin{equation*}\n\\left\\lvert j(i+\\delta)-1728-\\frac{j^{(2)}(i)}{2}\\delta^2\\right\\rvert \\leq 0.07\\lvert \\delta\\rvert^2,\n\\end{equation*}\nand\n\\begin{equation*}\n\\left\\lvert j\\left(\\frac{1+i\\sqrt{3}}{2}+\\delta\\right)-\\frac{j^{(3)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)}{3!}\\delta^3\\right\\rvert \\leq 0.07\\lvert \\delta\\rvert^3.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nWe first bound the coefficients of the Taylor series of $j$ at $z=i$ and $z=\\frac{1+i\\sqrt{3}}{2}$. Firstly, by Theorem 1 of \\cite{jcoefficients}, the coefficient of $e^{2\\pi in\\tau}$ in the $q$--expansion of $j$ is bounded by $4e^{4\\pi\\sqrt{n}}$, so the corresponding coefficient of the $k$'th derivative is bounded by $(2\\pi n)^ke^{4\\pi\\sqrt{n}}$. Further, for $n\\geq 1$, $e^{4\\pi\\sqrt{n}-\\sqrt{3}\\pi n}\\leq 100e^{-\\pi n}$, so we have the following bound of the derivatives at these two points,\n\\begin{align*}\n \\lvert j^{(k)}(i)\\rvert, \\left\\lvert j^{(k)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)\\right\\rvert&\\leq (2\\pi)^ke^{2\\pi}+744+400\\sum_{n=1}^\\infty(2\\pi n)^ke^{-\\pi n}.\n\\end{align*}\nWe now bound the sum in this expression. Firstly, we have the identity\n\\begin{equation*}\n \\sum_{n=1}^\\infty e^{-\\pi nx} = \\frac{1}{e^{\\pi x}-1},\n\\end{equation*}\nand so consider the derivatives of this function, expressing the derivative as sums of the form\n\\begin{equation*}\n \\sum \\alpha_i\\frac{e^{a\\pi x}}{(e^{\\pi x}-1)^b},\n\\end{equation*}\nwhere each term occurring in the expression for the $k$'th derivative is derived from taking the derivative of either the numerator or denominator of a term occurring in the $k-1$'th derivative, i.e. there is no collection of terms with $(a,b)$ equal. It is clear that there are at most $2^k$ terms, and, passing from one derivative to the next, $\\lvert\\alpha_i\\rvert$ may increase by at most $\\pi\\max\\{a,b\\}$, and that $a\\leq b\\leq k+1$, and $a\\leq k$. So $\\lvert\\alpha_i\\rvert\\leq \\pi^k(k+1)!$, and a bound for the whole expression evaluated at $1$ is therefore\n\\begin{equation*}\n 2^k\\pi^k(k+1)!\\frac{e^{k\\pi}}{(e^{\\pi}-1)^k}\\leq 14^kk!.\n\\end{equation*}\nNow we have the bounds\n\\begin{align*}\n \\lvert j^{(k)}(i)\\rvert, \\left\\lvert j^{(k)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)\\right\\rvert&\\leq (2\\pi)^ke^{2\\pi}+744+400\\cdot14^kk!\\\\\n &\\leq1700\\cdot14^kk!.\n\\end{align*}\nAt $z=i$ and $z=\\frac{1+i\\sqrt{3}}{2}$, the Taylor series expansions for $j(z)$ are\n\\begin{align*}\n j(z) = 1728 + \\sum_{n=2}^{\\infty}c_n(z-i)^n,\\\\\n j(z) = \\sum_{n=3}^{\\infty}c'_n\\left(z-\\frac{1+i\\sqrt{3}}{2}\\right)^n.\n\\end{align*}\nwhere $c_n$ and $c_n'$ are bounded in absolute value by $1400\\cdot14^k$.\n We now bound the tails of the sums of the Taylor series,\n \\begin{equation*}\n  \\sum_{n=3}^\\infty \\lvert c_n\\rvert\\lvert\\delta\\rvert^n\\leq \\lvert\\delta\\rvert^3\\sum_{n=0}^\\infty1700\\cdot14^{n+3}\\lvert\\delta\\rvert^n,\n \\end{equation*}\nand \n\\begin{equation*}\n  \\sum_{n=4}^\\infty \\lvert c_n'\\rvert\\lvert\\delta\\rvert^n\\leq \\lvert\\delta\\rvert^4\\sum_{n=0}^\\infty1700\\cdot14^{n+4}\\lvert\\delta\\rvert^n.\n \\end{equation*}\n Now as $\\lvert\\delta\\rvert\\leq 2^{-30}$,\n \\begin{equation*}\n  \\sum_{n=0}^\\infty1700\\cdot14^{n+4}\\lvert\\delta\\rvert^n=\\frac{1700\\cdot14^4}{1-14\\lvert\\delta\\rvert}\\leq 7\\cdot10^7,\n \\end{equation*}\nso that\n \\begin{align*}\n\\left\\lvert j(i+\\delta)-1728-\\frac{j^{(2)}(i)}{2}\\delta^2\\right\\rvert &\\leq 2^{-30}\\cdot7\\cdot10^7\\lvert\\delta\\rvert^2\\leq 0.07\\lvert \\delta\\rvert^2,\\\\\n\\left\\lvert j\\left(\\frac{1+i\\sqrt{3}}{2}+\\delta\\right)-\\frac{j^{(3)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)}{3!}\\delta^3\\right\\rvert &\\leq 2^{-30}\\cdot7\\cdot10^7\\lvert\\delta\\rvert^3\\leq 0.07\\lvert \\delta\\rvert^3.\n \\end{align*}\n\\end{proof}\nFurthermore, we note that\n$49700\\geq\\lvert j^{(2)}(i)\\rvert\\geq 49600$, and $275000\\geq\\left\\lvert j^{(3)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)\\right\\rvert\\geq 274000$, so if $\\lvert\\tau - i\\rvert\\leq 2^{-30}$, by the previous lemma, we have\n\\begin{equation*}\n2.4\\cdot10^4\\lvert\\delta\\rvert^2\\leq \\lvert j-1728\\rvert\\leq2.5\\cdot10^4\\lvert\\delta\\rvert,\n\\end{equation*}\nand if $\\left\\lvert\\tau - \\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\leq 2^{-30}$,\n\\begin{equation*}\n 4.5\\cdot10^4\\lvert\\delta\\rvert^3\\leq\\lvert j\\rvert\\leq4.6\\cdot10^4\\lvert\\delta\\rvert^3,\n\\end{equation*}\nfrom which we deduce the following,\n\\begin{lemma}\\label{j_to_tau_small}\nIf $\\lvert\\tau-i\\rvert\\leq 2^{-30}$ and $P\\geq 300$, then:\n\\begin{enumerate}\n \\item If $\\lvert j(\\tau) - 1728\\rvert\\leq 2^{-P}$, then $\\lvert\\tau - i\\rvert\\leq 2^{-P\/2-7}$.\n \\item If $\\lvert j(\\tau) - 1728\\rvert\\geq 2^{-P}$, then $\\lvert\\tau - i\\rvert\\geq 2^{-P\/2-8}$.\n \\end{enumerate}\n If $\\left\\lvert\\tau-\\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-30}$ and $P\\geq 300$, then:\n \\begin{enumerate}\n \\item If $\\lvert j\\rvert\\leq 2^{-P}$, then $\\left\\lvert\\tau - \\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-P\/3-5}$.\n \\item If $\\lvert j\\rvert\\geq 2^{-P}$, then $\\left\\lvert\\tau - \\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\geq2^{-P\/3-6}$.\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{lemma}\\label{taylor_series_at_2i}\n Suppose that $\\lvert\\delta\\rvert\\leq 2^{-28}$. Then\n \\begin{equation*}\n\\left\\lvert j\\left(2i+\\delta\\right)-j\\left(2i\\right)-j'\\left(2i\\right)\\delta\\right\\rvert \\leq 0.2\\lvert \\delta\\rvert,\n\\end{equation*}\nand \n\\begin{equation*}\n\\left\\lvert j\\left(i\\sqrt{3}+\\delta\\right)-j\\left(i\\sqrt{3}\\right)-j'\\left(i\\sqrt{3}\\right)\\delta\\right\\rvert \\leq 0.2\\lvert \\delta\\rvert.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nWe proceed similarly to the previous lemma -- as $17e^{-2\\pi n}\\geq e^{4\\pi n-2\\sqrt{3}\\pi n}$, \n\\begin{equation*}\n \\lvert j^{(k)}(2i)\\rvert,\\lvert j^{(k)}(\\sqrt{3}i)\\rvert \\leq (2\\pi)^ke^{4\\pi}+744+68\\sum_{n=1}^{\\infty}(2\\pi n)^ke^{-2\\pi n},\n\\end{equation*}\nand a similar analysis to the previous lemma bounds the sum in this inequality by\n\\begin{equation}\n (2\\pi)^ke^{4\\pi}+744+68\\cdot 13^kk! \\leq 300000\\cdot13^kk!.\n\\end{equation}\n So when $\\lvert\\delta\\rvert\\leq 2^{-28}$,\n \\begin{align*}\n\\left\\lvert j\\left(2i+\\delta\\right)-j(2i)-j'\\left(2i\\right)\\delta\\right\\rvert &\\leq \\sum_{n=2}^{\\infty}300000\\cdot13^n\\delta^n\\\\\n&\\leq \\sum_{n=2}^{\\infty}300000\\cdot13^n\\delta^n\\\\\n&\\leq 2^{-28}\\cdot300000\\cdot13^2\\lvert\\delta\\rvert\\sum_{n=0}^{\\infty}13^n2^{-28n}\\\\\n&\\leq0.2\\lvert \\delta\\rvert.\n\\end{align*}\n and as our bound for the terms of the Taylor series applies to both expansions, we similarly have\n \\begin{align*}\n\\left\\lvert j\\left(i\\sqrt{3}+\\delta\\right)-j\\left(i\\sqrt{3}\\right)-j'\\left(i\\sqrt{3}\\right)\\delta\\right\\rvert \\leq 0.2\\lvert \\delta\\rvert.\n\\end{align*}\n\\end{proof}\n\nWe now give two lemmas on the separation and closeness of the three preimages of roots of the modular polynomial $\\Phi_2(j(\\tau),z)$, for $\\tau$ near $i$ and $\\frac{1+i\\sqrt{3}}{2}$.\n\\begin{lemma}\\label{preimage_closeness}\n Suppose that $\\tau\\in\\mathcal{F}$. Then, if $\\tau = i+\\delta$,\n \\begin{align*}\n  \\lvert 2\\tau - 2i\\rvert&\\leq 2\\lvert\\delta\\rvert,\\\\\n  \\left\\lvert -\\frac{2}{\\tau}- 2i\\right\\rvert&\\leq 2\\lvert\\delta\\rvert,\n \\end{align*}\nand if $\\tau = \\frac{1+i\\sqrt{3}}{2}+\\delta$,\n\\begin{align*}\n  \\lvert \\left(2\\tau-1\\right) - \\sqrt{3}i\\rvert&\\leq 2\\lvert\\delta\\rvert,\\\\\n  \\left\\lvert \\left(1-\\frac{2}{\\tau}\\right)- \\sqrt{3}i\\right\\rvert&\\leq 2\\lvert\\delta\\rvert,\\\\\n  \\left\\lvert \\left(-1-\\frac{2}{\\tau-1}\\right)- \\sqrt{3}i\\right\\rvert&\\leq 2\\lvert\\delta\\rvert.\n \\end{align*}\n\\end{lemma}\n\\begin{proof}\nFirst note that as $\\tau\\in\\mathcal{F}$, $\\lvert\\tau\\rvert\\geq 1$. Let $2\\tau = 2i+\\delta_1$, and $-\\frac{2}{\\tau}=2i+\\delta_2$. For $\\delta_1$, $\\lvert2\\tau-2i\\rvert = 2\\lvert \\delta\\rvert$, and for $\\delta_2$,\n\\begin{align*}\n \\lvert\\delta_2\\rvert=\\left\\lvert -\\frac{2}{\\tau}-2i\\right\\rvert &= \\left\\lvert \\frac{-2-2i\\tau}{\\tau}\\right\\rvert\\\\\n &=\\frac{\\lvert2i\\delta\\rvert}{\\lvert\\tau\\rvert}\\\\\n &\\leq 2\\lvert\\delta\\rvert\n\\end{align*}\nIf $\\tau = \\frac{1+i\\sqrt{3}}{2}+\\delta$, let $2\\tau-1 = \\sqrt{3}i+ \\delta_1$, $\\left(1-\\frac{2}{\\tau}\\right) = \\sqrt{3}i+\\delta_2$, and $\\left(1-\\frac{2}{\\tau-1}\\right)=\\sqrt{3}i+\\delta_3$. For $\\delta_1$,\n\\begin{align*}\n \\lvert\\delta_1\\rvert = \\left\\lvert 2\\tau-1-\\sqrt{3}i\\right\\rvert\\leq 2\\lvert\\delta\\rvert,\n\\end{align*}\nfor $\\delta_2$,\n\\begin{align*}\n \\lvert\\delta_2\\rvert = \\left\\lvert \\left(1-\\frac{2}{\\tau}\\right)- \\sqrt{3}i\\right\\rvert& = \\left\\lvert\\frac{\\tau-2-\\sqrt{3}i\\tau}{\\tau}\\right\\rvert\\\\\n &=\\left\\lvert\\frac{(1-\\sqrt{3}i)\\delta}{\\tau}\\right\\rvert\\\\\n &\\leq2\\lvert\\delta\\rvert,\n\\end{align*}\nand for $\\delta_3$,\n\\begin{align*}\n \\lvert\\delta_3\\rvert = \\left\\lvert \\left(-1-\\frac{2}{\\tau-1}\\right)- \\sqrt{3}i\\right\\rvert &= \\left\\lvert\\frac{-\\tau+1-2-\\sqrt{3}i\\tau+\\sqrt{3}i}{\\tau}\\right\\rvert\\\\\n &=\\frac{\\lvert(1+\\sqrt{3}i)\\delta\\rvert}{\\lvert\\tau-1\\rvert}\\\\\n &\\leq 2\\lvert\\delta\\rvert.\n\\end{align*}\n\\end{proof}\n\n\\begin{lemma}\\label{preimage_separation}\n If $\\tau = i+\\delta$, $\\tau \\in \\mathcal{F}$, with $\\lvert \\delta\\rvert \\leq 2^{-30}$, then the distance between $2\\tau$ and $-\\frac{2}{\\tau}$ is at least $3.99\\lvert\\delta\\rvert$, and the distance between either of these and the point in $\\mathcal{F}$ which is $\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent to $\\frac{\\tau+1}{2}$ is at least $0.99$ in magnitude. If $\\tau = \\frac{1+i\\sqrt{3}}{2}+\\delta$, $\\tau\\in\\mathcal{F}$, then the distance between any pair of the three points lying in $\\mathcal{F}$ which are $\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent to $2\\tau$, $\\frac{\\tau}{2}$, and $\\frac{\\tau+1}{2}$ is at least $3.46\\lvert\\delta\\rvert$.\n\\end{lemma}\n\\begin{proof}\n Firstly, for the distance between $\\tau$ and $-\\frac{2}{\\tau}$, we have\n  \\begin{align*}\n   2\\tau +\\frac{2}{\\tau} &= \\frac{2(\\tau^2+1)}{\\tau}\\\\\n   &=\\frac{4i\\delta+2\\delta^2}{\\tau}.\n  \\end{align*}\n  As $\\lvert\\delta\\rvert\\leq 2^{-30}$, $\\lvert\\delta\\rvert^2\\leq2^{-30}\\lvert\\delta\\rvert$, and $\\lvert i+\\delta\\rvert\\leq1+2^{-30}$, so\n  \\begin{equation*}\n   \\left\\lvert2\\tau-\\left(-\\frac{2}{\\tau}\\right)\\right\\rvert\\geq3.99\\lvert\\delta\\rvert.\n  \\end{equation*}\n  Now for $-\\frac{2}{\\tau+1}+1$, we have\n  \\begin{align*}\n   -\\frac{2}{\\tau+1}+1-i &= \\frac{-2+\\tau+1-i\\tau-i}{\\tau+1}\\\\\n   &=\\frac{\\delta-i\\delta}{1+i+\\delta}\n  \\end{align*}\n  so that \n\\begin{equation}\n \\left\\lvert\\left(-\\frac{2}{\\tau+1}+1\\right)-i\\right\\rvert \\leq 2\\lvert\\delta\\rvert\n\\end{equation}\nand so either $-\\frac{2}{\\tau+1}+1$ or $-\\left(-\\frac{2}{\\tau+1}+1\\right)^{-1}$ is $\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent to $\\frac{1+\\tau}{2}$ and in $\\mathcal{F}$, and its distance from $i$ is bounded by $\\left\\lvert 1+\\frac{2\\lvert\\delta\\rvert}{1-2\\lvert\\delta\\rvert}\\right\\rvert\\leq 2^{-28}$. By Lemma \\ref{preimage_closeness}, the distance of either $2\\tau$ or $-\\frac{2}{\\tau}$ from $2i$ is at most $2\\lvert\\delta\\rvert\\leq2\\cdot2^{-30}$, so both of these are separated from either $-\\frac{2}{\\tau+1}+1$ or $-\\left(-\\frac{2}{\\tau+1}+1\\right)^{-1}$ by a distance of at least $0.99$.\n\nNow for $\\tau = \\frac{1+i\\sqrt{3}}{2}+\\delta$, the $\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent points to $2\\tau$, $\\frac{\\tau}{2}$, and $\\frac{\\tau+1}{2}$ are $2\\tau-1$, $1-\\frac{2}{\\tau}$, and $-1-\\frac{2}{\\tau-1}$ respectively. For their differences, we have,\n \\begin{align*}\n   2\\tau-1 -\\left(1-\\frac{2}{\\tau}\\right) &= \\frac{2(\\tau^2-\\tau+1)}{\\tau}\\\\\n   &=\\frac{i2\\sqrt{3}\\delta+2\\delta^2}{\\tau}.\n  \\end{align*}\n  So as $\\lvert \\tau\\rvert\\leq1+2^{-30}$,\n  \\begin{equation*}\n   \\left\\lvert 2\\tau-1 -\\left(1-\\frac{2}{\\tau}\\right)\\right\\rvert\\geq 3.46\\lvert \\delta\\rvert\n  \\end{equation*}\n  For the first and third points,\n  \\begin{align*}\n   2\\tau-1 -\\left(-1-\\frac{2}{\\tau-1}\\right) &= \\frac{2(\\tau^2-\\tau+1)}{\\tau-1}\\\\\n   &=\\frac{i2\\sqrt{3}\\delta+2\\delta^2}{\\tau-1}.\n  \\end{align*}\nAs $\\lvert\\tau-1\\rvert\\leq1+2^{-30}$, we have the lower bound\n\\begin{equation*}\n  \\left\\lvert2\\tau-1 -\\left(-1-\\frac{2}{\\tau-1}\\right)\\right\\rvert\\geq3.46\\lvert \\delta\\rvert.\n\\end{equation*}\nFor the second and third points,\n\\begin{align*}\n   \\left(1-\\frac{2}{\\tau}\\right) -\\left(-1-\\frac{2}{\\tau-1}\\right) &= \\frac{2(\\tau^2-\\tau+1)}{\\tau(\\tau-1)}\\\\\n   &=\\frac{i2\\sqrt{3}\\delta+2\\delta^2}{\\tau(\\tau-1)}.\n\\end{align*}\nSo we obtain the lower bound\n\\begin{equation*}\n  \\left\\lvert\\left(1-\\frac{2}{\\tau}\\right) -\\left(-1-\\frac{2}{\\tau-1}\\right)\\right\\rvert\\geq3.46\\lvert \\delta\\rvert.\n\\end{equation*}\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{bounds_for_images}\n If $\\tau = i+\\delta$, with $\\lvert\\delta\\rvert\\leq2^{-30}$, then \n \\begin{equation*}\n  7.18\\cdot10^{6}\\lvert\\delta\\rvert\\leq\\left\\lvert j(2\\tau)-j\\left(\\frac{\\tau}{2}\\right)\\right\\rvert\\leq7.25\\cdot10^6\\lvert\\delta\\rvert ,\n \\end{equation*}\n and \n \\begin{equation*}\n   2.4\\cdot10^5\\leq\\left\\lvert j(2\\tau)-j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert,  \\left\\lvert j\\left(\\frac{\\tau}{2}\\right)-j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert\\leq3.1\\cdot10^5,\n \\end{equation*}\n and if $\\tau = \\frac{1+i\\sqrt{3}}{2}+\\delta$, with $\\lvert\\delta\\rvert\\leq 2^{-30}$, then for $\\tau_1,\\tau_2\\in\\{2\\tau,\\frac{\\tau}{2},\\frac{\\tau+1}{2}\\}$, $\\tau_1\\neq\\tau_2$,\n \\begin{equation*}\n  1.15\\cdot10^6\\lvert\\delta\\rvert\\leq\\lvert j(\\tau_1)-j(\\tau_2)\\rvert\\leq 1.34\\cdot10^6\\lvert\\delta\\rvert.\n \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n We first use the Taylor series expansion of $j(z)$ at $2i$. By Lemma \\ref{preimage_closeness}, we may apply Lemma \\ref{taylor_series_at_2i}, and using also Lemma \\ref{preimage_separation}, we obtain the lower bound\n \\begin{align*}\n  \\left\\lvert j(2\\tau)-j\\left(-\\frac{2}{\\tau}\\right)\\right\\rvert&\\geq \\left\\lvert j(2i)+j'(2i)\\delta_1-j(2i)-j'(2i)\\delta_2\\right\\rvert - 0.4\\lvert\\delta\\rvert\\\\\n  &=\\left\\lvert j'(2i)\\right\\rvert\\left\\lvert \\delta_1- \\delta_2\\right\\rvert - 0.4\\lvert\\delta\\rvert\\\\\n  &=\\left\\lvert j'(2i)\\right\\rvert\\left\\lvert 2\\tau+\\frac{2}{\\tau}\\right\\rvert - 0.4\\lvert\\delta\\rvert\\\\\n  &\\geq 1.8\\cdot10^6\\cdot3.99\\rvert\\delta\\lvert - 0.4\\lvert\\delta\\rvert\\\\\n  &\\geq 7.18\\cdot10^{6}\\lvert\\delta\\rvert.\n \\end{align*}\n Similarly we have an upper bound of \n \\begin{equation*}\n  \\left\\lvert j(2\\tau)-j\\left(-\\frac{2}{\\tau}\\right)\\right\\rvert\\leq1.81\\cdot10^6\\cdot4\\rvert\\delta\\lvert+ 194\\lvert\\delta\\rvert\\leq7.25\\cdot10^6\\lvert\\delta\\rvert.\n \\end{equation*}\n By Lemma \\ref{preimage_separation}, the $\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent point in $\\mathcal{F}$ to $\\frac{\\tau+1}{2}$ is at most $2\\lvert\\delta\\rvert\\leq2^{-29}$ from $i$, so \n \\begin{equation*}\n  \\left\\lvert j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert\\leq e^{2.01\\pi}+2079\\leq e^{7.9},\n \\end{equation*}\n and by Lemma \\ref{preimage_closeness}, $2\\tau,-\\frac{\\tau}{2}$ are at most $2^{-29}$ from $2i$,\n \\begin{equation*}\n  \\left\\lvert j\\left(2\\tau\\right)\\right\\rvert,\\left\\lvert j\\left(\\frac{\\tau}{2}\\right)\\right\\rvert\\geq e^{3.99\\pi}-2079\\geq e^{12.5},\n \\end{equation*}\n so that \n \\begin{align*}\n  3.1\\cdot10^5\\geq\\left\\lvert j(2\\tau)-j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert,  \\left\\lvert j\\left(-\\frac{\\tau}{2}\\right)-j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert\\geq 2.4\\cdot10^5.\n \\end{align*}\nNow using the Taylor series of $j(z)$ at $i\\sqrt{3}$, by Lemma \\ref{preimage_closeness}, we may apply Lemma \\ref{taylor_series_at_2i}, and using also Lemma \\ref{preimage_separation}, with $\\tau_1$, $\\tau_2$ any distinct pair of $2\\tau$ ($\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent to $2\\tau-1$), $\\frac{\\tau}{2}$, ($\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent to $1-\\frac{2}{\\tau}$), and $\\frac{\\tau+1}{2}$ ($\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent to $-1-\\frac{1}{\\tau-1}$), with $\\tau_j = i\\sqrt{3}+\\delta_j$,\n\\begin{align*}\n \\left\\lvert j(\\tau_1)-j(\\tau_2)\\right\\rvert &\\geq \\left\\lvert j(\\sqrt{3}i)+j'(\\sqrt{3}i)\\delta_1-(j(\\sqrt{3}i)+j'(\\sqrt{3}i)\\delta_2)\\right\\rvert - 0.4\\lvert\\delta\\rvert\\\\\n &=\\lvert j'(\\sqrt{3}i)\\rvert\\lvert\\tau_1-\\tau_2\\rvert-0.4\\lvert\\delta\\rvert\\\\\n &\\geq 334500\\cdot3.46\\lvert\\delta\\rvert-0.4\\lvert\\delta\\rvert\\\\\n &\\geq1.15\\cdot10^6\\lvert\\delta\\rvert,\n\\end{align*}\nand similarly for an upper bound we have\n\\begin{align*}\n \\left\\lvert j(\\tau_1)-j(\\tau_2)\\right\\rvert &\\leq \\left\\lvert j(\\sqrt{3}i)+j'(\\sqrt{3}i)\\delta_1-(j(\\sqrt{3}i)+j'(\\sqrt{3}i)\\delta_2)\\right\\rvert + 0.4\\lvert\\delta\\rvert\\\\\n &=\\lvert j'(\\sqrt{3}i)\\rvert\\lvert\\tau_1-\\tau_2\\rvert+0.4\\lvert\\delta\\rvert\\\\\n &\\leq 334600\\cdot4\\lvert\\delta\\rvert+0.4\\lvert\\delta\\rvert\\\\\n &\\leq1.34\\cdot10^6\\lvert\\delta\\rvert.\n\\end{align*}\n\\end{proof}\n\n\\subsubsection{$j$ near $1728$}\nWe now bound the discrepancy between the roots of $\\Phi_2(j,z)$ and $\\Phi_2(\\tilde{j},z)$ when $\\lvert\\tau-i\\rvert\\leq2^{-20}$ and $\\lvert \\tilde{j}-1728\\rvert\\geq 2^{-P\/3}$.\n\\begin{lemma}\\label{phi2discrepancy_points_1728}\n Suppose that $\\tilde{j}$ is an approximation to $j(\\tau)$ of relative precision $2^{-P}$, with $P\\geq 300$, $\\lvert\\tau-i\\rvert\\leq2^{-30}$, and $\\left\\lvert \\tilde{j}-1728\\right\\rvert\\geq 2^{-P\/3}$. Then the relative precision of any root of $\\Phi_2(\\tilde{j},z)$ to the closest root of $\\Phi_2(j,z)$ is at least $2^{-P\/3+10}$\n\\end{lemma}\n\\begin{proof}\nFirstly, as in the proof of Lemma \\ref{phi2discrepancy}, with $f(z) = \\Phi_2(j,z)$ and $g(z) = \\Phi_2(\\tilde{j},z)$, and $\\tilde{j}=j+\\delta$, we will bound the size of the coefficients of $f-g$. We note that $\\lvert j\\rvert\\geq1700$. For $z^2$, we have\n\\begin{equation*}\n \\lvert 2j\\delta+\\delta^2+1448\\delta\\rvert\\leq 3\\lvert\\delta\\rvert\\lvert j\\rvert,\n\\end{equation*}\nfor $z$, we have\n\\begin{equation*}\n \\lvert 2976\\delta j+1488\\delta^2+40773375\\delta\\rvert\\\\\n  \\leq 30000\\lvert\\delta\\rvert\\lvert j\\rvert,\n\\end{equation*}\nand for the constant term\n\\begin{equation*}\n  \\left\\lvert 3\\delta j^2+3\\delta^2j+\\delta^3-16200\\delta j-16200\\delta^2+8748000000\\delta\\right\\rvert\\leq 3100\\lvert\\delta\\rvert\\lvert j\\rvert^2.\n \\end{equation*}\nNow evaluating $g$ at $j(2\\tau)$, we have, noting that $\\lvert j(2\\tau)\\rvert\\leq 0.1\\lvert j\\rvert^2$,\n\\begin{align*}\n \\lvert g(j(2\\tau))\\rvert &= \\lvert f(j(2\\tau))-g(j(2\\tau))\\rvert\\\\\n &\\leq 3\\lvert\\delta\\rvert\\lvert j\\rvert(0.1\\lvert j\\rvert^2)^2+30000\\lvert\\delta\\rvert\\lvert j\\rvert(0.1\\lvert j\\rvert^2)+3100\\lvert\\delta\\rvert\\lvert j\\rvert^2\\\\\n &\\leq 0.04\\lvert j\\rvert^5\n\\end{align*}\nLetting $\\beta_0,\\beta_1,\\beta_2$ be the roots of $g$, where $\\beta_0$ is the closest root of $g(z)$ to $j(2\\tau)$,\n\\begin{align*}\n \\lvert j(2\\tau)-\\beta_0\\rvert\\lvert j(2\\tau)-\\beta_1\\rvert\\lvert j(2\\tau)-\\beta_2\\rvert&\\leq0.04\\lvert\\delta\\rvert\\lvert j\\rvert^5\\\\\n \\lvert j(2\\tau)-\\beta_0\\rvert&\\leq10^5\\lvert\\delta\\rvert^{1\/3}\\\\\n &\\leq 1.2\\cdot10^6\\cdot2^{-P\/3},\n\\end{align*}\nand similarly for the nearest root of $g(z)$ to each of the roots of $f(z)$. As $\\lvert\\tilde{j}-1728\\rvert\\geq 2^{-P\/3}$, $\\lvert j-1728\\rvert\\geq 2^{-P\/3}-1800\\cdot2^{-P}\\geq 2^{-P\/3-1}$, and by Lemma \\ref{j_to_tau_small}, $\\lvert \\tau - i\\rvert\\geq 2^{-P\/6-9}$, so that by Lemma \\ref{bounds_for_images}, \n\\begin{equation*}\n \\left\\lvert j(2\\tau) - j\\left(\\frac{\\tau}{2}\\right)\\right\\rvert\\geq7.18\\cdot10^7\\cdot2^{- P\/6-9}.\n\\end{equation*}\nNow letting $\\beta_i$ and $\\beta_j$ be the nearest roots to $j(2\\tau)$ and $j\\left(\\frac{\\tau}{2}\\right)$ respectively,\n\\begin{align*}\n \\lvert\\beta_i-\\beta_j\\rvert &= \\left\\lvert (\\beta_i-j(2\\tau))-j(2\\tau)-\\left(\\left(\\beta_j-j\\left(\\frac{\\tau}{2}\\right)\\right)-j\\left(\\frac{\\tau}{2}\\right)\\right)\\right\\rvert\\\\\n &\\geq \\left\\lvert j(2\\tau) - j\\left(\\frac{\\tau}{2}\\right)\\right\\rvert - \\lvert j(2\\tau)-\\beta_i\\rvert - \\left\\lvert j\\left(\\frac{\\tau}{2}\\right)-\\beta_j\\right\\rvert\\\\\n &\\geq 7.18\\cdot10^7\\cdot 2^{-P\/6-9} - 1.2\\cdot10^6\\cdot2^{-P\/3}\\\\\n &\\geq 7.18\\cdot10^7\\cdot 2^{-P\/6-9} - 0.01\\cdot2^{-P\/6}\\\\\n &\\geq 1.4\\cdot10^5\\cdot2^{-P\/6}.\n\\end{align*}\nIn particular, $\\beta_i$ and $\\beta_j$ are distinct, and given that, by Lemma \\ref{bounds_for_images}, $\\left\\lvert j(2\\tau)-j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert\\geq2.4\\cdot10^5\\geq7.18\\cdot10^7\\cdot2^{-P\/6-9}$, one may similarly deduce a separation of $2\\cdot10^{5}$ between the closest root to $j\\left(\\frac{\\tau+1}{2}\\right)$ and either of the other two. So each root $j_i$ of $\\Phi_2(j,z)$ has a unique associated closest root $\\beta_i$ of $\\Phi_2(\\tilde{j},z)$, which satisfies\n\\begin{equation*}\n \\lvert j_i - \\beta_i\\rvert\\leq 1.2\\cdot10^6\\cdot2^{-P\/3}.\n\\end{equation*}\nThe relative precision of $\\beta_i$ to its closest root is then bounded by \n\\begin{equation*}\n \\frac{1.2\\cdot10^6\\cdot2^{-P\/3}}{\\lvert j(2\\tau)\\rvert}, \\frac{1.2\\cdot10^6\\cdot2^{-P\/3}}{\\lvert j\\left(\\frac{\\tau}{2}\\right)\\rvert}\\leq 2^{-P\/3 + 3}\n\\end{equation*}\nif $\\beta_i$ is the root close to $j(2\\tau)$ or $j\\left(\\frac{\\tau}{2}\\right)$, and \n\\begin{equation*}\n \\frac{1.2\\cdot10^{6}\\cdot2^{-P\/3}}{\\lvert j(\\frac{\\tau+1}{2})\\rvert}\\leq 2^{-P\/3+10}\n\\end{equation*}\nfor the root close to $ j\\left(\\frac{\\tau+1}{2}\\right)$.\n\\end{proof}\nFinally we show that Newton's method may be used to obtain an approximation to $j(2\\tau)$.\n\\begin{proposition}\nLet $j=j(\\tau)$, where $\\lvert\\tau-i\\rvert\\leq2^{-30}$, and $\\tilde{j}$ be an approximation to $j$ of relative precision $2^{-P}$, $P\\geq 300$, such that $\\lvert\\tilde{j}-1728\\rvert\\geq2^{-P\/3}$. Let\n\\begin{equation*}\n \\tau_0 = i+\\sqrt{\\frac{\\tilde{j}-1728}{j^{(2)}(i)}},\n\\end{equation*}\nwhere the sign of the square root is chosen arbitrarily.\nThen Newton's method applied to $\\Phi_2(\\tilde{j},z)$, with starting point $j(2\\tau_0)$ will converge to either the root of $\\Phi_2(\\tilde{j},z)$ closest to $j(2\\tau)$, or the root of $\\Phi_2(\\tilde{j},z)$ closest to $j\\left(\\frac{\\tau}{2}\\right)$, and after $[2\\log P]$ steps will produce an approximation either $j(2\\tau)$ or $j\\left(\\frac{\\tau}{2}\\right)$ of relative precision $2^{-P\/3+11}$.\n\\end{proposition}\n\\begin{proof}\n Firstly, we bound the difference between $\\tau$ and $\\tau_0$. We let $\\tau = i+\\delta$. By Lemma \\ref{taylor_series_at_i},\n \\begin{equation*}\n  \\left\\lvert j(\\tau)-1728-\\frac{j^{(2)}(i)}{2}\\delta^2\\right\\rvert\\leq0.07\\lvert\\delta\\rvert^2,\n \\end{equation*}\nso that\n\\begin{equation*}\n \\left\\lvert\\sqrt{\\frac{2(\\tilde{j}-1728)}{j^{(2)}(i)}}-\\delta\\right\\rvert\\left\\lvert\\sqrt{\\frac{2(\\tilde{j}-1728)}{j^{(2)}(i)}}+\\delta\\right\\rvert\\leq \\frac{388}{j^{(2)}(i)}\\lvert\\delta\\rvert^2+\\frac{2^{-P+1}\\lvert j\\rvert}{j^{(2)}(i)}\\leq 3\\cdot10^{-6}\\lvert\\delta\\rvert^2.\n\\end{equation*}\nNow we will show that, taking some branch of the square root above, we will obtain a good starting point for Newton's method. Let $\\epsilon = \\sqrt{\\frac{2(\\tilde{j}-1728)}{j^{(2)}(i)}}$, where the branch of the square root is arbitrary. Firstly, one of $\\left\\lvert\\epsilon-\\delta\\right\\rvert$ or $\\left\\lvert\\epsilon+\\delta\\right\\rvert$ is $\\geq\\lvert\\delta\\rvert$, and so the other is $\\leq 3\\cdot10^{-6}\\lvert\\delta\\rvert$. In the first case, $\\lvert\\tau_0-\\tau\\rvert\\leq3\\cdot10^{-6}\\lvert\\delta\\rvert$, and in the second case,\n\\begin{equation*}\n \\left\\lvert\\tau_0-\\left(-\\frac{1}{\\tau}\\right)\\right\\rvert=\\left\\lvert\\frac{-1+i\\delta+i\\epsilon+1+\\delta\\epsilon}{\\tau}\\right\\rvert\\leq\\lvert\\delta+\\epsilon\\rvert+\\lvert\\delta\\epsilon\\rvert\\leq3.1\\cdot10^{-6}\\lvert\\delta\\rvert.\n\\end{equation*}\nNow as $\\lvert\\delta\\rvert\\leq2^{-30}$, $j'(z)$ is bounded in absolute value by $1.81\\cdot10^6$ between $2\\tau_0$ and either of $2\\tau$ or $-\\frac{2}{\\tau}$, and the distance from $2\\tau_0$ to the closest of these is bounded by $6.2\\cdot10^{-6}\\lvert\\delta\\rvert$, so either\n\\begin{equation*}\n \\left\\lvert j(2\\tau_0)-j(2\\tau)\\right\\rvert\\leq 12\\lvert\\delta\\rvert\n\\end{equation*}\nor\n\\begin{equation*}\n \\left\\lvert j(2\\tau_0)-j\\left(-\\frac{2}{\\tau}\\right)\\right\\rvert\\leq 12\\lvert\\delta\\rvert.\n\\end{equation*}\nNow we bound the terms occurring in Kantorovich's criterion. Let $\\beta_0,\\beta_1,\\beta_2$ be the roots of $\\Phi_2(\\tilde{j},z)$, with $\\beta_0$ the closest root to $j(2\\tau_0)$, and $\\beta_1$ the other root near $j(2i)$. Firstly, as $\\lvert j\\rvert\\geq2^{-P\/2}$, by Lemma \\ref{bounds_for_images}, the above, and Lemma \\ref{phi2discrepancy_points_1728}, we have, for $2\\tau_0$,  \n\\begin{align*}\n  \\left\\lvert j(2\\tau_0)-\\beta_0\\right\\rvert&\\leq 12\\lvert\\delta\\rvert + 2^{-P\/3+10}\\leq12.1\\lvert\\delta\\rvert\\\\\n \\left\\lvert j(2\\tau_0)-\\beta_1\\right\\rvert&\\leq 7.25\\cdot10^6\\lvert\\delta\\rvert+2^{-P\/3+10}\\leq 7.26\\cdot10^6\\lvert\\delta\\rvert\\\\\n \\left\\lvert j(2\\tau_0)-\\beta_2\\right\\rvert&\\leq3.1\\cdot10^5+2^{-P\/3+10}\\leq3.11\\cdot10^5\\\\\n \\left\\lvert j(2\\tau_0)-\\beta_1\\right\\rvert&\\geq7.18\\cdot10^6\\lvert\\delta\\rvert-2^{-P\/3+10}\\geq 7.17\\cdot10^6\\lvert\\delta\\rvert\\\\\n \\left\\lvert j(2\\tau_0)-\\beta_2\\right\\rvert&\\geq2.4\\cdot10^5-2^{-P\/3+10}\\geq2.39\\cdot10^5,\n \\end{align*}\n and similarly, for any $\\tau'$ satisfying $\\lvert\\tau'-2\\tau\\rvert\\leq35\\lvert\\delta\\rvert$ (which we take to ensure the condition on $r$ is satisfied), we have, as $\\lvert j'(z)\\rvert\\leq1.81\\cdot10^{6}$ between $2\\tau_0$ and $\\tau'$, and as $\\lvert\\delta\\rvert\\leq2^{-30}$,\n \\begin{align*}\n  \\left\\lvert j(\\tau')-\\beta_0\\right\\rvert&\\leq 6.4\\cdot10^{7}\\lvert\\delta\\rvert+12\\lvert\\delta\\rvert\\leq0.06,\\\\\n \\left\\lvert j(\\tau')-\\beta_1\\right\\rvert&\\leq 6.4\\cdot10^{7}\\lvert\\delta\\rvert+7.26\\cdot10^6\\lvert\\delta\\rvert\\leq0.06,\\\\\n \\left\\lvert j(\\tau')-\\beta_2\\right\\rvert&\\leq6.4\\cdot10^{7}\\lvert\\delta\\rvert+3.11\\cdot10^5\\leq3.2\\cdot10^5.\n \\end{align*}\nFor our bound on $\\Phi_2(\\tilde{j},z)$ evaluated at $j(\\tau')$, we have\n\\begin{align*}\n \\left\\lvert\\Phi_2(\\tilde{j},j(2\\tau_0))\\right\\rvert &= \\lvert j(\\tau')-\\beta_0\\rvert\\lvert j(\\tau')-\\beta_1\\rvert\\lvert j(\\tau')-\\beta_2\\rvert\\\\\n &\\leq2.8\\cdot10^{13}\\lvert\\delta\\rvert^2.\n\\end{align*}\nFor the first derivative, we have\n\\begin{align*}\n \\left\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\right\\rvert &= ( j(\\tau')-\\beta_0)( j(\\tau')-\\beta_1)+( j(\\tau')-\\beta_0)( j(\\tau')-\\beta_2)+( j(\\tau')-\\beta_1)( j(\\tau')-\\beta_2)\\\\\n &\\geq \\lvert j(\\tau')-\\beta_1\\rvert\\lvert j(\\tau')-\\beta_2\\rvert-\\lvert j(\\tau')-\\beta_0\\rvert\\lvert j(\\tau')-\\beta_1\\rvert-\\lvert j(\\tau')-\\beta_0\\rvert\\lvert j(\\tau')-\\beta_2\\rvert\\\\\n &\\geq 7.17\\cdot10^6\\cdot2.39\\cdot10^5\\lvert\\delta\\rvert - 12.1\\cdot7.26\\cdot10^6\\lvert\\delta\\rvert^2- 12\\cdot3.11\\cdot10^5\\lvert\\delta\\rvert\\\\\n &\\geq 1.7\\cdot10^{12}\\lvert\\delta\\rvert,\n\\end{align*}\nand for the second derivative,\n\\begin{align*}\n \\left\\lvert\\Phi_2''(\\tilde{j},j(\\tau'))\\right\\rvert &\\leq 2\\lvert j(\\tau')-\\beta_0\\rvert+2\\lvert j(\\tau')-\\beta_1\\rvert+2\\lvert j(\\tau')-\\beta_2\\rvert\\\\\n &\\leq 6.5\\cdot10^5.\n\\end{align*}\nThese now give\n\\begin{equation*}\n \\frac{\\lvert\\Phi_2(\\tilde{j},j(2\\tau_0))\\rvert\\lvert\\Phi_2''(\\tilde{j},j(\\tau'))\\rvert}{\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\rvert^2}\\leq\\frac{2.8\\cdot10^{13}\\lvert\\delta\\rvert^2\\cdot6.5\\cdot10^5}{(1.7\\cdot10^{12}\\lvert\\delta\\rvert)^2}\\leq 2^{-17}<\\frac{1}{2},\n\\end{equation*}\nand for the condition on $r$, we have\n\\begin{align*}\n r&=35\\lvert\\delta\\rvert,\\\\\n 2\\eta&=2\\frac{\\lvert\\Phi_2(\\tilde{j},j(2\\tau_0))\\rvert}{\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\rvert}\\leq2\\frac{2.8\\cdot10^{13}\\lvert\\delta\\rvert^2}{1.7\\cdot10^{12}\\lvert\\delta\\rvert}\\leq34\\lvert\\delta\\rvert,\n\\end{align*}\nwhich ensures convergence. \nFor the rate of convergence, we need a lower bound on the second derivative and an upper bound on the first derivative, which we have as follows,\n\\begin{align*}\n \\left\\lvert\\Phi_2''(\\tilde{j},j(\\tau'))\\right\\rvert &\\geq 2\\lvert j(\\tau')-\\beta_2\\rvert-2\\lvert j(\\tau')-\\beta_0\\rvert-2\\lvert j(\\tau')-\\beta_1\\rvert\\\\\n &\\geq 4.7\\cdot10^5,\n\\end{align*}\nand\n\\begin{align*}\n \\left\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\right\\rvert &\\leq \\lvert j(2\\tau_0)-\\beta_1\\rvert\\lvert j(2\\tau_0)-\\beta_2\\rvert+\\lvert j(2\\tau_0)-\\beta_0\\rvert\\lvert j(2\\tau_0)-\\beta_1\\rvert\\\\\n &\\quad\\quad\\quad+\\lvert j(2\\tau_0)-\\beta_0\\rvert\\lvert j(2\\tau_0)-\\beta_2\\rvert\\\\\n &\\leq 7.25\\cdot10^6\\cdot3.11\\cdot10^5\\lvert\\delta\\rvert + 12.1\\cdot7.25\\cdot10^6\\lvert\\delta\\rvert^2+ 12.1\\cdot3.11\\cdot10^5\\lvert\\delta\\rvert\\\\\n &\\leq 2.3\\cdot10^{12}\\lvert\\delta\\rvert,\n\\end{align*}\nwhich gives a bound on the convergence, for $k\\geq 1$, of\n\\begin{equation*}\n \\frac{1}{2^k}2^{-17\\cdot2^k}\\frac{\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\rvert}{\\lvert\\Phi_2''(\\tilde{j},j(\\tau'))\\rvert}\\leq2^{-17\\cdot2^k}.\n\\end{equation*}\nSo in order to obtain an absolute precision of $2^{-P}$, $[2\\log P]$ steps will suffice. By Lemma \\ref{phi2discrepancy_points_1728}, this approximation to $\\beta_0$ will then be an approximation to either $j(2\\tau)$ or $j\\left(-\\frac{2}{\\tau}\\right)$ of relative precision $2^{-P\/3+11}$.\n\\end{proof}\n\n\\subsubsection{$j$ near $0$}\n\n\nWe now bound the discrepancy between the roots of $\\Phi_2(j,z)$ and $\\Phi_2(\\tilde{j},z)$ when $\\left\\lvert\\tau-\\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-30}$ and $\\lvert \\tilde{j}\\rvert\\geq 2^{-P\/2}$.\n\\begin{lemma}\\label{phi2discrepancy_points_0}\n Let $j=j(\\tau)$, where $\\left\\lvert\\tau-\\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-30}$, and suppose that $\\tilde{j}$ is an approximation to $j$ of absolute precision $2^{-P}$, with $P\\geq 300$, and $\\left\\lvert \\tilde{j}\\right\\rvert\\geq 2^{-P\/2}$. Then relative precision of any root of $\\Phi_2(\\tilde{j},z)$ to its closest root of $\\Phi_2(j,z)$ is at most $2^{-P\/3 + 2}$, and the roots of $\\Phi_2(\\tilde{j},z)$ are separated by at least $2^{-P\/6+8}$.\n\\end{lemma}\n\\begin{proof}\nFirstly, as in the proof of the Lemma \\ref{phi2discrepancy_points_1728}, with $f(z) = \\Phi_2(j,z)$ and $g(z) = \\Phi_2(\\tilde{j},z)$, we will bound the size of the coefficients of $f-g$. Let $\\tilde{j}=j+\\delta$. For the coefficient of $z^2$, we have\n\\begin{equation*}\n \\lvert 2j\\delta+\\delta^2+1448\\delta\\rvert\\leq 1500\\lvert\\delta\\rvert,\n\\end{equation*}\nfor the coefficient of $z$,\n\\begin{equation*}\n \\lvert 2976\\delta j+1488\\delta^2+40773375\\delta\\rvert\\\\\n  \\leq 4.1\\cdot10^7\\lvert\\delta\\rvert,\n\\end{equation*}\nand for the constant term\n\\begin{equation*}\n  \\left\\lvert 3\\delta j^2+3\\delta^2j+\\delta^3-16200\\delta j-16200\\delta^2+8748000000\\delta\\right\\rvert\\leq 8.8\\cdot10^9\\lvert\\delta\\rvert.\n \\end{equation*}\n Now evaluating $g(z)$ at $j(\\tau')$, for $\\tau'\\in\\{2\\tau,\\frac{\\tau}{2},\\frac{\\tau+1}{2}\\}$, as $\\lvert j(\\tau')\\rvert\\leq60000$, we have\n \\begin{align*}\n  \\lvert g(j(2\\tau))\\rvert&\\leq60000^2\\cdot1500\\lvert\\delta\\rvert + 60000\\cdot4.1\\cdot10^7\\lvert\\delta\\rvert + 60000\\cdot8.8\\cdot10^9\\lvert\\delta\\rvert\\\\\n  &\\leq 6\\cdot10^{14}\\lvert\\delta\\rvert.\n \\end{align*}\nLetting $\\beta_0,\\beta_1,\\beta_2$ be the roots of $g(z)$, we have, as $\\lvert\\tilde{j}\\rvert\\geq2^{-P\/2}$, which impels $\\lvert j\\rvert\\geq 2^{-P\/2-1}$,\n\\begin{align*}\n \\lvert\\beta_0-j(\\tau_i)\\rvert\\lvert\\beta_1-j(\\tau_i)\\rvert\\lvert\\beta_2-j(\\tau_i)\\rvert&\\leq6\\cdot10^{14}\\lvert\\delta\\rvert\\\\\n &\\leq2^{-P+50}\\\\\n\\end{align*}\nNow for any $\\beta_i$, letting $j(\\tau_j)$ be the nearest root of $f(z)$, we have\n\\begin{equation*}\n \\lvert\\beta_i-j(\\tau_j)\\rvert\\leq2^{-P\/3+17}.\n\\end{equation*}\nBy Lemma \\ref{j_to_tau_small}, as $\\lvert j\\rvert\\geq2^{-P\/2}$,  \n\\begin{equation*}\n \\left\\lvert \\tau - \\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\geq 2^{-P\/6-9 },\n\\end{equation*}\nwhich yields, by Lemma \\ref{bounds_for_images}, for $i\\neq j$,\n\\begin{equation*}\n \\lvert j(\\tau_i)-j(\\tau_j)\\rvert\\geq 2^{-P\/6+9}.\n\\end{equation*}\nSimilarly to the previous lemma, we now observe that, with $\\beta_i$ the closest root to $j(\\tau_i)$,\n\\begin{align*}\n \\lvert\\beta_i-\\beta_j\\rvert &\\geq \\lvert j(\\tau_i) - j(\\tau_j)\\rvert - 2\\cdot2^{-P\/3+17}\\\\\n &\\geq 2^{-P\/6+9} - 2^{-P\/3+17}\\\\\n &\\geq 2^{-P\/6+8}\n\\end{align*}\nand so $\\beta_i$, $\\beta_j$ are distinct. So each root has a unique closest associated root, with relative precision\n\\begin{equation*}\n \\frac{2^{-P\/3+17}}{\\lvert j(\\tau_j)\\rvert} \\leq 2^{-P\/3+2}.\n\\end{equation*}\n\\end{proof}\n\n\n\n\n\\begin{proposition}\nLet $j=j(\\tau)$, where $\\left\\lvert\\tau-\\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-31}$, and $\\tilde{j}$ be an approximation to $j$ of relative precision $2^{-P}$ such that $\\lvert\\tilde{j}\\rvert\\geq2^{-P\/2}$. Let\n\\begin{equation*}\n \\tau_0 = \\frac{1+i\\sqrt{3}}{2}+\\sqrt[3]{\\frac{6\\tilde{j}}{j^{(3)}(i)}},\n\\end{equation*}\nwhere the branch of the cube root is chosen arbitrarily.\nThen Newton's method applied to $\\Phi_2(\\tilde{j},z)$, with starting point $j(2\\tau_0)$ will converge to either the root of $\\Phi_2(\\tilde{j},z)$ closest to $j(2\\tau)$, the root of $\\Phi_2(\\tilde{j},z)$ closest to $j\\left(\\frac{\\tau}{2}\\right)$, or the root of $\\Phi_2(\\tilde{j},z)$ closest to $j\\left(\\frac{\\tau+1}{2}\\right)$, and after $[2\\log P]$ steps will produce an approximation to either $j(2\\tau)$, $j\\left(\\frac{\\tau}{2}\\right)$, or $j\\left(\\frac{\\tau+1}{2}\\right)$ of relative precision $2^{-P\/3+3}$.\n\\end{proposition}\n\\begin{proof}\n Let $\\tau = \\frac{1+i\\sqrt{3}}{2}+\\delta$. By Lemma \\ref{taylor_series_at_i},\n \\begin{equation*}\n  \\left\\lvert j(\\tau) - \\frac{j^{(3)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)}{6}\\delta^3\\right\\rvert\\leq 0.07\\lvert\\delta\\rvert^3,\n \\end{equation*}\nso that\n\\begin{equation*}\n  \\left\\lvert \\frac{6\\tilde{j}}{j^{(3)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)} -\\delta^3\\right\\rvert\\leq 1.6\\cdot10^{-6}\\lvert\\delta\\rvert^3.\n\\end{equation*}\nWe let $\\epsilon = \\sqrt[3]{\\frac{6\\tilde{j}}{j^{(3)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)}}$. Considering the geometry of the cube roots of $\\delta^3$, the product of the two furthest from $\\epsilon$ is at least $\\lvert\\delta\\rvert^2$, so letting the closest be $\\delta_0$, we have\n\\begin{equation*}\n\\lvert\\epsilon - \\delta_0\\rvert\\leq 1.6\\cdot10^{-6}\\lvert\\delta\\rvert.\n\\end{equation*}\nWe now show that any branch of the cube root taken for $\\epsilon$ will result $2\\tau_0-1$ being very close to one of the $\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent elements of $\\mathcal{F}$ to one of $2\\tau,\\frac{\\tau}{2}, \\frac{\\tau+1}{2}$. We have\n\\begin{align*}\n \\left\\lvert\\frac{1+i\\sqrt{3}}{2}+\\epsilon- \\left(-\\frac{1}{\\tau}+1\\right)\\right\\rvert&=\\frac{\\left\\lvert \\tau+i\\sqrt{3}\\tau+2+2\\epsilon\\tau-2\\tau\\right\\rvert}{2\\lvert\\tau\\rvert}\\\\\n &=\\frac{\\left\\lvert -\\frac{1}{2}-i\\frac{\\sqrt{3}}{2}-\\delta+i\\frac{\\sqrt{3}}{2} - \\frac{3}{2}+i\\sqrt{3}\\delta+2+(1+i\\sqrt{3})\\epsilon+2\\epsilon\\delta\\right\\rvert}{2\\lvert\\tau\\rvert}\\\\\n &=\\frac{\\left\\lvert(1+i\\sqrt{3})\\epsilon-(1-i\\sqrt{3})\\delta+2\\epsilon\\delta\\right\\rvert}{\\left\\lvert2\\tau\\right\\rvert}\\\\\n &\\leq\\lvert\\epsilon - e^{4i\\pi\/3}\\delta\\rvert+\\frac{\\lvert\\delta\\epsilon\\rvert}{2},\n\\end{align*}\nand \n\\begin{align*}\n \\left\\lvert\\frac{1+i\\sqrt{3}}{2}+\\epsilon- \\left(-\\frac{1}{\\tau-1}\\right)\\right\\rvert&=\\frac{\\left\\lvert \\tau-1+i\\sqrt{3}(\\tau-1)+2+2\\epsilon(\\tau-1)\\right\\rvert}{2\\lvert\\tau\\rvert}\\\\\n &=\\frac{\\left\\lvert -\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}+(1+i\\sqrt{3})\\delta-i\\frac{\\sqrt{3}}{2}-\\frac{3}{2}+2+(-1+i\\sqrt{3})\\epsilon +2\\epsilon\\delta\\right\\rvert}{2\\lvert\\tau\\rvert}\\\\\n &=\\frac{\\left\\lvert(1-i\\sqrt{3})\\epsilon-(1+i\\sqrt{3})\\delta-2\\epsilon\\delta\\right\\rvert}{\\left\\lvert2\\tau\\right\\rvert}\\\\\n &\\leq\\lvert\\epsilon - e^{2i\\pi\/3}\\delta\\rvert+\\frac{\\lvert\\delta\\epsilon\\rvert}{2}.\n\\end{align*}\nIn particular, the difference between $2\\tau_0-1$ and the closest of the $\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent elements of $\\mathcal{F}$ to $2\\tau,\\frac{\\tau}{2}, \\frac{\\tau+1}{2}$, which we let be $\\tau_1$, and the others $\\tau_2,\\tau_3$, is bounded by\n\\begin{equation*}\n 2\\left\\lvert\\epsilon-\\delta_0\\right\\rvert+\\lvert\\delta\\epsilon\\rvert\\leq3.3\\cdot10^{-6}\\lvert\\delta\\rvert.\n\\end{equation*}\nNow as the absolute value of the derivative of $j(z)$ is bounded in absolute value by $3.4\\cdot10^5$ between $\\tau_0$ and $\\tau_1$, we have\n\\begin{equation*}\n \\left\\lvert j(2\\tau_0)-j(\\tau_1)\\right\\rvert\\leq 1.2\\lvert\\delta\\rvert.\n\\end{equation*}\nNow by the bound of Lemma \\ref{phi2discrepancy_points_0} on the discrepancy of the roots of $\\Phi_2(\\tilde{j},z)$ and $\\Phi_2(j,z)$, and the bounds on the distances between the distinct roots of Lemma \\ref{bounds_for_images}, letting $\\beta_0$ be the closest root of $\\Phi_2(\\tilde{j},z)$ to $j(2\\tau_0)$, and $\\beta_1,\\beta_2$ the other two roots, we collect the relevant bounds for Kantorovich's criterion,\n\\begin{align*}\n \\left\\lvert j(2\\tau_0) - \\beta_0\\right\\rvert&\\leq 1.2\\lvert\\delta\\rvert\\\\\n \\left\\lvert j(2\\tau_0) - \\beta_1\\right\\rvert, \\left\\lvert j(2\\tau_0) - \\beta_2\\right\\rvert&\\leq 1.35\\cdot10^6\\lvert\\delta\\rvert\\\\\n \\left\\lvert j(2\\tau_0) - \\beta_1\\right\\rvert, \\left\\lvert j(2\\tau_0) - \\beta_2\\right\\rvert&\\geq 1.14\\cdot10^6\\lvert\\delta\\rvert,\n\\end{align*}\nand with $r=4\\lvert\\delta\\rvert$, for any $\\tau'$ such that $\\lvert\\tau'-2\\tau_0\\rvert\\leq4\\lvert\\delta\\rvert$, we have, as the derivative of $j$ is bounded in absolute value by $3.4\\cdot10^{5}$ here,\n\\begin{align*}\n \\left\\lvert j(\\tau') - \\beta_0\\right\\rvert&\\leq 1.37\\cdot10^6\\lvert\\delta\\rvert\\\\\n \\left\\lvert j(\\tau') - \\beta_1\\right\\rvert, \\left\\lvert j(\\tau') - \\beta_2\\right\\rvert&\\leq 2.71\\cdot10^6\\lvert\\delta\\rvert\\\\\n\\end{align*}\nNow we bound the various terms of Kantorovich's criterion. Firstly,\n\\begin{align*}\n \\lvert\\Phi_2(\\tilde{j},j(2\\tau_0))\\rvert&\\leq 1.2\\lvert\\delta\\rvert\\cdot(1.35\\cdot10^6\\lvert\\delta\\rvert)^2\\\\\n &\\leq 2.2\\cdot10^{12}\\lvert\\delta\\rvert^3,\n\\end{align*}\nfor the first derivative, we have\n\\begin{align*}\n \\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\rvert&\\geq (1.14\\cdot10^6\\lvert\\delta\\rvert)^2 - 2\\cdot1.2\\lvert\\delta\\rvert\\cdot1.35\\cdot10^6\\lvert\\delta\\rvert\\\\\n &\\geq 1.2\\cdot10^{12}\\lvert\\delta\\rvert^2,\n\\end{align*}\nand for the second derivative,\n\\begin{align*}\n \\lvert\\Phi_2''(\\tilde{j},j(\\tau'))\\rvert&\\leq 2\\cdot1.37\\cdot10^6\\lvert\\delta\\rvert+4\\cdot2.71\\cdot10^6\\lvert\\delta\\rvert\\\\\n &\\leq1.36\\cdot10^7\\lvert\\delta\\rvert,\n\\end{align*}\nso that we have\n\\begin{equation*}\n \\frac{\\lvert\\Phi_2(\\tilde{j},j(2\\tau_0))\\rvert\\lvert\\Phi_2''(\\tilde{j},j(2\\tau_0))\\rvert}{\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\rvert^2}\\leq\\frac{2.2\\cdot10^{12}\\lvert\\delta\\rvert^3\\cdot1.36\\cdot10^7\\lvert\\delta\\rvert}{(1.2\\cdot10^{12}\\lvert\\delta\\rvert^2)^2}\\leq 2^{-15}<\\frac{1}{2},\n\\end{equation*}\nand for the condition on $r$ we have\n\\begin{align*}\n r &=4\\lvert\\delta\\rvert\\\\\n 2\\eta&=2\\frac{\\lvert\\Phi_2(\\tilde{j},j(2\\tau_0))\\rvert}{\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\rvert}\\leq3.8\\lvert\\delta\\rvert,\n\\end{align*}\nensuring convergence. For the rate of convergence, we have the upper bound on the first derivative as follows,\n\\begin{align*}\n \\left\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\right\\rvert \\leq 1.4\\cdot10^{12}\\lvert\\delta\\rvert^2.\n\\end{align*}\nFor the lower bound on the second derivative, we have that\n\\begin{equation*}\n \\lvert\\Phi_2''(\\tilde{j},j(\\tau'))\\rvert = \\lvert6 j(\\tau') - 2(\\beta_0+\\beta_1+\\beta_2)\\rvert.\n\\end{equation*}\nNow the second term in the absolute value is the coefficient of $z^2$ of $\\Phi_2(\\tilde{j},z)$, which is equal to \n\\begin{equation*}\n -\\tilde{j}^2+1488\\tilde{j}-162000.\n\\end{equation*}\nBy Lemma \\ref{taylor_series_at_i}, $\n\\lvert j\\rvert\\leq\\left(\\frac{\\left\\lvert j\\left(\\frac{1+i\\sqrt{3}}{2}\\right)\\right\\rvert}{6}+0.07\\right)\\lvert\\delta\\rvert^3$, and as $\\lvert\\tilde{j}-j\\rvert\\leq2^{-P}\\leq\\lvert j\\rvert^2$, $\\lvert\\tilde{j}\\rvert\\leq1.01\\lvert j\\rvert$, so\n\\begin{equation*}\n \\lvert-\\tilde{j}^2+1488\\tilde{j}\\rvert\\leq4.13\\cdot10^8\\lvert\\delta\\rvert^3\\leq0.1\\lvert\\delta\\rvert.\n\\end{equation*}\nIn particular, $-(\\beta_1+\\beta_2+\\beta_3)=162000+\\theta_1$, where $\\lvert\\theta_1\\rvert\\leq 0.1\\lvert\\delta\\rvert$.\nAs $\\lvert\\delta\\rvert\\leq2^{-31}$, $\\lvert\\tau'-2\\tau_0\\rvert\\leq4\\lvert\\delta\\rvert$, and $\\lvert2\\tau_0-1-i\\sqrt{3}\\rvert\\leq(2+3.3\\cdot10^{-6})\\lvert\\delta\\rvert$, \n\\begin{equation*}\n \\lvert\\tau'-1-i\\sqrt{3}\\rvert\\leq 6.1\\lvert\\delta\\rvert\\leq2^{-28},\n\\end{equation*}\nand hence by Lemma \\ref{taylor_series_at_2i},\n\\begin{equation*}\n \\lvert j(\\tau') - j(i\\sqrt{3})\\rvert\\geq \\lvert  j'(i\\sqrt{3})\\delta\\rvert-1.3\\lvert\\delta\\rvert\\geq 334000\\lvert\\delta\\rvert.\n\\end{equation*}\nIn particular, as $j(i\\sqrt{3})=54000$,  $j(2\\tau_0) = 54000+\\theta_2$, where $\\lvert\\theta_2\\rvert\\geq334000\\lvert\\delta\\rvert$.\nNow combining these bounds, we have\n\\begin{align*}\n  \\lvert 6 j(\\tau') - 2(\\beta_0+\\beta_1+\\beta_2)\\rvert &=\\lvert 324000+6\\theta_2-324000+2\\theta_1\\rvert\\\\\n  &\\geq 6\\lvert\\theta_2\\rvert - 2 \\lvert\\theta_1\\rvert\\\\\n  &\\geq6\\cdot334000\\lvert\\delta\\rvert - 0.2\\lvert\\delta\\rvert\\\\\n  &\\geq2\\cdot10^6\\lvert\\delta\\rvert.\n\\end{align*}\nReturning to the convergence, we have\n\\begin{equation*}\n \\frac{\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\rvert}{\\lvert\\Phi_2''(\\tilde{j},j(\\tau'))\\rvert}\\leq\\frac{1.4\\cdot10^{12}\\lvert\\delta\\rvert^2}{2\\cdot10^6\\lvert\\delta\\rvert}\\leq2^{-10}, \n\\end{equation*}\nso a bound for the rate of convergence is\n\\begin{equation*}\n 2^{-15\\cdot2^k}.\n\\end{equation*}\nIn particular, to obtain an absolute precision of $2^{-P}$, $[2\\log P]$ steps will suffice.  The approximant obtained will then, by Lemma \\ref{phi2discrepancy_points_0}, be an approximation of one of $j(2\\tau),j\\left(\\frac{\\tau}{2}\\right)$, or $j\\left(\\frac{\\tau+1}{2}\\right)$, of relative precision at least $2^{-P\/3+3}$.\n\\end{proof}\n\n\\subsubsection{Running time}\nNow we have shown the time required to obtain an approximation to one of $j(2\\tau),j\\left(\\frac{\\tau}{2}\\right),$ or $j\\left(\\frac{\\tau+1}{2}\\right)$ is $O(M(P)\\log P)$, and the obtained value may then be used as an input to Newton's method on the compact set described in the subsequent section. In order to determine which of $j(2\\tau),j\\left(\\frac{\\tau}{2}\\right),j\\left(\\frac{\\tau+1}{2}\\right)$ was computed above, after obtaining an inverse $\\tau_0$, we compute $j(2\\tau_0)$ and $j\\left(\\frac{\\tau_0}{2}\\right)$, and return the argument corresponding to whichever was closest to our initial approximation $\\tilde{j}$ to $j$, after applying elements of $\\text{\\normalfont SL}_2(\\mathbb{Z})$ to move it to the fundamental domain (which takes $O(P)$ time). In particular we will obtain an approximation of precision at least $2^{-P\/3+12}$.\n\n\\subsection{Newton iteration on the compact set}\n\nFor $\\tau$ such that $\\tau \\in \\mathcal{F}$, $\\text{\\normalfont Im}(\\tau)\\leq 3.1$, $\\lvert \\tau - i\\rvert\\geq2^{-32}$, $\\left\\lvert\\tau-\\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\geq 2^{-32}$, we make use of an algorithm due to Dupont, \\cite{fastj} for the quasilinear evaluation of $j(\\tau)$ to relative precision, in order to invert $j(z)$ by the secant method. As we are considering a compact set, there is some fixed precision our starting points for the secant method may be in order to obtain convergence for any $\\tau$.\nWe compute the low precision inverses of $j$ by the formula\n\\begin{align*}\n \\tau_0 &= i\\frac{_2F_1\\left(\\frac{1}{6},\\frac{5}{6},1,\\frac{1}{2}+\\frac{1}{2}\\sqrt{1-\\frac{1728}{\\tilde{j}}}\\right)}{_2F_1\\left(\\frac{1}{6},\\frac{5}{6},1,\\frac{1}{2}-\\frac{1}{2}\\sqrt{1-\\frac{1728}{\\tilde{j}}}\\right)},\\\\\n \\tau_1 &= \\tau_0 - \\frac{j(\\tau_0)-j}{j'(\\tau_0)},\n\\end{align*}\nand we note that as $j$ is bounded away from $1728$, the arguments of the Gaussian hypergeometric functions are bounded away from their singularities at $0$ and $1$, and $j'$ is bounded away from $0$, so computation of the two points to a fixed precision takes a fixed amount of time. Upon a failure of convergence (or slow convergence), we may simply increase the precision of our starting points and repeat the process -- as there is a uniform bound on the required precision, this requires only constant time. As the secant method converges quadratically,  and the evaluation of $j(z)$ via the algorithm of $\\cite{fastj}$ takes $O(M(P)\\log P)$ time, the computational complexity of obtaining an approximation of precision $2^{-P+1}$ requires time $O(M(P)(\\log P)^2)$. \n\nA similar analysis to that of Lemma \\ref{taylor_series_at_i} gives a bound of \\begin{equation*}\n\\left\\lvert j^{k}(z_0)\\right\\rvert\\leq3\\cdot10^8\\cdot14^kk!                                                                                  \\end{equation*}\nfor $z_0$ in our compact set, so that if $z_1=z_0+\\lvert\\delta\\rvert$, with $\\lvert\\delta\\rvert\\leq2^{-150}$, \n\\begin{align*}\n \\lvert j(z_0)+j'(z_0)\\delta-j(z_1)\\rvert&\\leq3\\cdot10^8\\sum_{n=2}^\\infty\\lvert\\delta\\rvert^n14^n\\\\\n &\\leq3\\cdot10^8\\cdot14^2\\cdot2^{-150}\\cdot\\lvert\\delta\\rvert\\sum_{n=0}^\\infty2^{-150n}14^n\\\\\n &\\leq5\\cdot10^{-35}\\lvert\\delta\\rvert.\n\\end{align*}\nIt may be verified that $\\lvert j'(z)\\rvert\\geq10^{-19}$ in our compact set, so\n\\begin{equation*}\n \\lvert j(z_0)-j(z_1)\\rvert\\geq \\lvert j'(z_0)\\rvert\\lvert\\delta\\rvert-5\\cdot10^{-35}\\lvert\\delta\\rvert\\geq 10^{-20}\\lvert\\delta\\rvert,\n\\end{equation*}\nand in particular once we have computed by the secant method $\\tau^*$ such that $\\lvert \\tilde{j}-j(\\tau^*)\\rvert\\leq2^{-P}$, $\\lvert j-j(\\tau^*)\\rvert\\leq3\\cdot10^8\\cdot2^{-P}$, and so, with $j(\\tau)=j$, and $\\tau^*=\\tau+\\delta$, we have \n\\begin{equation*}\n \\lvert\\delta\\rvert\\leq 3\\cdot10^8\\cdot10^{20}\\cdot2^{-P}\\leq2^{-P+100}.\n\\end{equation*}\n\\subsection{$j$ very close to $0$ or $1728$}\nNow if $\\lvert\\tilde{j}\\rvert\\leq 2^{-P\/2}$, or $\\lvert\\tilde{j}-1728\\rvert\\leq2^{-P\/3}$, we simply return $\\frac{1+i\\sqrt{3}}{2}$ or $i$ respectively, and this will, by Lemma \\ref{j_to_tau_small}, be an approximation to the inverse of $j$ of absolute, and as $\\lvert\\tau\\rvert\\geq 1$, relative precision $2^{-P\/6}$.\n\n\\iffalse\nWe do not perform an analysis of the required precision for the secant method, but note that an initial precision of $2^{-90}$ would suffice for Newton's method in this situation, so the constant time required does not seem excessive.\n\\fi\n\n\\iffalse\n The following bounds hold,\n\\begin{align*}\n \\left\\lvert j(\\tau)\\right\\rvert &\\leq 6\\cdot10^5\\\\\n \\left\\lvert j'(\\tau)\\right\\rvert &\\leq 4\\cdot10^6\\\\\n \\left\\lvert j''(\\tau)\\right\\rvert &\\leq 3\\cdot10^7\\\\\n \\left\\lvert j'(\\tau)\\right\\rvert &\\geq 3\\cdot10^{-5}\\\\\n \\left\\lvert j''(\\tau)\\right\\rvert &\\geq 4\n\\end{align*}\n\\fi\n\\iffalse\nso that by  (KATOROVICH), if we start with a low--precision approximation $\\tau_0$ to $\\tau$ of absolute precision $2^{-90}$, then Newton's method applied to the function $j(z)-\\tilde{j}$ will converge to the pre--image of $\\tilde{j}$ under $j(z)$, as\n\\begin{equation}\n \\frac{(j(\\tau_0)-\\tilde{j})j''(\\tau_0)}{j'(\\tau_0)^2}\\leq \n\\end{equation}\n\\fi\n\n\\iffalse\n\\begin{equation*}\n j(z) = 32\\frac{\\theta_{00}(0;z)^8+\\theta_{01}(0;z)^8+\\theta_{10}(0;z)^8}{\\theta_{00}(0;z)^8\\theta_{01}(0;z)^8\\theta_{10}(0;z)^8},\n\\end{equation*}\n\\fi\n\n\\section{Testing CM}\n\\iffalse\nHere we consider the problem of determining whether an elliptic curve has complex multiplication. We take as input an elliptic curve in the form\n\\begin{equation*}\n y^2=4x^3+\\alpha x+\\beta\n\\end{equation*}\nwith the following data,\n\\begin{itemize}\n \\item Approximations to $\\alpha$ and $\\beta$ of relative precision ,\n \\item The Mahler measures $M_\\alpha$, $M_\\beta$ and degrees $d_\\alpha$, $d_\\beta$ of $\\alpha$ and $\\beta$.\n\\end{itemize}\n\nWe first compute the $j$-invariant of the elliptic curve,\n\\begin{equation*}\n j = 1728\\frac{\\alpha^3}{\\beta^3-27\\alpha^3}.\n\\end{equation*}\nAs $\\beta^3\\neq27\\alpha^3$, by Liouville's inequality, we have\n\\begin{equation*}\n \\lvert\\beta^3-27\\alpha^3\\rvert\\geq28^{-d_\\alpha d_\\beta}M_\\alpha^{-3d_\\beta}M_\\beta^{-3d_\\alpha}.\n\\end{equation*}\nFirst note that as $\\lvert\\alpha\\rvert\\leq M_\\alpha$, and $\\lvert\\beta\\rvert\\leq M_\\beta$, $\\lvert\\delta_1\\rvert,\\lvert\\delta_2\\rvert\\leq1$. Let our approximations to $\\alpha$, $\\beta$ be $\\tilde{\\alpha}=\\alpha+\\delta_1$, and $\\tilde{\\beta}=\\alpha+\\delta_2$, with $\\lvert\\delta_1\\rvert,\\lvert\\delta_2\\rvert\\leq\\delta$. The discrepancy between our approximation of $\\beta^3-27\\alpha^3$ and its true value is bounded as follows,\n\\begin{align*}\n \\lvert(\\beta+\\delta_1)^3-27(\\alpha+\\delta_2)^3 - \\beta^3-27\\alpha^3\\rvert&\\leq 3\\delta_2\\lvert\\beta\\rvert^2+3\\delta_1^2\\lvert\\beta\\lvert+\\delta_1^3+81\\delta_2\\lvert\\alpha\\rvert+81\\lvert\\delta_1\\rvert^2\\lvert\\alpha\\rvert+27\\lvert\\delta_2\\rvert^3\\\\\n &\\leq250\\delta\\max\\{M_\\alpha,M_\\beta\\},\n\\end{align*}\nand so on inverting this, if the difference of our approximation \n\nSo as our approximations to $\\beta$ and $\\alpha$ have precision at least\n\\begin{equation*}\n \\left(250^{-1}28^{-3d_\\alpha d_\\beta}M_\\alpha^{-3d_\\beta-3}M_\\beta^{-3d_\\alpha-3}\\right)^4 =: \\epsilon^4,\n\\end{equation*}\nour approximation of the inverse of $\\beta^3-27\\alpha^3$ is\n\\begin{align*}\n \\left\\lvert\\frac{1}{\\beta^3-27\\alpha^3}-\\frac{1}{\\tilde{\\beta}^3-27\\tilde{\\alpha}^3}\\right\\rvert &\\leq \\frac{250\\delta\\max\\{M_\\alpha,M_\\beta\\}}{28^{-d_\\alpha d_\\beta}M_\\alpha^{-3d_\\beta}M_\\beta^{-3d_\\alpha}\\left(28^{-d_\\alpha d_\\beta}M_\\alpha^{-3d_\\beta}M_\\beta^{-3d_\\alpha}-250\\delta\\max\\{M_\\alpha,M_\\beta\\}\\right)}\\\\\n &\\leq 2\\cdot250^{-2}M_\\alpha^{-5}M_\\beta^{-5}\\epsilon^2,\n\\end{align*}\nand we let this approximation be $\\gamma_1$.\nOur approximation to $\\alpha^3$ satisfies\n\\begin{equation*}\n \\lvert(\\alpha+\\delta_2)^3-\\alpha^3\\rvert\\leq7\\delta\\lvert\\alpha\\rvert^2,\n\\end{equation*}\nand we let this be $\\gamma_2$. So on multiplying this with the inverse of $\\tilde{\\beta}^3-27\\tilde{\\alpha}^3$, we obatin an approximation $\\tilde{j}$ to $j$ with error bounded as follows,\n\\begin{equation*}\n \\left\\lvert1728\\gamma_1\\gamma_2-1728\\frac{\\alpha^3}{\\beta^3-27\\alpha^3}\\right\\rvert\\leq 1728(250^{-2}M_\\alpha^{-5}M_\\beta^{-5}\\epsilon^2\\lvert\\gamma_2\\rvert+7\\delta M_\\alpha\\lvert\\gamma_1\\rvert+250^{-1}7\\delta M_\\alpha\\epsilon).\n\\end{equation*}\nas \n\\begin{align*}\n \\lvert\\gamma_1\\rvert&\\leq 2\\cdot28^{d_\\alpha d_\\beta}M_\\alpha^{3d_\\beta}M_\\beta^{3d_\\alpha},\\\\\n \\lvert\\gamma_2\\rvert&\\leq 7M_\\alpha^3,\n\\end{align*}\nour approximation to $j$ has absolute precision at least $\\epsilon^2$. Furthermore,\n\\begin{equation*}\n \\lvert j\\rvert \\geq 28^{-d_\\alpha d_\\beta}M_\\alpha^{-3d_\\beta}M_\\beta^{-3d_\\alpha}\\cdot M_\\alpha^{-3}\\geq \\epsilon,\n\\end{equation*}\nso\n\\begin{equation*}\n \\frac{\\left\\lvert\\tilde{j}-j\\right\\rvert}{\\lvert j\\rvert}\\leq \\frac{\\epsilon^2}{\\epsilon}\\leq \\epsilon,\n\\end{equation*}\nso our approximation has relative precision at least $\\epsilon$. As $\\lvert j\\rvert\\leq \\frac{1}{\\epsilon}$, \n\\begin{equation*}\n \\lvert j-e^{2\\pi i\\tau}\\rvert\\leq 2079,\n\\end{equation*}\nand $\\epsilon \\leq 250^{-4}$, $\\lvert \\tau\\rvert\\leq \\frac{-\\log\\epsilon}{2\\pi}+2$.\nApplying our algorithm to calculate the inverse of $j$, we obtain an approximation to its inverse $\\tau$ of relative precision $\\epsilon^{1\/4}\\log\\frac{1}{\\epsilon}$, and so an approximation of absolute precision $\\epsilon^{1\/4}$.\n \\fi\n \n Firstly, given the input of the $j$--invariant of an elliptic curve $E$, its degree $d$, and a bound on its height $H\\geq e^e$, we may bound the maximum discriminant from which $j$ may arise, if $E$ were to have complex multiplication. We first consider $\\lvert D\\rvert\\geq16$. The principal form of a discriminant has an associated $\\tau$ of imaginary part $\\frac{i \\sqrt{\\lvert D\\rvert}}{2}$, so as $\\lvert D\\rvert\\geq 16$, $M(j(\\tau))\\geq e^{\\pi\\sqrt{\\lvert D\\rvert}}-2079 \\geq e^{3.13\\sqrt{\\lvert D\\rvert}}$. In particular, as $M(j)\\leq H^d$, \n \\begin{equation*}\n  \\lvert D\\rvert\\leq \\frac{d^2(\\log H)^2}{9.7}.\n \\end{equation*}\n Now the $\\tau$ associated to a binary quadratic form of discriminant of absolute value $\\leq N$ has real part bounded in height by $2N$, and the square of the imaginary part bounded in height by $4N^2$, so we will determine if the preimage of $j$ is a quadratic irrational satisfying these conditions. \n \n \n \n We first bound the degree of  $j(z)$ at a quadratic irrational of discriminant $D$. For a fundamental discriminant $D$, we have the bound (Proposition 2.2, \\cite{paulin}),\n \\begin{equation*}\n  h(D)\\leq\\frac{1}{\\pi}\\sqrt{\\lvert D\\rvert}(2+\\log\\lvert D\\rvert),\n \\end{equation*}\n and in the case of non--fundamental discriminants, by Theorem 7.4 of \\cite{buell}, for an odd prime $p$,\n \\begin{equation*}\n  h(p^2D)\\leq (p+1)h(D)\\leq\\frac{p+1}{p\\pi}\\sqrt{\\lvert p^2D\\rvert}(2+\\log\\lvert D\\rvert)\\leq\\frac{4}{3\\pi}\\sqrt{\\lvert p^2D\\rvert}(2+\\log\\lvert p^2D\\rvert)\n \\end{equation*}\n and for $2$, if $D\\not\\equiv0(8)$,\n \\begin{equation*}\n h(4D)\\leq 3h(D)\\leq \\frac{3}{2\\pi}\\sqrt{\\lvert 4D\\rvert}(2+\\log\\lvert D\\rvert)\\leq \\frac{3}{2\\pi}\\sqrt{\\lvert 4D\\rvert}(2+\\log\\lvert 4D\\rvert)\n \\end{equation*}\nand if $D\\equiv 0(4)$,\n\\begin{equation*}\n h(4D)\\leq 2h(D)\\leq\\frac{1}{\\pi}\\sqrt{\\lvert 4D\\rvert}(2+\\log\\lvert D\\rvert)\\leq\\frac{1}{\\pi}\\sqrt{\\lvert 4D\\rvert}(2+\\log\\lvert 4D\\rvert).\n\\end{equation*}\nIn particular, we have the bound\n\\begin{equation*}\n h(D)\\leq\\frac{3}{2\\pi}\\sqrt{\\lvert D\\rvert}(2+\\log\\lvert D\\rvert).\n\\end{equation*}\n For the Mahler measure of $j(\\tau)$, as $\\lvert j(\\tau)\\rvert\\leq e^{2\\pi\\text{\\normalfont Im}(\\tau)}+2079\\leq9.1e^{2\\pi\\text{\\normalfont Im}(\\tau)}$, we have the upper bound in terms of the reduced binary quadratic forms $ax^2+bxy+cy^2$ of discriminant $D$,\n \\begin{equation*}\n  M(j(\\tau))\\leq9.1^{h(D)}\\exp\\left(2\\pi\\sqrt{\\lvert D\\rvert}{\\sum_{\\substack{(a,b,c)\\\\ \\text{ reduced}}}\\frac{1}{a}}\\right).\n \\end{equation*}\n The number of times each $a$ may occur is bounded by the number of solutions of $b^2\\equiv d(2a)$, which is bounded by twice the number of distinct divisors $r(a)$ of $a$. By \\cite{divisor_problem}, we have the bound\n \\begin{equation*}\n  A(x) = 2\\sum_{1\\leq a\\leq x}r(A)\\leq 2x\\log x+0.4x+2x^{1\/2},\n \\end{equation*}\n so that by Abel's summation formula,\n \\begin{align*}\n  \\sum_{1\\leq a\\leq h(D)}\\frac{r(a)}{a} &= \\frac{A(h(D))}{h(D)}+\\int_{1}^{h(D)}\\frac{A(y)}{y^2}dy\\\\\n  &\\leq \\frac{(\\log h(D))^2}{2}+1.2\\log(h(D))+2.2\\\\\n  &\\leq 1.3\\log\\lvert D\\rvert^2,\n \\end{align*}\n yielding a bound of\n \\begin{equation*}\n  M(j(\\tau))\\leq9.1^{h(D)}e^{5.2\\pi\\sqrt{\\lvert D\\rvert}(\\log\\lvert D\\rvert)^2}\\leq e^{5.9\\pi\\sqrt{\\lvert D\\rvert}(\\log\\lvert D\\rvert)^2}.\n \\end{equation*}\n In particular, by Liouville's inequality, if $j\\neq j(\\tau)$, where $\\tau$ is of discriminant $D$,\n \\begin{align*}\n  \\lvert j - j(\\tau)\\rvert &\\geq 2^{-\\frac{3d}{2\\pi}\\sqrt{\\lvert D\\rvert}(2+\\log\\lvert D\\rvert)} H^{-\\frac{3d}{2\\pi}\\sqrt{\\lvert D\\rvert}(2+\\log\\lvert D\\rvert)}e^{5.9\\pi d\\sqrt{\\lvert D\\rvert}(\\log\\lvert D\\rvert)^2},\n \\end{align*}\n and substituting our bound on $\\lvert D\\rvert$, we obtain the lower bound\n\\begin{equation*}\n \\exp\\left(-30d^2\\log H(\\log d+\\log\\log H)^2\\right).\n\\end{equation*}\n  Now as $\\lvert j\\rvert\\leq H^d$, letting $j=j(z_0)$, where $z_0\\in\\mathcal{F}$, as $\\lvert j(z_0)\\rvert\\geq e^{2\\pi\\text{\\normalfont Im}(z_0)}-2079$, we have $\\text{\\normalfont Im}(z_0)\\leq \\frac{d\\log H}{2\\pi}+2$. By differentiating the $q$--expansion of $j$, and taking an upper bound, (note that the sum of the absolute values of the terms of the tail is decreasing), we have, as $d\\log H\\geq0$,\n  \\begin{equation*}\n   \\lVert j'\\rVert_{B(z_0,1)}\\leq 2\\pi e^{d\\log H+6\\pi}+800\\leq e^{d\\log H+21},\n  \\end{equation*}\nso that if $\\lvert z_0 - \\tau\\rvert\\leq1$, \n\\begin{equation*}\n \\lvert j - j(\\tau)\\rvert\\leq\\lvert z_0 - \\tau\\rvert e^{d\\log H+21}.\n\\end{equation*}\nCombining this with our separation result, noting that $\\log d+\\log\\log H\\geq1$, if\n\\begin{equation*}\n \\lvert z_0 - \\tau\\rvert\\leq \\exp\\left(-31d^2\\log H(\\log d+\\log\\log H)^2-21\\right),\n\\end{equation*}\nthen $j=j(\\tau)$, and so $j$ is a singular modulus. Now as our method of inverting $j$ from an input of regulated precision $2^{-P}$ obtains its inverse with relative precision at least $2^{-P\/6}$, to obtain a result of absolute precision $2^{-Q}$, as $\\lvert \\tau\\rvert\\leq\\frac{d\\log H}{2\\pi}+2$, we will need an input of relative precision $2^{-6Q-2\\log(d\\log H+2)}$. In particular, with an input of relative precision\n\\begin{equation*}\n 2^{-300d^2\\log H(\\log d+\\log\\log H)^2-200},\n\\end{equation*}\nour computed approximation $\\tilde{z_0}$ to $z_0$ will have sufficient precision to determine whether\n\\begin{equation*}\n \\lvert z_0 - \\tau\\rvert\\leq\\lvert z_0 - \\tilde{z_0}\\rvert+\\lvert\\tau-\\tilde{z_0}\\rvert\n\\end{equation*}\nis sufficiently small.\nIn order to obtain our candidate $\\tau$, we develop the continued fractions of the real part and the square of the imaginary part of our computed inverse $\\tilde{\\tau}$ -- as the discriminants under consideration are bounded by $\\lvert D\\rvert$, the distance between the real parts of any two preimages of the singular moduli is at least \n\\begin{equation*}\n 19^{-1}d^{-4}(\\log H)^{-4},\n\\end{equation*}\nand of the squares imaginary parts is at least\n\\begin{equation*}\n 19^{-2}d^{-8}(\\log H)^{-8}.\n\\end{equation*}\nSo we develop the continued fractions of the real part of $\\tilde{z_0}$ and the square of the imaginary part of $\\tilde{z_0}$ until their difference from the convergents is bounded by  \n\\begin{equation*}\n 19^{-3}d^{-8}(\\log H)^{-8},\n\\end{equation*}\nfor once this holds of the convergents, they will form the only possible quadratic irrational inverse of $j$.\nIf the height of the real convergent $c_r$ is greater than $\\frac{d^2(\\log H)^2}{9.7}$, or the height of the square of the imaginary convergent $c_i$ is greater than $\\frac{d^4(\\log H)^4}{90}$, or the square root of $c_i$ is a rational number, then we may conclude that $j$ is not a singular modulus -- otherwise, letting $\\tau = c_r+i\\sqrt{c_i}$, we may test if $\\lvert \\tilde{z_0}-\\tau\\rvert$ satisfies our condition for $j=j(\\tau)$. If so, then $j$ is a singular modulus, and otherwise $j$ is not a singular modulus for $\\tau$ of discriminant $\\lvert D\\rvert\\geq16$. It is clear that the computational complexity of the calculation of the convergents and other operations in this algorithm is dominated by that of inverting $j$, so we obtain a running time, with $T=d^2\\log H(\\log d+\\log\\log H)^2$ of \n\\begin{equation*}\n O(M(T)(\\log T)^2).\n\\end{equation*}\nFor testing discriminants with $\\lvert D\\rvert\\leq16$, we may simply take the list of all $\\tau$ of discriminant $\\leq16$, and test whether $j$ is equal to them. The degree of $j(\\tau)$ for such $\\tau$ is bounded by $2$, and its Mahler measure is bounded by $3\\cdot10^6$, so if \n\\begin{equation*}\n \\lvert j-j(\\tau)\\rvert\\leq2^{-2d}H^{-2d}\\left(3\\cdot10^6\\right)^{-d}\\leq\\exp(2d(33+\\log H)),\n\\end{equation*}\nthen $j=j(\\tau)$. So with an approximation $\\tilde{j(\\tau)}$ to $j(\\tau)$ of absolute precision $2^{-4d(33+\\log H)+2}$, then we will be able to establish whether $j=j(\\tau)$. By the algorithm to compute $j$ of \\cite{fastj}, which requires time $O(M(P)\\log P)$ for relative precision of $P$, as $\\lvert j(\\tau)\\rvert\\leq 3\\cdot10^{6}$, computing $j$ to the required absolute precision possible in time\n\\begin{equation*}\n O(M(d\\log H)(\\log d+\\log\\log H)),\n\\end{equation*}\nwhich again is $O\\left(M(T)(\\log T)^2\\right)$.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Key notions}\n\nA decay of a particle $A$ is a process of the type\n $A \\rightarrow B_1 + B_2 + \\ldots B_n$, where  the particles  $B_i$ are  the decay products. Decays are ubiquitous in physics. Examples include the emission of photons from excited atoms or nuclei, alpha and beta emission, decays of composite subatomic particles (for example, neutrons or  pions) and decays of elementary particles (for example, a muon decaying to one electron and two neutrinos).\n\nMost decays follow an exponential law.  The probability that a decay takes place within the time-interval\n  $[t, t+ \\delta t]$, for $t > 0 $,  equals  $p(t) \\delta t$, where the probability density $p(t)$ is given by\n\\begin{eqnarray}\np(t) = \\Gamma e^{- \\Gamma t}. \\label{probdec}\n\\end{eqnarray}\nThe {\\em decay constant} $\\Gamma$ is positive and has dimensions of inverse time.\n\nWe note that for any two instants of time $t_1$ and $t_2$,\n\\begin{eqnarray}\np(t_2) = p(t_1) e^{-\\Gamma(t_2-t_1)},\n\\end{eqnarray}\nHence, the exponential law remains invariant under a shift of the     initial moment of time $t = 0$. The decay of an ensemble of $A$ particles after a moment of time $t$ carries no memory of any properties prion to $t$. In classical  probability theory, exponential decays correspond to {\\em Markovian processes}.\n\nExperiments typically involve a large number of decaying particles with identical preparation and the detection of  some of  the decay products. Recording the number of detection events within given time intervals $[t, t+\\delta t]$, we can reconstruct the probability density $p(t)$ associated to the decay. Hence,  $p(t)$ is a {\\em directly observable quantity}.\n\nThe focus of this paper is the derivation of the probability density $p(t)$ from the rules of quantum mechanics. We present different approaches to the problem, we apply them to specific physical problems and we analyze their underlying assumptions and their limitations.\n\n\n\n\n\n\n\n\n\n\\subsection{The quantum description of decays}\n\nThe aim of the quantum mechanical description of a decay process is to construct the probability density $p(t)$ from first principles. This probability density is different from the ones usually considered in quantum theory, because  the random variable $t$ is   temporal.  We cannot use Born's rule, because there is no self-adjoint operator for time in quantum theory \\cite{Pauli}. Indeed, the construction of quantum probabilities in which time is a random variable is an old problem---for reviews, see,  Ref.   \\cite{ToAbooks}. There are several different approaches that lead to different results, even for elementary problems, for example, constructing probabilities for the time of arrival \\cite{ML} or   specifying the time it takes a particle to tunnel through a potential barrier \\cite{tunt}.\n\n \\medskip\n {\\em The persistence amplitude method.}\nMost studies of decays avoid  a direct construction of $p(t)$.  Instead they focus on a slightly different issue,  namely, on finding the probability that the quantum system persists in its initial  configuration. In quantum theory, the notion of a configuration refers to the set of all quantum states compatible with a specific property. Mathematically, it corresponds  to a subspace of the system's Hilbert space, and it is represented by the associated projection operator. For example, if the defining property of the initial configuration is that a particle is confined by a potential well in a spatial region $U$, the relevant subspace corresponds to the projection operator $\\hat{P}_U = \\int_U dx |x\\rangle \\langle x|$ that describes the property $\"x \\in U\"$ for the particle position $x$.\n\n In some unstable systems, the subspace of the initial configuration is one-dimensional, and it coincides with the quantum state  $|\\psi\\rangle$ at which the system has been initially prepared. For such systems,\nit is convenient to employ the\n   {\\em persistence amplitude} (or {\\em survival amplitude})\n\n\\begin{eqnarray}\n{\\cal A}_{\\psi}(t) = \\langle \\psi|e^{-i\\hat{H}t}|\\psi\\rangle. \\label{persiss}\n\\end{eqnarray}\nwhere $\\hat{H}$ is the Hamiltonian of the system---for the mathematical properties of the persistence amplitude, see Refs. \\cite{GhiFo, Peres80}.\n\n\nThe modulus square of\n  ${\\cal A}_{\\psi}(t)$ is the {\\em persistence probability} (or {\\em survival probability}), i.e., the probability  a system prepared at the state $|\\psi\\rangle$ at $t = 0$ will still  be found at $|\\psi\\rangle$ by a measurement at a later time $t$. The persistence probability is identical with the {\\em fidelity} between the initial state and its time-evolution.\n\n\n   It is then suggested that the decay probability density be defined as\n\\begin{eqnarray}\np(t) = - \\frac{d}{dt} |{\\cal A}_{\\psi}(t)|^2, \\label{decprob2}\n\\end{eqnarray}\ni.e., the decay is assumed to happen when the system leaves  $|\\psi\\rangle$.\n\nEq. (\\ref{decprob2}) is a reasonable candidate probability density provided that (i) all states  normal to $|\\psi\\rangle$ that can be reached via Hamiltonian evolution correspond to decay products, and (ii) the probability of the reverse  process to the decay is negligible.\n\nIf condition (i) is not satisfied, then the propositions A = \"the system left $|\\psi\\rangle$\" and B = \"the decay happened\" are not identical: B implies A, but A does not imply B. Hence,  Eq. (\\ref{decprob2}) cannot be identified with the decay probability. For example,  if\n $|\\psi\\rangle$ evolves to a different state $|\\phi\\rangle$ that describes the initial particle $A$,  a part of the persistence amplitude will describe  Rabi-type oscillations between $|\\psi\\rangle $ and $|\\phi \\rangle$. The persistence probability may not be a decreasing function of $t$, and, hence, $p(t)$ may take negative values. It will not be a genuine probability density: Eq. (\\ref{decprob2}) will not predict the number of particles detected at each  moment of time.\n\nCondition (ii) is satisfied if there are many more states available to the decay products than to the initial particle. Furthermore, the configuration of the experiment must be such as to allow the decay products to `explore' their available states. This means that the decay products  leave the locus of their production and they are measured far away. Consider for example an excited atom contained in a cavity. For some cavity geometries,  the emitted photon does not exit the cavity immediately, and it may be reabsorbed by the atom at a later time. The persistence amplitude of the excited atomic state may has an oscillating component. In this case, the candidate probability density (\\ref{decprob2}) does not correlate with  photodetection records.\n\n\nFor systems that satisfy conditions (i) and (ii), Eq. (\\ref{decprob2}) provides a  reasonable  construction of the decay probability density. It works best in model systems that incorporate the conditions (i) and (ii)  above into their definition. On such example is   the Lee model \\cite{Lee} that is presented in   Sec. 3. However, even in such models the candidate probability density (\\ref{persiss}) may become negative outside the exponential decay regime---see, Sec. 3.1.3.\n\n\nWe note that the interpretation of the persistence probability as the fidelity of the initial state makes it a useful tool also for the study of a broader class of physical phenomena than decays \\cite{BPSZ}. Hence, the quantity (\\ref{decprob2}) may be of physical interest, even if it takes negative values and cannot be identified with the probability density for decay.\n \\medskip\n\n {\\em Probability currents.}\n An alternative elementary description of decays is available, whenever we can  associate a probability-current operator $\\hat{\\pmb J}({\\pmb x}, t)$ to one of the decay products. For example, for non-relativistic particles of mass $m$ satisfying Schr\\\"odinger's equation with Hamiltonian $\\hat{H}$, the current operator is\n\\begin{eqnarray}\n \\hat{\\pmb J}({\\pmb x}, t) = \\frac{1}{2m}e^{i\\hat{H} t}\\left[ \\hat{\\pmb p} \\delta^3(\\hat{\\pmb x} - {\\pmb x}) + \\delta^3(\\hat{ \\pmb x} - {\\pmb x}) \\hat{\\pmb p} \\right] e^{-i\\hat{H}t},\n\\end{eqnarray}\nwhere $\\hat{\\pmb x}$ and $\\hat{\\pmb p}$ are the standard position and momentum operators, respectively.\nThe expectation value of  $\\hat{\\pmb J}({\\pmb x}, t)$ on a   state $|\\psi\\rangle $ reproduces the standard expression for the probability current $\\frac{1}{m}[\\mbox{Im} \\psi^*({\\pmb x}, t) {\\pmb \\nabla} \\psi({\\pmb x}, t)]$  associated to the solution $\\psi({\\pmb x}, t) := \\langle {\\pmb x}|e^{-i \\hat{H}t}|\\psi\\rangle$ of Schr\\\"odinger's equation.\n\nGiven a current operator  $\\hat{\\pmb J}({\\pmb x}, t)$ for one of the decay products, we can evaluate the flux  $\\Phi_C(t) = \\int_C d^2{\\pmb \\sigma} \\cdot \\langle \\psi| \\hat{\\pmb J}({\\pmb x}, t)|\\psi\\rangle$ through any surface $C$. In many set-ups, the probability flux  coincides with the particle flux through this surface, which is  a directly measurable quantity. Then, the flux $\\Phi_S(t)$ over a two-sphere surrounding the unstable system is expected to be proportional to the decay probability density $p(t)$.\n\n The main limitation of this method is that it cannot be consistently applied to relativistic systems, because of difficulties in defining probability currents that are both  Lorentz covariant and causal \\cite{RoHo}. For this reason, it has mainly been used only in non-relativistic settings, in particular, for decays that can be described in terms of tunneling \\cite{LL, Perel}.\n\n Another problem of the probability current method is the possibility of back-flow: the current operator has  negative eigenvalues even for states with strictly positive momentum \\cite{Brack}. Hence, in some cases the probability flux may turn out to be negative for some times $t$--especially when the current is evaluated near the locus of the decay \\cite{Winter}. In such cases the flux is not a reliable measure of detected outgoing particles.\n\n \\medskip\n\n {\\em Detector models.} A more rigorous description of quantum decays requires the incorporation of  the measurement process  \\cite{EkSi, GhiFo}. In fact, the idea that quantum decays cannot be explained solely in terms of unitary time evolution was  crucial to the early interpretational discussions of quantum theory \\cite{Heismem}. We have to take into account   the irreversibility of the quantum measurements on the decay products  in order to avoid theoretical discrepancies. This is the case, for example, in systems characterized by competing decay channels. If one ignores the effects of the apparatus, the persistence amplitude  exhibits large oscillations, thereby contradicting the standard classical description of sequential decays \\cite{FoGhi}.\n\nIn quantum measurement theory, the consistent treatment of the measuring apparatus allows of the description of quantum observables in terms of Positive Operator Valued Measures (POVM), i.e., a set of positive operators $\\hat{\\Pi}(a)$, where $a$ are the values of the physical magnitude recorded by the apparatus. Then, given an initial state $\\hat{\\rho}$ of the system, the probability that the result $a$ is obtained is given by $Tr\\left[\\hat{\\rho}_0 \\hat{\\Pi}(a)\\right]$.\n\n In recent years, a new class of model for particle detection has been developed, allowing    for  the construction of quantum observables for the time $t$ of a detection event \\cite{AnSav}---see, also \\cite{An08} for an early application to the decay problem. In this paper, we present a study of decays, using one such\nobservable $\\hat{\\Pi}(t)$ for detection time  that can be obtained by  elementary arguments similar to the ones of standard photodetection theory \\cite{Glauber}.\n\n\n\\subsection{This paper}\nThe aim of this paper is to provide the\n reader with tools for addressing a large class of decay problems. We present the most common approximation schemes, as well as criteria for checking when they fail. We prefer to work with models that admit simple solutions and controlled approximations, but we introduce more complex models for the explicit demonstration of important physical issues.\n\nA key motivation for this paper is to provide explicit arguments that the traditional methods for describing decays (persistence amplitude and probability current) are inapplicable for many problems of current physical interest.  These methods work excellently for exponential decays, but they lead to negative values of the decay probability $p(t)$ outside the exponential regime. The problem is not a failure of some approximation,  the  very definition of such methods cannot guarantee positivity. We believe that a first-principles construction of decay probabilities is essential for describing decays in novel regimes, for example, in entangled systems \\cite{AnHu2, CVS17, C18}, attosecond tunneling ionization \\cite{atto} or decay oscillations in nuclear physics \\cite{GSI}.\n\n\n\nThe paper is organized as follows.\n  Sec. 2 presents the simplest method for the study of decays, namely, the evaluation of the persistence amplitude as a line integral on the complex energy space. This method is particularly suitable for perturbative decays, i.e., systems in which the decay originates from a small term in the Hamiltonian of the system. It also applies to relativistic systems described by quantum field theory.\n\n  We show that   exponential decay is common in perturbative decays: it originates  from the separation of energy scales in the persistence amplitude. Nonetheless, it is neither exact nor generic. Exponential decay fails at very early and very late times, and also if the energy of the initial state is close to a resonance of the system.\n\n In  Sec. 3, we present the Lee model that describes the decaying particle as a  two-level system. This model can easily be adapted to problems in different branches of physics, including high energy physics, nuclear physics, atom optics, and condensed matter. Here, we present its applications to photo-emission and beta decay. We also employ the model in order to demonstrate the breakdown of the persistence amplitude method outside the exponential decay regime.\n\n Sec. 4 describes the decays of an atom in a cavity, providing an explicit example of decays that are non-exponential at all times because of resonance.\n\nIn Sec. 5, we study non-perturbative decays due to quantum tunneling, using the probability current method. We show that exponential decays originate from a decoherence condition, namely the lack of interferences between different attempts of the particle to tunnel through the barrier.\n\nIn Sec. 6, we revisit the Lee  model using probabilities constructed from a simple temporal observable, analogous to the one used in photodetection theory. We show that the detection probability reproduces the results of the persistence amplitude method, but can also be used in regimes where the latter fails.\n\nThe material in this article is mostly at the level of a graduate course in quantum mechanics. Familiarity with quantum many-particle systems is presupposed.\nSecs. 2, 3 and  5 provide a minimal   introduction to quantum decays, explaining the emergence of the exponential decay law and its limitations,  developing methods that can be applied to different branches of physics, and presenting some important    physical examples.\nThe Appendix contains a sketch of  additional results that can be used as exercises.\n\n\\section{Perturbative evaluation of the persistence amplitude}\n\nIn this section, we study decays in which the persistence amplitude can be evaluated perturbatively. All physical properties of the decays are encoded into a function defined on the complex energy plane, the {\\em self-energy function}. We identify the conditions under which the exponential law emerges, and we identify regimes for non-exponential decays.\n\n\\subsection{ Preliminaries}\nBy definition, the initial state\n $|\\psi\\rangle$ of an unstable system is not an eigenstate of the Hamiltonian $\\hat{H}$. However, in many problems of physical interest, $|\\psi\\rangle$ is close to an eigenstate of $\\hat{H}$, in the sense that  $\\hat{H}$  is of the form\n\\begin{eqnarray}\n\\hat{H} = \\hat{H}_0 + \\hat{V}, \\label{Hamper}\n\\end{eqnarray}\nwhere $|\\psi\\rangle$ is an eigenstate of $\\hat{H}_0$ and  $\\hat{V}$ is a small perturbation. For example, $\\hat{H}_0$ may be the Hamiltonian of an atom, and $\\hat{V}$ the interaction Hamiltonian of the atom with the quantum electromagnetic field.\nIn this case, we evaluate the persistence amplitude   perturbatively.\n\nLet us denote by $|b\\rangle$ the eigenstates of $\\hat{H}_0$ and by $E_b$ the corresponding eigenvalues. Then,\n\\begin{eqnarray}\n\\hat{H}_0 |b\\rangle = E_b|b\\rangle.\n\\end{eqnarray}\nWe will take the initial state $|\\psi\\rangle$ to be one of the eigenstates, say $|a\\rangle$, and we will write  $V_{ab} := \\langle a|\\hat{V}|b\\rangle$.\n\nWithout loss of generality, we     assume that  $\\langle a|\\hat{V}|a\\rangle =0$. If $\\langle a|\\hat{V}|a\\rangle \\neq 0$,\n  we can always  redefine\n \\begin{eqnarray}\n \\hat{H}_0' = \\hat{H}_0 + \\sum_a \\langle a|\\hat{V}|a\\rangle |a\\rangle \\langle a| \\hspace{1cm}  \\hat{V}' = \\hat{V} - \\sum_a \\langle a|\\hat{V}|a\\rangle |a\\rangle \\langle a|,\n  \\end{eqnarray}\nso that  $|a\\rangle$ is an eigenstate of $\\hat{H}_0'$ and $\\langle a|\\hat{V}'|a\\rangle = 0$.\n\n\\medskip\n\n\\noindent {\\em The resolvent.} The perturbative calculation of the persistence amplitude is simplified by using the {\\em resolvent} associated to the Hamiltonian operator:\n  $(z - \\hat{H})^{-1}$, for $z \\in {\\pmb C}$. We will often write the resolvent  as $\\frac{1}{z - \\hat{H}}$.\n\n\n\nThe resolvent is related to the evolution operator $e^{-i\\hat{H}t}$ by\n\n\n\\begin{eqnarray}\ne^{-i\\hat{H}t} = \\lim_{\\epsilon \\rightarrow 0} \\frac{i}{2\\pi  } \\int_{-\\infty }^{\\infty } \\frac{dE e^{-iEt}}{E+ i \\epsilon - \\hat{H}},  \\label{propagator2}\n\\end{eqnarray}\nwhere $\\epsilon > 0$  and $t > 0$.\n\nEq. (\\ref{propagator2}) follows from  the identity\n\\begin{eqnarray}\ne^{- i \\omega t} =  \\lim_{\\epsilon \\rightarrow 0} \\frac{i}{2\\pi  } \\int_{-\\infty }^{\\infty } \\frac{dE e^{-iEt}}{E+ i \\epsilon - \\omega}. \\label{idtycmplx}\n\\end{eqnarray}\n To prove Eq. (\\ref{idtycmplx}) we evaluate the line integral\n\n \\begin{eqnarray}\nI(t) = \\oint_{C_+} dz \\frac{e^{-izt}}{z - \\omega}, \\label{cont}\n\\end{eqnarray}\nalong the negative-oriented contour $C_+$ of Fig. 1. At the lower half of the imaginary plane $Im z  =  - y$, for $y >0$. Therefore, the integrand along the semicircle is proportional to $e^{-yt}$ and vanishes as the radius of the semi-circle tends to infinity. Hence,\n\\begin{eqnarray}\nI(t) = \\lim_{\\epsilon \\rightarrow 0} \\int_{-\\infty }^{\\infty } \\frac{dE e^{-iEt}}{E+ i \\epsilon - \\omega}.\n\\end{eqnarray}\nThe contour $C_+$ includes the single pole of the integrand (\\ref{cont}), at $z = \\omega$. Using Cauchy's residue theorem, we arrive at Eq. (\\ref{idtycmplx}).\n\nBy integrating along the contour $C_-$, we can similarly prove that\n\\begin{eqnarray}\n\\lim_{\\epsilon \\rightarrow 0}   \\int_{-\\infty }^{\\infty } \\frac{dE e^{-iEt}}{E -  i \\epsilon - \\hat{H}} = 0 ,  \\label{propagator2b}\n\\end{eqnarray}\nfor $\\epsilon > 0$  and $t > 0$. The   integral vanishes because the contour $C_-$ does not enclose any pole.\n\n\n\\begin{figure}[]\n\\includegraphics[height=6cm]{complexcurve} \\caption{ \\small Integration contours $C_{\\pm}$ for calculating the integrals  (\\ref{idtycmplx}) and (\\ref{propagator2b}).  The integrals are evaluated at the limit where the radius $r$ of the semi-circle goes to infinity. Both curves are traversed clock-wise and therefore have negative orientation.}\n\\end{figure}\n\n\nA crucial property of the resolvent is that it can be expanded in a perturbative series. For a Hamiltonian $\\hat{H}$ of the form (\\ref{Hamper}),\n \\begin{eqnarray}\n  (z - \\hat{H})^{-1}\n = (z - \\hat{H}_0 - \\hat{V})^{-1} = [(z-\\hat{H}_0) (1 - (z-\\hat{H}_0)^{-1}\\hat{V})]^{-1}\\nonumber \\\\\n  = (1 - (z-\\hat{H}_0)^{-1}\\hat{V})^{-1}(z-\\hat{H}_0)^{-1}. \\nonumber\n \\end{eqnarray}\n Using the geometric series formula $(1 - \\hat{A})^{-1} = \\sum_{n=0}^{\\infty} \\hat{A}^n$, we obtain\n\\begin{eqnarray}\n\\frac{1}{z - \\hat{H}} = \\frac{1}{z - \\hat{H}_0} + \\frac{1}{z - \\hat{H}_0}  \\hat{V} \\frac{1}{z - \\hat{H}_0}  + \\frac{1}{z - \\hat{H}_0}  \\hat{V}\\frac{1}{z - \\hat{H}_0}  \\hat{V} \\frac{1}{z - \\hat{H}_0}  + \\ldots \\label{seriesprop}\n\\end{eqnarray}\n\n\\subsection{The random phase approximation}\nWe construct the persistence amplitude  ${\\cal A}_a(t)$ for an initial state $|a\\rangle$ that is an eigenstate of $\\hat{H}_0$. By  Eq. (\\ref{propagator2}),\n \\begin{eqnarray}\n {\\cal A}_a(t) = \\lim_{\\epsilon \\rightarrow 0} \\frac{i}{2\\pi } \\int_{-\\infty }^{\\infty } dE e^{-iEt}  G_a(E+i\\epsilon) , \\label{ampll}\n \\end{eqnarray}\n where\n  \\begin{eqnarray}\n  G_a(z) =  \\langle a|(z - \\hat{H})^{-1}|a\\rangle.\n \\end{eqnarray}\n\n We evaluate $G_a(z)$ using the perturbative series (\\ref{seriesprop}). We assume that $\\hat{V}$ is of first order to  some  dimensionless parameter $\\lambda << 1$.  The zeroth-order contribution to $G_a(z)$ is\n  $ \\frac{1}{z - E_a}$. The first-order contribution is\n $ \\frac{1}{(z - E_a)^2} \\langle a|\\hat{V}|a\\rangle$ = 0.\n\n The second-order term is  $ \\frac{1}{(z - E_a)^2} \\langle a|\\hat{V} \\frac{1}{z - \\hat{H}_0} \\hat{V}|a\\rangle$. Writing $ \\frac{1}{z - \\hat{H}_0} =  \\sum_b \\frac{1}{z - E_b} |b\\rangle \\langle b|$, this term becomes $ \\frac{1}{(z - E_a)^2} \\Sigma_a(z) $, where\n\\begin{eqnarray}\n\\Sigma_a(z) = \\sum_b \\frac{|V_{ab}|^2}{z - E_b}, \\label{sigmaa}\n\\end{eqnarray}\nis the {\\em self-energy} function of the state $|a\\rangle$.\n The third-order term is\n \\begin{eqnarray}\n\\sum_{bc} \\frac{1}{(z-E_a)^2} \\frac{ V_{ab} V_{bc} V_{ca}}{(z-E_b)(z-E_c)}, \\nonumber\n\\end{eqnarray}\nand the fourth-order term is\n\\begin{eqnarray}\n\\sum_{bcd} \\frac{1}{(z-E_a)^2} \\frac{ V_{ab} V_{bc} V_{cd}V_{da}}{(z-E_b)(z-E_c)(z-E_d)}. \\nonumber\n\\end{eqnarray}\n The procedure can be continued ad infinitum.\n\nWe assume that the Hamiltonian $\\hat{H}_0$ has continuous spectrum for $E > \\mu$, for some parameter $\\mu$. The continuous spectrum corresponds to the kinetic energy of the decay products. With this assumption, we invoke the   {\\em Random Phase Approximation} (RPA), according to which\n\n\\begin{eqnarray}\n\\sum_c \\frac{V_{ac}V_{cb}}{z-E_c} \\simeq \\delta_{ab} \\Sigma_a(z). \\label{rpa}\n\\end{eqnarray}\nThe reasoning for Eq. (\\ref{rpa}) is the following. Suppose that the system is contained in a box of volume $V$ with periodic boundary conditions and that it contains a large number $N$ of degrees of freedom. The RPA is the assumption that the phases of the matrix elements $\\langle a| \\hat{V}|b\\rangle$, for $b \\neq a$ are {\\em randomized} in the continuous limit, i.e., in the limit where $N$ and $V$ goes to infinity, with $N\/V$ constant. By `randomized', we mean that the phases of $\\langle a| \\hat{V}|b\\rangle$ do not exhibit any periodicity, or quasi-periodicity as $b$ varies. Hence, the summation over $b$ is a sum of many random phases, and is expected to be much smaller than the term for $a = b$ that involves no such phases. Eq. (\\ref{rpa}) then follows.\n\n  The RPA was initially postulated in systems with  a large but finite number of degrees for freedom, in condensed matter \\cite{BohmPi, GeBr} and\nin nuclear physics \\cite{Bloch, Nami}. However, it  also applies to systems with an infinite number of degrees of freedom, i.e., quantum fields.  Indeed, the name `self-energy'  for the function (\\ref{sigmaa}) originates from quantum field theory.\n\n\n\n\n\nBy the RPA,     odd-order terms in the expansion of $G_a(z)$  vanish. Furthermore, the terms of order $2n$ for $n$ integer equal $\\Sigma_a(z)^n\/(z-E_a)^{n+1}$. Hence, $G_a(z)$ is give by  a geometric series,\n\\begin{eqnarray}\nG_a(z) = \\frac{1}{z-E_a} \\sum_{n=0}^{\\infty} \\frac{\\Sigma_a(z)^n}{(z-E_a)^n} = \\frac{1}{z-E_a} \\frac{1}{1 - \\frac{\\Sigma_a(z)}{z-E_a}} = \\frac{1}{z - E_a - \\Sigma_a(z)}. \\label{asas}\n\\end{eqnarray}\n\n Eq. (\\ref{asas}) is accurate to order $\\lambda^2$. Hence, it can be obtained without the RPA, solely by a perturbative analysis. Most treatments of RPA assume a weaker condition that Eq. (\\ref{rpa}), so that the resulting expression for  $G_a(z)$ involves a  self-energy function that coincides with Eq. (\\ref{sigmaa}) only to second order in $\\lambda$---for details, see,   Ref. \\cite{NNP96}. In the present treatment, the RPA gives the same results with a second-order perturbative expansion. In general, it has a larger domain of validity.\n\n\n\\subsection{Structure of the self-energy function}\nEqs. (\\ref{asas}) and (\\ref{ampll}) imply that\n\\begin{eqnarray}\n{\\cal A}_a(t) = \\frac{i}{2 \\pi  } \\lim_{\\epsilon \\rightarrow 0} \\int_{-\\infty }^{\\infty } \\frac{dE e^{-iEt}}{E + i \\epsilon- E_a - \\Sigma_a(E+i\\epsilon)}. \\label{ampldec}\n\\end{eqnarray}\nIf the self-energy function $\\Sigma_a(z)$ were analytic, the integral (\\ref{ampldec}) could be evaluated by integrating     along the contour of Fig. 1, and using Cauchy's theorem. However, this is not the case, the self-energy function is discontinuous and it may contain poles or branch points.\n\nTo see this, consider the definition     (\\ref{sigmaa}) of  $\\Sigma_a(z)$. Since the spectrum of $\\hat{H}_0$ is continuous for energies larger than $\\mu$, $\\Sigma_a(z)$ is\n     divergent along the half-line $D = \\{ z \\in {\\pmb C}|\\; \\mbox{Re} z > \\mu, \\mbox{Im}z = 0\\}$. Shifting the Hamiltonian by a constant term, we can always choose    $\\mu = 0$, so that $D = {\\pmb R}^+$.\n\nTo analyze the behavior of  $\\Sigma_a(z)$ near $D$, we define\n\\begin{eqnarray}\n\\Sigma_a(E^{\\pm}) = \\lim_{\\eta \\rightarrow 0} \\Sigma_a(E\\pm i\\eta),\n \\end{eqnarray}\nfor $\\eta > 0$. By definition,\n\\begin{eqnarray}\n\\mbox{Re} \\Sigma_a(E\\pm  i \\eta) = \\sum_b \\frac{|V_{ab}|^2(E-E_b)}{(E  - E_b)^2 +  \\eta^2}.\n\\end{eqnarray}\nIt follows that $\\mbox{Re} \\Sigma_a(E^+) = \\mbox{Re} \\Sigma_a(E^-)$, leading to the definition of the {\\em level-shift function}\n\n\\begin{eqnarray}\nF_a(E) := \\mbox{Re}  \\Sigma_a(E^{\\pm})  \\label{faE}\n\\end{eqnarray}\nOn the other hand, the imaginary part of $\\Sigma_a(z)$\n\\begin{eqnarray}\n\\mbox{Im} \\Sigma_a(E\\pm  i \\eta) = \\mbox{Im} \\sum_b \\frac{|V_{ab}|^2}{E  - E_b + i \\eta} = \\mp \\eta \\sum_b \\frac{|V_{ab}|^2}{(E  - E_b)^2 +  \\eta^2}, \\label{imsa}\n\\end{eqnarray}\n is discontinuous as the half-line $D$ is crossed.  Eq. (\\ref{imsa}) implies that   $\\mbox{Im} \\Sigma(E^-) = - \\mbox{Im} \\Sigma(E^+)\\geq 0$.\n\nWe define the {\\em decay function}\n\\begin{eqnarray}\n\\Gamma_a(E) := 2 \\mbox{Im}  \\Sigma(E^-) > 0, \\label{gammaE}\n\\end{eqnarray}\n so that\n \\begin{eqnarray}\n \\Sigma_a(E^{\\pm}) = F_a(E) \\mp \\frac{i}{2} \\Gamma_a(E).\n \\end{eqnarray}\n  The discontinuity of $\\Sigma_a$  across $D$  is\n$\\Delta \\Sigma_a(E) := \\Sigma_a(E^+) - \\Sigma_a(E^-) = - i \\Gamma_a(E)$.\nObviously,  $\\Gamma_a(E) = 0$ for $E < 0$.\n\n  Eq. (\\ref{ampldec}) becomes\n\\begin{eqnarray}\n{\\cal A}_a(t) = \\frac{i}{2 \\pi  }   \\int_{-\\infty }^{\\infty } \\frac{dE e^{-iEt}}{E  - E_a - F_a(E) + \\frac{i}{2} \\Gamma_a(E)}. \\label{ampldecb}\n\\end{eqnarray}\nSince $\\Gamma(E) = 0$ for $E < 0$,\n\\begin{eqnarray}\n{\\cal A}_a(t) = \\frac{i}{2 \\pi }  \\left( \\int_{-\\infty }^{0} \\frac{dE e^{-iEt}}{E  - E_a - F_a(E)}+ \\int_{0 }^{\\infty } \\frac{dE e^{-iEt}}{E  - E_a - F_a(E) + \\frac{i}{2} \\Gamma_a(E)} \\right). \\label{ampldecc}\n\\end{eqnarray}\n\nOn the other hand, Eq. (\\ref{propagator2b}) implies that\n\\begin{eqnarray}\n \\lim_{\\epsilon \\rightarrow 0} \\int_{-\\infty }^{\\infty } \\frac{dE e^{-iEt}}{E - i \\epsilon- E_a - \\Sigma_a(E^-)} = 0,\n\\end{eqnarray}\nhence,\n\\begin{eqnarray}\n  \\int_{-\\infty }^{0 } \\frac{dE e^{-iEt}}{E -   E_a - F_a(E)}  = -  \\int_{0 }^{\\infty } \\frac{dE e^{-iEt}}{E -   E_a - F_a(E) - \\frac{i}{2}  \\Gamma_a(E)}.\n\\end{eqnarray}\nSubstituting into Eq. (\\ref{ampldecb}), we obtain\n\\begin{eqnarray}\n{\\cal A}_a(t) &=& \\frac{i}{2 \\pi  }  \\int_{0}^{\\infty } dE e^{-iEt} \\left[ \\frac{1}{E  - E_a - F_a(E) + \\frac{i}{2} \\Gamma_a(E)} - \\frac{1}{E  - E_a - F_a(E) - \\frac{i}{2} \\Gamma_a(E)}\\right] \\nonumber \\\\\n&=&  \\frac{1}{2 \\pi }  \\int_{0 }^{\\infty } \\frac{dE \\Gamma_a(E) e^{-iEt}}{ [E - E_a - F_a(E)]^2 + \\frac{1}{4}[\\Gamma_a(E)]^2}. \\label{mainampl}\n\\end{eqnarray}\n\n\n\n\nEq. (\\ref{mainampl}) is the main result of this section, an explicit formula relating the persistence amplitude to the components of the self-energy function.\n\n\n\\subsection{The Wigner-Weisskopf approximation}\n\nThe integral  (\\ref{mainampl}) involves the functions $F_a(E)$ and $\\Gamma_a(E)$ in the denominator. These function are second-order with respect  the perturbation parameter $\\lambda << 1$. If  $|F_a(E)| << E_a$ and $\\Gamma(E) << E_a$ for $E$ in the vicinity of $E_a$, we can evaluate the persistence amplitude using the {\\em Wigner-Weisskopf Approximation} (WWA) \\cite{WWA}.\n\nThe WWA essentially postulates the substitution of the Lorentzian-like function of $E$ in Eq. (\\ref{mainampl}) with an actual Lorentzian. The justification is the following.  The integral (\\ref{mainampl}) is dominated by values of $E$  within distance of order $\\lambda^2$ from  $E \\simeq E_a$. For these values, the integrand is of order $\\lambda^{-2}$, otherwise it is of order $\\lambda^0$.   Hence, with an error of order $\\lambda^2$,\n we   can substitute the energy-shift function $F_a(E)$ with the constant\n\\begin{eqnarray}\n\\delta E := F_a(E_a)\n\\end{eqnarray}\n and the decay function $\\Gamma_a(E)$ with the constant\n \\begin{eqnarray}\n \\Gamma  := \\Gamma_a(E_a).\n \\end{eqnarray}\n Within an error of the same order of magnitude,\n we extend the range of integration to $(-\\infty, \\infty)$. Thus, we  obtain an elementary integral\n\\begin{eqnarray}\n{\\cal A}_a(t) =  \\frac{\\Gamma_a}{2 \\pi }  \\int_{-\\infty}^{\\infty } \\frac{dE   e^{-iEt}}{ (E - E_a -  \\delta E )^2 + \\frac{1}{4}\\Gamma^2} =  e^{ -i (E_a +\\delta E ) t - \\frac{\\Gamma }{2}t},  \\label{ampl3}\n\\end{eqnarray}\n\n\n Substituting Eq.  (\\ref{ampl3}) in Eq. (\\ref{decprob2}), we conclude\n\\begin{eqnarray}\np(t) = \\Gamma  e^{-\\Gamma t}.\n\\end{eqnarray}\n\n Hence, the WWA leads to an exponential decay law with a decay constant $\\Gamma $ that is determined by the imaginary part of the self-energy function. The real part of the self-energy function leads to a shift\n $\\delta E $ of the energy level $E_a$, usually referred to as the {\\em Lamb shift}.\n\n\n\n\n The same expression for the decay constant $\\Gamma $ is  given by {\\em Fermi's golden rule}. To see this, we use\nEq. (\\ref{imsa}),\n\\begin{eqnarray}\n\\Gamma  =  \\lim_{\\eta \\rightarrow 0} \\sum_b \\frac{2\\eta |V_{ab}|^2}{(E_a  - E_b)^2 +  \\eta^2}.\n\\end{eqnarray}\nSince $\\lim_{\\eta \\rightarrow 0} \\frac{\\eta}{x^2 +\\eta^2} = \\pi \\delta(x)$, we obtain Fermi's decay rate\n\\begin{eqnarray}\n\\Gamma  = 2 \\pi \\sum_{b, E_b = E_a} |V_{ab}|^2. \\label{fgg}\n\\end{eqnarray}\n\n\n\n\n\n\nA more rigorous derivation of Eq. (\\ref{ampl3}) from Eq. (\\ref{ampldecb}) employs the notion of the  {\\em van Hove limit} \\cite{vanHo}.  This limit is obtained as follows. First, we change the time variable to $\\tilde{t} = \\lambda^2 t$, and we define $x = (E-E_a)\/\\lambda^2$. Then, Eq. (\\ref{ampldecb}) becomes\n\\begin{eqnarray}\ne^{iE_at} {\\cal A}_a(t) =  \\frac{i}{2 \\pi  } \\lim_{\\bar{\\epsilon} \\rightarrow 0} \\int_{-\\infty }^{\\infty } \\frac{dx e^{-ix \\tilde{t}}}{x + i \\bar{\\epsilon}-  \\tilde{F}_a(E_a+\\lambda^2x ) + \\frac{i}{2} \\tilde{\\Gamma}_a(E_a+\\lambda^2x)}, \\label{aaavH}\n\\end{eqnarray}\nwhere $\\tilde{\\Gamma} (E) = \\lambda^{-2}\\Gamma(E)$ and $\\tilde{F} (E) = \\lambda^{-2}F (E)$ are of order $\\lambda^0$; $\\bar{\\epsilon} = \\epsilon\/\\lambda^2$ can still be chosen arbitrarily small. The van-Hove limit consists of taking the limit $\\lambda \\rightarrow 0$ in the r.h.s. of Eq. (\\ref{aaavH}), while keeping $\\tilde{t}$ constant. Then,\n\\begin{eqnarray}\ne^{iE_at} {\\cal A}_a(t) =  \\frac{i}{2 \\pi  }  \\int_{-\\infty }^{\\infty } \\frac{dx e^{-ix \\tilde{t}}}{x   - \\tilde{F}_a(E_a) + \\frac{i}{2} \\tilde{\\Gamma}_a(E_a)}, \\label{aaavH2}\n\\end{eqnarray}\nEq. (\\ref{aaavH2}) can be straightforwardly evaluated using the contour integral of Fig. 1. It leads to  Eq. (\\ref{ampl3}). Hence, the WWA  is equivalent to the imposition of the van Hove limit on the decay amplitude.\n\nWhile the van Hove limit of the persistence amplitude is always well defined, we have to keep in mind that in physical systems $\\lambda^2$ is always finite and non-zero. In specific systems, the van Hove limit may misrepresent the form of the persistence amplitude. This is the case if $\\Gamma_a(E)$ strongly varies with $E$ within a distance of order $\\lambda^2$ from $E_a$.  Hence, taking the van Hove limit is justified only if the self-energy function is sufficiently `flat' in the vicinity of $E_a$ \\cite{BaRa}.\n\n\n\n\n\n\\subsection{Beyond   exponential decay }\nWe derived  exponential decay as a consequence of two approximations, the RPA and WWA. Since the RPA is redundant for a second-order approximation to the self-energy function,  WWA is the only approximation that needs to be considered in the weak coupling regime.\n\n\\medskip\n\n\\noindent {\\em Very early times.} First, we note that exponential decay cannot be valid at very early times. This is a general statement that originates from the definition (\\ref{decprob2}).\nWe Taylor-expand the persistence amplitude around $t = 0$, to obtain    ${\\cal A}_{\\psi} = 1 - i t \\langle \\hat{H} \\rangle - \\frac{t^2}{2} \\langle \\hat{H}^2 \\rangle + \\ldots$. Keeping terms up to order    $t^2$,\nthe probability density becomes\n\\begin{eqnarray}\n|{\\cal A}_{\\psi} |^2 = 1 - (\\Delta H)^2 t^2 + \\ldots.\n\\end{eqnarray}\nEq.  (\\ref{decprob2}) implies that\n\\begin{eqnarray}\np(t) =  (\\Delta H)^2 t.\n\\end{eqnarray}\n It follows that $p(0) =0$, while in   exponential decays,  $p(0) = \\Gamma$.  This violation of exponential decay at early times is a special case of the  so called {\\em quantum Zeno effect} \\cite{MiSu, FP08}, and it has been verified experimentally  \\cite{Zeno}.\n\nThe early time behavior of a decaying system can also be identified by an uncertainty relation for the persistence probability, first derived by Mandelstam and Tamm \\cite{MaTa45}, see, also \\cite{Bhatta, Dodo}. For a system in a state $|\\psi\\rangle$ and Hamiltonian $\\hat{H}$, the Kennard-Robertson uncertainty relation gives,\n \\begin{eqnarray}\n \\Delta H \\Delta A \\geq \\frac{1}{2} |\\langle \\psi| [\\hat{H}, \\hat{A}]|\\psi\\rangle|, \\label{kerob}\n \\end{eqnarray}\n for any observable $\\hat{A}$. For the Heisenberg evolved observable $\\hat{A}(t) = e^{i\\hat{H}t} \\hat{A} e^{-i\\hat{H}t}$,  Eq. (\\ref{kerob}) implies the so called Mandelstam-Tamm inequality\n\\begin{eqnarray}\n\\Delta H \\Delta A(t) \\geq \\frac{1}{2} \\large|\\langle \\psi| \\frac{\\partial \\hat{A}(t)}{\\partial t} |\\psi\\rangle \\large| \\label{mata1}\n\\end{eqnarray}\nChoosing   $\\hat{A} = |\\psi\\rangle \\langle \\psi|$, the expectation $ \\langle \\psi|\\hat{A}(t)|\\psi\\rangle$ coincides with the survival probability of $|\\psi\\rangle$, which we will denote by $a(t)$.  Hence, Eq.\n (\\ref{mata1}) becomes\n  \\begin{eqnarray}\n  \\frac{|\\dot{a}|}{ \\sqrt{a-a^2}} \\leq 2\\Delta H. \\label{mata2}\n  \\end{eqnarray}\n\n  Eq. (\\ref{mata2}) holds for all quantum states and Hamiltonians. It is easy to verify that it is  not compatible with the exponential decay law at early times. Setting $a(t) = e^{-\\Gamma t}$, we find that Eq. (\\ref{mata2} is satisfied only if $t > \\Gamma^{-1} \\ln \\left[ 1 + \\frac{\\Gamma^2}{4 (\\Delta H)^2}\\right]$.\n\n  Assuming that $\\dot{a} \\leq 0$, we can integrate inequality   (\\ref{mata2}), to obtain\n\\begin{eqnarray}\n\\cos^{-1} \\sqrt{a(t)} \\leq  \\Delta H t  \\label{mata3}\n\\end{eqnarray}\nEq. (\\ref{mata3})  is satisfied trivially for\n $t > \\frac{\\pi}{2\\Delta H}$; for $t < \\frac{\\pi}{2\\Delta H}$, it implies that  $a(t) \\geq    \\cos^2(\\Delta H t)$. The half-life $\\tau_h$  of the state is defined by the requirement\n  $a(\\tau_h) = \\frac{1}{2}$. Hence, Eq.  (\\ref{mata3}) leads to an uncertainty relation between energy spread and half-life\n\\begin{eqnarray}\n\\Delta H \\tau_h \\geq \\frac{\\pi}{4},\n\\end{eqnarray}\n that applies even in regimes where exponential decay fails\\footnote{For  $\\tau$ such that $a(\\tau) = 0$, Eq.  (\\ref{mata3}) leads to the uncertainty relation $\\Delta H \\tau \\geq \\frac{\\pi}{4}$. The time $\\tau$ is interpreted as the minimum time required for the system to arrive to a state orthogonal to $|\\psi\\rangle$, and the uncertainty relation is said to define a limit to the `speed of quantum evolution', and, consequently, to the speed of quantum computation. The inequality has been improved \\cite{MaLe, LeTo} in order to also incorporate the case of $\\Delta H \\rightarrow \\infty$. It has then been broadly generalized, for example, to open-system dynamics \\cite{MLOS} and classical systems \\cite{Shan}---for a review, see, Ref. \\cite{DeCa}.  }.\n\n\\medskip\n\n\\noindent {\\em The persistence amplitude in terms of a contour integral. } In order to establish the  range of validity of the WWA, we must evaluate the survival amplitude (\\ref{ampldec}) without approximations. To this end, we analytically continue  $\\Gamma_a(E)$ to the lower imaginary plane, and we define   $\\Sigma^-_a(z) := \\Sigma_a(z)$ and $ \\Sigma^+_a(z) := \\Sigma_a(z) - i \\Gamma_a(z)$.   The functions $\\Sigma^{\\pm}_a(z)$ can be viewed as components of a multi-valued complex function: $\\Sigma^+$ corresponds to  the first Riemann sheet, and $\\Sigma^-$ corresponds to the second Riemann sheet \\cite{colth}.\n\nWe define a line integral over the contour $C_-$ of Fig. 1.  The contribution from the  circle at infinity vanishes, hence, taking the limit $\\epsilon \\rightarrow 0$, we obtain\n\n \\begin{eqnarray}\n{\\cal A}_a(t) = \\frac{i}{2 \\pi } \\oint_{C_-} dz e^{-izt} \\left(\\frac{1}{z-E_a - \\Sigma^+_a(z)} -  \\frac{1}{z-E_a - \\Sigma^-_a(z)}  \\right)+  I_a(t),  \\label{amplnew}\n\\end{eqnarray}\nwhere $I_a(t)$ is the contribution of the negative real axis\\footnote{We can use a different integration contour, consisting of the positive real axis, an arc of the circle at infinity, and a half line $N$ that starts from the latter and ends at the origin---for an example, see, Fig. 5. As long as the contour encloses the physically relevant poles near the positive real axis, the analysis remains unchanged. Then, $I_a(t)$ is a line integral along $N$.\nThe choice   of the negative imaginary axis for $N$ is particularly useful for calculating the long time limit of Eq. (\\ref{amplnew}). In fact, we use it implicitly in the derivation of Eq. (\\ref{iat0}).}.\n\\begin{eqnarray}\nI_a(t) = \\frac{1}{2\\pi }\\int_0^{\\infty} \\frac{dx e^{ixt} \\Gamma_a(-x)}{[x+  E_a  +F_a(-x)]^2 + \\frac{1}{4}\\Gamma_a(-x)^2}. \\label{remainder}\n\\end{eqnarray}\n\n\n If $\\Sigma^{\\pm}_a(z)$ are meromorphic functions in the region of the complex plane enclosed by the contour $C_-$, the line integral in Eq. (\\ref{amplnew}) is evaluated by finding the poles of the integrand inside $C_-$. To this end, we must  solve the equation\n\\begin{eqnarray}\nz-E_a - \\Sigma_a^{\\pm}(z) = 0. \\label{poleq}\n\\end{eqnarray}\nWe will refer to this contribution to the survival amplitude as the {\\em pole term}; we will refer to $I_a(t)$ as the {\\em remainder term}.\n\n\n\\medskip\n\n\\noindent {\\em The pole term.}   Unless $E_a$ is very close to a point of divergence of $\\Sigma_a^{\\pm} $, we expect that $|\\Sigma_a(E_a^+)|\/ E_a$  is  much smaller than unity. Hence, there  exist a solution to Eq. (\\ref{poleq}) within a distance of order $\\lambda^2$ from the point $z = E_a$.  Setting $z = E + \\lambda^2 x$, we find that $ \\lambda ^2 x = \\Sigma^{\\pm}_a(E_x + \\lambda^2x) = \\Sigma_a^{\\pm} (E_a) + O(\\lambda^4)$. Hence, Eq. (\\ref{poleq}) is satisfied for\n\n\\begin{eqnarray}\nz = E_a + F_a(E_a) \\mp \\frac{i}{2} \\Gamma_a(E_a) + O(\\lambda^4). \\label{BBpole}\n\\end{eqnarray}\nThe pole with the plus sign is outside $C_-$, hence, it does not contribute to the contour integral. The pole with the minus sign reproduces the result of the WWA\\footnote{Hence, the WWA justifies a common statement of scattering theory, that unstable particles correspond to  poles of the $S$-matrix on the second Riemann sheet \\cite{colth, RNewt}. Indeed, the poles of the S-matrix provide a simple way for identifying the basic properties of an unstable state. However, the $S$-matrix formalism  gives a coarse description of a decay process. It  enforces the constancy of transition rates \\cite{Weinberg} by averaging over a long time interval, and, for this reason, it cannot discern transient phenomena, including deviation from exponential decay.}. The associated residue is $[1 - F_a'(E_a) + \\frac{i}{2}\\Gamma'(E_a)]^{-1}$.\n\nLet all other solutions to Eq.   (\\ref{poleq}) inside $C_-$ be  $z = \\alpha_i$ and $R_i$ the associated residues. Then, by Cauchy's theorem,\n\n\\begin{eqnarray}\n{\\cal A}_a(t)  = \\frac{e^{ -i (E_a +\\delta E ) t - \\frac{\\Gamma }{2}t}}{1- F_a'(E_a) + \\frac{i}{2}\\Gamma_a'(E_a)}  + K_a(t) + I_a(t), \\label{accuracy}\n \\end{eqnarray}\n where\n $K_a(t) = \\sum_i R_i e^{-i\\alpha_i t}$. Hence, the WWA is valid if both terms $K_a(t)$ and $I_a(t)$ are negligible.\n\n  If the self-energy function is analytic in the region enclosed by the contour $C_-$, then, typically,  other roots $z = \\alpha_i$ to Eq. (\\ref{poleq}) are further away from the real axis than the perturbative root (\\ref{BBpole}). This means that   $|Im \\alpha_i| >> \\Gamma$. Hence, $K_a(t)$  vanishes much faster than the WWA term, and can be neglected after an initial transient period.\n\n   If\n $\\Sigma^{\\pm}_a(z)$ has divergences near or on the real axis, then there may exist other roots, as close to the real axis as the perturbative root  (\\ref{BBpole}).  There is no general rule here, and, in principle, a case-by-case study is required.\n   Of particular importance is the possibility of {\\em resonance}. If $E_a$ is sufficiently close to   a pole of $\\Sigma_a(z)$, then the condition $|\\Sigma_a(E_a^+)|\/ E_a <<1$ will not hold. Then, there is no perturbative pole, the WWA fails, and decays are likely to be non-exponential.\n\n\\medskip\n\n \\noindent{\\em Late times.} The contribution of the pole term to the persistence amplitude drops exponentially.  The remainder term $I_a(t)$ drops as an inverse power law, hence, it dominates at sufficiently late times \\cite{Helund, Nami2}.\n\nTo see this, we change  variables to $y = x t$, and write Eq. (\\ref{remainder}) as\n\n\\begin{eqnarray}\nI_a(t) = \\frac{1}{2\\pi t}\\int_0^{\\infty} \\frac{e^{iy} \\Gamma_a(-y\/t)}{[y\/t + E_a +F_a(-y\/t)]^2 + \\frac{1}{4}\\Gamma_a(-y\/t)^2  }, \\label{eqyb}\n\\end{eqnarray}\n As $t \\rightarrow \\infty $, the denominator becomes $[E_a+F_a(0)]^2 + \\frac{1}{4}\\Gamma_a(0)^2 = E_a^2 +O(\\lambda^2)$.\n  Hence,\n\n\\begin{eqnarray}\nI_a(t) \\simeq    \\frac{1}{2\\pi E_a^2 t} \\int_0^\\infty dy e^{i y} \\Gamma_a(-y\/t) =  \\frac{1 }{2\\pi E_a^2  }  \\int_0^\\infty dx e^{i xt} \\Gamma_a(-x). \\label{iat3}\n\\end{eqnarray}\n\n\nThe long -time limit of $I_a(t)  $ is determined by  the behaviour of $\\Gamma_a(z)$ near zero.\n  Let $\\Gamma_a(z) \\simeq A z^{n}$ as $x \\rightarrow 0$, for some positive constant $A$ and integer $n$. We evaluate the integral $\\int_0^\\infty dx e^{i xt} (-x)^n$ for imaginary time $t = i \\tau$, with $\\tau >0$, to obtain $(-1)^n n!\\tau^{1+n}$. We analytically continue back to $t$, to obtain\n\n\\begin{eqnarray}\n{\\cal A}_a(t)=  - \\frac{A n! }{2\\pi E_a^2 i^{n+1} } \\frac{1}{t^{n+1}}. \\label{iat0}\n\\end{eqnarray}\n\n\nThe persistence amplitude  drops as an inverse power of   $t$. Hence, in the long-time limit, decays are characterized by an inverse-power law and not an exponential one. In most systems, the inverse-power behavior appears is time-scales too large to be measurable. Nonetheless, it has been experimentally confirmed in luminescence decays of dissolved organic\nmaterial \\cite{longtime}, with the exponent depending on the material.\n\n\n\\bigskip\n\n\\noindent We summarize the results of our analysis.\n\\\\\n\\noindent 1. The exponential law always fails at very short and very long times.\n\\\\\n\\noindent 2. If the self-energy function has no divergences near or on the real axis,  the exponential law is very accurate at all intermediate times.\n\\\\\n\\noindent 3. If the self-energy function has divergences near or on the real axis,  there may be corrections to exponential decay from other poles.\n\\\\\n\\noindent 4. Exponential decay likely fails on resonance, i.e., for energies near a divergence point of the self-energy function.\n\n\n\n\n\n\\section{Lee's model}\nIn this section, we present a general model for decays that can be applied to many different physical situations. This model originates from the work of T.D. Lee \\cite{Lee}. We shall consider two versions of the model, one where the decay is accompanied by the emission of a bosonic particle and one where the decay is accompanied by the emission of two fermions. The former describes  spontaneous emission of photons, the latter describes beta decay.\n\\subsection{Bosonic emission}\n\\subsubsection{Definitions and properties}\nWe consider decays of the form $A' \\rightarrow A + B$, in which the emitted bosonic particle $B$ is much lighter than the particles $A'$ and $A$. Examples of this type of decay are the following.\n\\begin{itemize}\n\\item  $A'$ is the excited state of a nucleus, or of an atom, or of a molecule, $A$ is the corresponding ground state and   $B$ is a photon.\n\\item  $A'$ is a heavy nucleus that decays to  the nucleus $A$ and an alpha particle.\n\\item  $A'$ and $A$ are baryons and  $B$ is a light meson.\n\\end{itemize}\nThe key idea in Lee's model is to ignore all degrees of freedom pertaining to the motion of the  heavy particles. The heavy particles are then treated as a two-level system (2LS).The ground state $|g\\rangle$ corresponds to the particle $A$ and the excited state $|e\\rangle$ to the particle   $A'$. The $B$ particle is described by a bosonic Fock space ${\\cal F}_B$. We   denote the vacuum of ${\\cal F}_B$ by\n  $|0\\rangle$ and the creation and annihilation operators by $\\hat{a}_r$ and $\\hat{a}^{\\dagger}_r$. The latter satisfy the canonical commutation relations\n  \\begin{eqnarray}\n  [\\hat{a}_r, \\hat{a}_s] = 0, \\hspace{0,3cm}   [\\hat{a}^{\\dagger}_r, \\hat{a}^{\\dagger}_s] = 0, \\hspace{0,3cm}    [\\hat{a}_r, \\hat{a}^{\\dagger}_s] = \\delta_{rs}.\n  \\end{eqnarray}\n  The indices $r, s, \\ldots$ denote the possible states of a single $B$ particle, i.e., they label a (generalized) basis on the associated single-particle  Hilbert space.\n\nThe Hilbert space of the total system is  ${\\pmb C}^2 \\otimes {\\cal F}_B$. The Hamiltonian\n  $\\hat{H}_L$ of Lee's model consists of three terms\n\\begin{eqnarray}\n\\hat{H}_L = \\hat{H}_A + \\hat{H}_B + \\hat{V} ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\hat{H}_A = \\frac{1}{2} \\Omega (\\hat{1} + \\hat{\\sigma}_3) , \\label{vlee1}\n\\end{eqnarray}\nis the 2LS Hamiltonian, and $\\Omega$ stands for the energy difference of the two levels;\n\\begin{eqnarray}\n\\hat{H}_B =  \\sum_r \\omega_r \\hat{a}^{\\dagger}_r \\hat{a}_r, \\label{vlee2}\n\\end{eqnarray}\nis the   Hamiltonian for non-interacting particles $B$ particles;\n\\begin{eqnarray}\n\\hat{V} = \\sum_r \\left(g_r \\hat{\\sigma}_+   \\hat{a}_r + g^*_r  \\hat{\\sigma}_-   \\hat{a}_r^{\\dagger} \\right), \\label{vlee3}\n\\end{eqnarray}\n  is the interaction term. It describes the excitation of the 2LS accompanied by the absorption of a B particle, and the decay of the 2LS accompanied by the emission of a $B$ particle.\n   The coefficients  $g_r$ depend upon the physical system under consideration.\n\nThe initial state for Lee's model is\n\\begin{eqnarray}\n|A'\\rangle = |e\\rangle \\otimes |0\\rangle,\n\\end{eqnarray}\ni.e., the 2LS is excited and no $B$-particle is present.\n\n We evaluate the survival amplitude\n  ${\\cal A}(t) := \\langle A' |e^{-i\\hat{H}_L t}|A'\\rangle$ using the series (\\ref{seriesprop}) for $\\hat{H}_0 = \\hat{H}_A + \\hat{H}_B$.  We find that\n\\begin{eqnarray}\n(z-\\hat{H}_0)^{-1}\\hat{V}(z-\\hat{H}_0)^{-1}|A'\\rangle =  \\frac{1}{z - \\Omega} |g\\rangle \\otimes \\sum_r \\frac{g_r}{z- \\omega_r} \\hat{a}^{\\dagger}_r|0\\rangle,\n\\end{eqnarray}\nhence,  the  matrix elements\n $\\langle \\psi|\\hat{V}|A'\\rangle$ are non-zero only for $|\\psi\\rangle$\nare of the form $|g\\rangle \\otimes \\hat{a}^{\\dagger}_r|0\\rangle$.  In particular, $\\langle A'|\\hat{V}|A'\\rangle = 0$.\n\nThe next term in the perturbative series is\n\\begin{eqnarray}\n(z-\\hat{H}_0)^{-1}\\hat{V} (z - \\hat{H}_0)^{-1} \\hat{V}(z - \\hat{H}_0)^{-1}|A'\\rangle = \\frac{\\Sigma(z)}{(z- \\Omega)^2}  |A'\\rangle, \\label{leetwo}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\Sigma(z) = \\sum_r \\frac{|g_r|^2}{z-\\omega_r}, \\label{selee}\n\\end{eqnarray}\nis the self-energy function for the initial state  $|A'\\rangle$.\n\nSince the second-order term is proportional to the zero-th order one, the above expressions are reproduced to all orders of perturbation. We obtain\n\\begin{eqnarray}\n[(z - \\hat{H}_0)^{-1}\\hat{V}]^{2n}(z - \\hat{H}_0)^{-1} |A'\\rangle = \\frac{1}{z- \\Omega} \\left(\\frac{ \\Sigma(z)}{z- \\Omega}\\right)^n|A'\\rangle \\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n[(z - \\hat{H}_0)^{-1}\\hat{V}]^{2n+1} (z - \\hat{H}_0)^{-1} |A'\\rangle = \\frac{1}{z- \\Omega} \\left(\\frac{ \\Sigma(z)}{z- \\Omega}\\right)^n|g\\rangle \\otimes \\sum_r \\frac{g_r \\hat{a}^{\\dagger}_r}{z- \\omega_r } |0\\rangle. \\nonumber\n\\end{eqnarray}\nHence,\n\\begin{eqnarray}\n\\frac{1}{z -\\hat{H}}|A'\\rangle =  \\frac{1}{z-\\Omega} \\sum_{n=0}^{\\infty} \\frac{\\Sigma(z)^n}{(z-\\Omega)^n} \\left(|A'\\rangle + |g\\rangle \\otimes \\sum_r \\frac{g_r \\hat{a}^{\\dagger}_r}{z- \\omega_r} |0\\rangle\\right)\\nonumber \\\\\n = \\frac{1}{z - \\Omega - \\Sigma(z)}\\left(|A'\\rangle + |g\\rangle \\otimes \\sum_r \\frac{g_r \\hat{a}^{\\dagger}_r}{z- \\omega_r} |0\\rangle\\right). \\label{decay555}\n\\end{eqnarray}\n\nWe conclude that\n\\begin{eqnarray}\nG(z)   =  \\frac{1}{z - \\Omega - \\Sigma(z)}. \\label {asas2}\n\\end{eqnarray}\n\nWe obtained Eq.  (\\ref{asas}) by resumming the full perturbative series (\\ref{seriesprop}), without any approximation. This means that\n Lee's model incorporates the RPA in its definition.\n\n\n\\subsubsection{Spontaneous emission from atoms}\nWe will evaluate the self-energy function in a simple model where the emitting particles have zero spin and zero rest mass, i.e., scalar photons \\cite{AnHu}. This model ignores the effects of polarization, but otherwise it describes well the emission of photons by excited atoms. The inclusion of polarization changes $\\Sigma(z)$  only by a multiplicative factor that  can be absorbed in a redefinition of the coupling constant.\n\nIn this model, the basis $r$ corresponds to photon momenta ${\\pmb k}$, $\\omega_r$ corresponds to $\\omega_{\\pmb k } = |{\\pmb k}|$, and the summation over $r$ corresponds to integration with   measure $\\frac{d^3k}{(2\\pi)^3}$. If  the size $a_0$ of the emitting atom is much smaller than the wavelengths of the emitted radiation, we can describe the atom-radiation interaction in the dipole approximation  \\cite{QO}. Then, the coupling coefficients are  $g_{\\pmb k} = \\frac{\\lambda }{\\sqrt{\\omega_{\\pmb k}}}e^{i {\\pmb k}\\cdot{\\pmb r}}$, where $\\lambda $ is a dimensionless constant and ${\\pmb r}$ is the position vector of the atom\\footnote{The coupling originates from an interaction Hamiltonian of the form $\\lambda[\\hat{\\sigma}_-\\hat{\\phi}^{(-)}({\\pmb r}) + \\hat{\\sigma}_+\\hat{\\phi}^{(+)}({\\pmb r})] $, where the scalar field components $\\hat{\\phi}^{(\\pm)}  ({\\pmb r}) $ are given by Eqs. (\\ref{scalar}, \\ref{chik}).}.\n\n We substitute into Eq. (\\ref{selee}) for the self-energy function, to obtain\n\\begin{eqnarray}\n\\Sigma(z) = \\frac{\\lambda^2}{2 \\pi^2} \\int_0^{\\infty} \\frac{k dk}{z - k}. \\label{seel4}\n\\end{eqnarray}\n\nThe integral in Eq. (\\ref{seel4}) diverges as $k \\rightarrow \\infty$.  However, photon energies much larger than $a_0^{-1}$, where $a_0$ is the size of the atom, are not physically relevant. Hence, we regularize the integral (\\ref{seel4}) by introducing a high-frequency cut-off $\\Lambda >> \\Omega$,\n\n  \\begin{eqnarray}\n   \\Sigma (z) = \\frac{\\lambda^2}{2 \\pi^2}  \\int_0^{\\Lambda} \\frac{k  dk}{z-k} = - \\frac{\\lambda^2}{2 \\pi^2} \\left[ \\Lambda + z (\\ln(\\Lambda - z) - \\ln (-z)) \\right]. \\label{seel5}\n  \\end{eqnarray}\n\n  There are many other ways to regularize the integral (\\ref{seel4}), for example, by inserting an exponential cut-off function  $e^{-k\/\\Lambda}$. The choice of regularization does not affect the form of $\\Sigma(z)$ for the physical range of values of $z$, i.e., $|z|<< \\Lambda$. However, it introduces an arbitrariness in the behaviour of $\\Sigma(z)$ for $z$ of the order of $\\Lambda$. In particular,   the apparent branch-point at $z = \\Lambda$ in Eq. (\\ref{seel5}) is a artefact of the regularization. As far as the physically relevant values of $z$ are  concerned,  there is no error in substituting\n   $\\ln(\\Lambda - z) \\simeq \\ln \\Lambda$ in Eq. (\\ref{seel5}). Furthermore, it is convenient to absorb the constant term in $\\Sigma(z)$ into a redefinition of the frequency  $\\tilde{\\Omega} = \\Omega -  \\frac{\\lambda^2\\Lambda}{2 \\pi^2}$.\n   With these modifications, Eq. (\\ref{seel5}) becomes\n\n      \\begin{eqnarray}\n   \\Sigma (z) =   \\frac{\\lambda^2  }{2 \\pi^2} \\left[ -  (\\ln\\Lambda) z  + z \\ln (-z) \\right]. \\label{seel6}\n  \\end{eqnarray}\n\n   The logarithm in Eq. (\\ref{seel6}) is defined in the principal branch,  i.e., its argument lies in  $(-\\pi, \\pi]$. When evaluating $\\Sigma(E^-)$, we substitute $z = E - i \\eta$, for $\\eta > 0$. Hence, $\\ln(-z) = \\ln E + \\ln[-(1-i\\eta\/E)]$. As $\\eta \\rightarrow 0$,  $1-i\\eta\/E \\simeq e^{-i\\eta\/E}$. We have two options for the $-1$ term in the logarithm, we can express it either as $e^{i \\pi}$ or as $e^{-i \\pi}$. The first choice gives $\\ln (e^{i (\\pi-\\eta\/E)})$, hence, the argument lies in the principal branch. The second choice gives $\\ln (e^{i (-\\pi-\\eta\/E)})$, and the argument lies outside the principal branch. Only the first choice is acceptable. Hence, $\\ln[-(1-i\\eta\/E)] = i(\\pi-\\eta\/E)$, and  $\\lim_{\\eta \\rightarrow 0} \\ln[-(E-i\\eta)] = \\ln E + i \\pi$. It follows that\n\n\\begin{eqnarray}\n\\Sigma(E^-) = - \\frac{\\lambda^2}{2 \\pi^2} \\left[  E \\ln(\\Lambda\/E) - i \\pi E ) \\right].\n\\end{eqnarray}\nBy Eqs. (\\ref{faE}) and (\\ref{gammaE}),\n\\begin{eqnarray}\nF(E) &=& - \\frac{\\lambda^2}{2 \\pi^2}     E \\ln(\\Lambda\/E)   \\\\\n\\Gamma(E) &=& \\frac{\\lambda^2}{ \\pi}E.  \\label{gammaE2}\n\\end{eqnarray}\n\n\nHence, the decay constant in the WWA is\n\\begin{eqnarray}\n\\Gamma = \\Gamma(\\tilde{\\Omega}) = \\frac{\\lambda^2}{ \\pi}\\tilde{\\Omega}.\n\\end{eqnarray}\nThe Lamb shift depends on the cut-off parameter $\\Lambda$ and it induces a renormalization of the frequency $\\tilde{\\Omega}$.\n\n\n\\bigskip\n\n\\noindent{\\em Validity of exponential decay.}\nWe examine the domain of validity of the WWA. First, we consider the contribution from other poles. To this end, we look for solutions to equation\n \\begin{eqnarray}\n z - \\tilde{\\Omega} - \\frac{\\lambda^2}{2 \\pi^2} z \\ln(1 - \\Lambda\/z) = 0.  \\label{sol3}\n \\end{eqnarray}\n Eq. (\\ref{sol3}) admits one solution  for $z$ near  $\\Lambda$. To see this, we write $z = \\Lambda ( 1 +x)$ for $|x| << 1$. We obtain\n$x   \\simeq  e^{-\\frac{2\\pi^2}{\\lambda^2}} << 1$ to leading order in $\\lambda^2$.  This solution is possibly an artefact of the regularization, but in any case it corresponds to oscillations much more rapid than any physically relevant time scale. It averages to zero in any measurement with  temporal resolution of order $\\sigma_T >> \\Lambda^{-1}$. We conclude that the contribution of other poles to the decay probability is negligible.\n\nNext, we evaluate the remainder term, Eq. (\\ref{iat0}). By Eq. (\\ref{gammaE2}), $A = \\frac{\\lambda^2}{\\pi}$ and $n = 1$. Hence,\n\\begin{eqnarray}\nI(t) = \\frac{\\Gamma}{2 \\pi \\tilde{\\Omega}^3 t^2}. \\label{asyem}\n\\end{eqnarray}\nFor sufficiently large $t$ the remainder term dominates over the exponential term $e^{- \\Gamma t\/2}$ in the persistence amplitude. The relevant time-scale $\\tau$ is found from the solution of the equation $|I(\\tau) |= e^{- \\Gamma \\tau\/2}$ at large $\\tau$. We set $\\Gamma \\tau\/ 2 =x$ and $\\alpha = \\frac{1}{8\\pi} (\\Gamma\/\\tilde{\\Omega})^3$, to obtain an equation for $x$,\n\\begin{eqnarray}\n2 \\ln x - x =  \\ln \\alpha. \\label{eqqq}\n\\end{eqnarray}\n Solutions to Eq. (\\ref{eqqq}) for different values of $\\Gamma\/\\tilde{\\Omega}$ are given in the following able.\n\n \\bigskip\n\n \\begin{tabular}{|c||c|c|c|c|c|c|}\n   \\hline\n  \n   $\\Gamma\/\\tilde{\\Omega}$ & $10^{-2}$ & $10^{-3}$ & $10^{-4}$ & $10^{-5}$ & $10^{-6}$ & $10^{-7}$ \\\\\n   \\hline\n   x & 23.3 & 30.8 & 38.1 & 45.4 & 52.5 & 59.8 \\\\\n   \\hline\n  $ e^{-\\Gamma\\tau}$ & $5\\cdot 10^{-21}$ & $2 \\cdot 10^{-27}$ & $8 \\cdot 10^{-34}$ & $4 \\cdot 10^{-40}$  & $2 \\cdot 10^{-46}$  &$ 10^{-52}$ \\\\\n   \\hline\n \\end{tabular}\n\n \\bigskip\n\nEven for values of $\\Gamma\/\\tilde{\\Omega}$ as large as $0.01$,    the exponential decay law breaks down at a time where less than $1:10^{20}$ of the initial   atoms remains in the excited state. We conclude that any deviations from exponential decay in photo-emission are negligible outside the quantum Zeno regime.\n\n\\subsubsection{Emission of a massive boson}\nWe adapt the model of Sec. 3.1.2 to describe the emission of a   bosonic particle of mass $\\mu$.  The model is identical to that of Sec. 3.1.2, except for the form of the energy function $\\omega_{\\pmb k}$. We assume non-relativistic energies for the product particle, hence,  $\\omega_{\\pmb k}= \\mu + \\frac{{\\pmb k}^2}{2 \\mu}$. This model is a variation of the one in Refs. \\cite{Ghiold, AA} that describes  the decay of a heavy baryon to a lighter one   with the emission of a pion. It allows for an analytic calculation of the persistence amplitude at all times. Hence,  we can witness the transition between the exponential and the power-law regimes without worrying about the validity of specific approximation schemes.\n\nThe self-energy function is\n\\begin{eqnarray}\n\\Sigma(z) = \\frac{\\lambda^2}{2 \\pi^2} \\int_0^{\\infty} \\frac{k^2 dk}{\\left(\\mu + \\frac{k^2}{2\\mu}\\right) \\left(z - \\mu - \\frac{k^2}{2\\mu}\\right)} = -  \\frac{\\sqrt{2}\\lambda^2\\mu}{\\pi^2}\\int_0^{\\infty} \\frac{dxx^2}{(1+x^2)(x^2 - \\zeta)},\n\\end{eqnarray}\nwhere we set $\\zeta = z\/\\mu - 1$ and $x = k\/(\\sqrt{2}\\mu)$.  The integral over $x$ is evaluated to $\\frac{\\pi}{2}(1 +\\sqrt{-\\zeta})^{-1}$. Hence,\n\\begin{eqnarray}\n\\Sigma(z) =  -  \\frac{\\lambda^2\\mu}{\\sqrt{2}\\pi} \\frac{1}{1 + \\sqrt{ - \\frac{z - \\mu}{\\mu}}}. \\label{sigmazint}\n\\end{eqnarray}\n\nIn the non-relativistic regime, the relevant values of $z$ satisfy $|z - \\mu| << \\mu$, so we can approximate $(1 + \\sqrt{ - \\frac{z - \\mu}{\\mu}})^{-1}$ with $1 -  \\sqrt{ -\\frac{z - \\mu}{\\mu}}$. We shift the  energy of  the ground state by $\\mu$, so that the energy of the  product  particle starts at  $z = 0$ rather than at $z = \\mu$. We also absorb the constant $\\Sigma(0)$ into $\\Omega$. Then, the  energy of the 2LS is $\\tilde{\\Omega} = \\Omega - \\mu  -  \\frac{\\lambda^2\\mu}{\\sqrt{2}\\pi}$, and the self-energy function becomes\n\\begin{eqnarray}\n\\Sigma(z) =  -  \\frac{\\lambda^2}{\\pi} \\sqrt{-\\frac{\\mu z}{2}} . \\label{sigmazin}\n\\end{eqnarray}\nFor $E > 0$, the level-shift function $F(E)$ vanishes because $\\Sigma(E^{\\pm})$ is purely imaginary. The decay function is\n\\begin{eqnarray}\n\\Gamma(E) = \\frac{\\lambda^2}{\\pi} \\sqrt{2m E},\n\\end{eqnarray}\nand the decay constant  $\\Gamma = \\frac{\\lambda^2}{\\pi} \\sqrt{2m \\tilde{\\Omega}}$.\n\nThe persistence amplitude (\\ref{mainampl}) becomes\n\\begin{eqnarray}\n{\\cal A}(t) = \\frac{\\Gamma}{2\\pi} \\int_0^{\\infty} \\frac{dE e^{-iEt} \\sqrt{E\/\\tilde{\\Omega}}}{(E - \\tilde{\\Omega})^2 + \\frac{\\Gamma^2E}{4\\tilde{\\Omega}}},\n\\end{eqnarray}\nwhere we wrote $\\Gamma(E) = \\Gamma \\sqrt{E\/\\tilde{\\Omega}}$. We change variables to $x = E\/\\tilde{\\Omega}$, to obtain\n\\begin{eqnarray}\n{\\cal A}(t) = \\frac{\\sqrt{2}\\gamma}{\\pi} \\int_0^{\\infty} \\frac{dx e^{-ix (\\tilde{\\Omega t})} \\sqrt{x}}{(x-1)^2+ 2 \\gamma^2 x},\n\\end{eqnarray}\nwhere $\\gamma = \\frac{\\Gamma}{2\\sqrt{2}\\tilde{\\Omega}} << 1$. The denominator is a binomial with two roots, at $x = x_{\\pm}:= 1- \\gamma^2 \\pm i \\gamma\\sqrt{2-\\gamma^2}$. We use the identity\n\\begin{eqnarray}\n\\frac{x_+-x_-}{(x-x_+)(x-x_-)} = \\frac{1}{x-x_+} - \\frac{1}{x-x_-},\n\\end{eqnarray}\nand change variables to $y = x \\tilde{\\Omega}t$, to obtain\n\n\\begin{eqnarray}\n{\\cal A}(t) = \\frac{R\\left(  (1- \\gamma^2 + i \\gamma\\sqrt{2-\\gamma^2})\\Omega t\\right) - R\\left(  (1- \\gamma^2 - i \\gamma\\sqrt{2-\\gamma^2})\\Omega t\\right)}{2\\pi i\\sqrt{1 - \\frac{1}{2} \\gamma^2}\\sqrt{\\tilde{\\Omega t}}}.\n\\end{eqnarray}\nWe defined the function\n\\begin{eqnarray}\nR(a) := \\int_0^{\\infty} \\frac{dy e^{-iy}\\sqrt{y}} {y - a}. \\label{fai}\n\\end{eqnarray}\nFor $\\mbox{Re} \\; a > 0$, the integral (\\ref{fai})  can be expressed analytically in terms of the Fresnel integrals $C(x) = \\int_0^x ds \\cos(s^2)$ and $S(x) =  \\int_0^x ds \\cos(s^2)$. We find \\cite{ASt},\n\\begin{eqnarray}\nR(a) = (1-i) \\sqrt{\\frac{\\pi }{2}}-\\pi  e^{-i a} \\sqrt{-a}-(1+i)e^{-i a}\\pi \\sqrt{a}\\left[C\\left(\\sqrt{\\frac{2}{\\pi }} \\sqrt{a}\\right)+i S\\left(\\sqrt{\\frac{2}{\\pi }} \\sqrt{a}\\right)\\right]. \\label{closedat}\n\\end{eqnarray}\nEq. (\\ref{closedat}) is a closed expression for the persistence amplitude that is valid for all times. The logarithm of the persistence probability $|{\\cal A}(t)|^2$ as a function of time is plotted in Fig. 2. The agreement with exponential decay is excellent until $t \\simeq 15 \\Gamma^{-1}$. The transition to power-law decay is accompanied by increasingly stronger oscillations that originate from the asymptotic behavior of the Fresnel integrals.\n\nIt is important to emphasize that the oscillations of Fig. 2 signify the breakdown not only of exponential decay, but of the  persistence probability method. They imply that the probability density (\\ref{decprob2}) takes negative values. Hence, the   method does not correlate with the experimental records, namely, the number of product particles recorded at each moment of time. The result strongly suggests that the persistence probability method is not reliable outside the regime of exponential decay.\n\n\\begin{figure}\n\\includegraphics[height=6cm]{per4}\n\\caption{The persistence probability as a function of $\\Gamma t$ for $\\gamma = 0.05$. }\n\\end{figure}\n\n\\subsection{Fermionic emission}\n\\subsubsection{The Hamiltonian}\nNext, we consider decays of the form $A' \\rightarrow A + B_1 + B_2$, in which   $B_1$ and $B_2$ are fermionic particles, much lighter than $A'$ and $A$. The most important example of this type is beta decay, where $A'$ and $A$ are nuclei, $B_1$ is an electron (or positron) and $B_2$ is an anti-neutrino (or a neutrino). Again the nucleus is described as a 2LS, with a ground state $|g \\rangle$ and an excited state $|e\\rangle$.\n\nA fermionic Fock space ${\\cal F}_1$ is associated to the  particle $B_1$ and a fermionic Fock space ${\\cal F}_2$ is associated to the particle $B_2$. The corresponding ground states are $|0\\rangle_1$ and $|0 \\rangle_2$, respectively. The creation and annihilation operators on ${\\cal F}_1$ will be denoted as $\\hat{c}_{r}$ and $\\hat{c}_{r}^{\\dagger}$, and the creation and annihilation operators  on ${\\cal F}_2$ as $\\hat{d}_{l}$ and $\\hat{d}_{l}^{\\dagger}$. They satisfy the canonical anti-commutation relations\n\\begin{eqnarray}\n[\\hat{c}_r, \\hat{c}_s]_+ = [\\hat{c}_r^{\\dagger}, \\hat{c}^{\\dagger}_s]_+ = 0, \\hspace{0.4cm} [\\hat{c}_r, \\hat{c}^{\\dagger}_s]_+ = \\delta_{rs}\n\\end{eqnarray}\n\\begin{eqnarray}\n[\\hat{d}_l, \\hat{d}_m]_+ = [\\hat{d}_l^{\\dagger}, \\hat{d}^{\\dagger}_m]_+ = 0, \\hspace{0.4cm} [\\hat{d}_l, \\hat{d}^{\\dagger}_m]_+ = \\delta_{lm}.\n\\end{eqnarray}\nIn the above, $r$ and $s$ are labels of a basis on the Hilbert space of a single $B_1$ particle;  $l$ and $m$   are labels of a basis on the Hilbert space of a single $B_2$ particle. The Hilbert space of the total system is  ${\\pmb C}^2 \\otimes {\\cal F}_1 \\otimes {\\cal F}_2$\n\nThe Hamiltonian for this system again consists of three parts $\\hat{H}_L = \\hat{H}_A + \\hat{H}_B + \\hat{V}$, where\n\\begin{eqnarray}\n\\hat{H}_A &=& \\frac{1}{2} \\Omega (\\hat{1} + \\hat{\\sigma}_3)  , \\label{vleeb1}\\\\\n\\hat{H}_B &=&   \\sum_r \\omega_r \\hat{c}^{\\dagger}_r \\hat{c}_r   +     \\sum_r \\tilde{\\omega}_l \\hat{d}^{\\dagger}_l \\hat{d}_l  , \\label{vleeb2}\\\\\n\\hat{V} &=& \\sum_{r,l} \\left(g_{r,l} \\hat{\\sigma}_+   \\hat{c}_r   \\hat{d}_l + g_{r,l}^*  \\hat{\\sigma}_-   \\hat{c}_r^{\\dagger}  \\hat{d}_l^{\\dagger} \\right). \\label{vleeb3}\n\\end{eqnarray}\nIn Eq. (\\ref{vleeb2}), $\\omega_r$ and $\\tilde{\\omega}_l$ are energy eigenvalues of a single $B_1$ and $B_2$ particle, respectively. The coefficients $g_{r, l}$ are model-dependent.\n\nThe self-energy function  for an initial state $|A'\\rangle = |e\\rangle \\otimes |0\\rangle_1 \\otimes |0\\rangle_2$ is\n\\begin{eqnarray}\n\\Sigma(z) = \\sum_{r, l} \\frac{|g_{r,l}|^2}{z - \\omega_r - \\tilde{\\omega}_l}. \\label{sig2f}\n\\end{eqnarray}\n\n\\subsubsection{Beta decay}\nWe consider a simplified model for beta decay, in which both emitted particles have zero spin and mass. This model is similar to the original Fermi theory of weak interactions \\cite{Fermi, Wilson}.\n The zero-mass approximation is reasonable, if the energy $\\Omega$ is much larger than the masses of the emitted particles, as is often the case in beta decay. The zero spin approximation is bad; spin is important in the weak interactions.\n\n\n\nIn this model, the basis $r$ corresponds to momenta ${\\pmb p}$, the basis $l$ to momenta ${\\pmb q}$, $\\omega_r$\n  corresponds to $\\omega_{\\pmb k } = |{\\pmb p}|$, and $\\tilde{\\omega}_l$\n  corresponds to $\\tilde{\\omega}_{\\pmb q } = |{\\pmb q}|$. The summation over $r$ corresponds to integration over  $\\frac{d^3p}{(2\\pi)^3}$ and the summation over $s$ corresponds to integration over  $\\frac{d^3q}{(2\\pi)^3}$. The appropriate constants $g_{r,l}$ for beta decay are\n \\begin{eqnarray}\n  g_{{\\pmb p}, {\\pmb q}} = \\frac{1}{\\mu^2}e^{i({\\pmb p}+{\\pmb q})\\cdot{\\pmb r}}\n  \\end{eqnarray}\n   where $\\mu$ has dimensions of mass\\footnote{In terms of the standard parameters of the theory of weak interactions, $\\mu^{-2} = G_F V_{ud} {\\cal M}_n$, where $G_F$ is Fermi's constant, $V_{ud}$ is an element of the Kobayashi-Maskawa matrix, and ${\\cal M}_n$ the amplitude for the nuclear transition \\cite{nuclear}. The dimensionality of $\\mu$ comes from the inverse mass squared dimension of Fermi's constant.   } and   ${\\pmb r}$ is the position vector of the atom.\n\n   We substitute to  Eq. (\\ref{sig2f}), to obtain\n  \\begin{eqnarray}\n  \\Sigma(z) = \\frac{1}{64 \\pi^6 \\mu^4} \\int \\frac{d^3p d^3q}{z - |\\pmb p| - |{\\pmb q}|} = \\frac{1}{16 \\pi^4 \\mu^4} \\int_0^{\\infty}p^2dp \\int_0^{\\infty} q^2 dq  \\frac{1}{z-p-q}.\n  \\end{eqnarray}\nWe change the integration  variables to $y = p+q, \\xi = p-q$. Then,\n  \\begin{eqnarray}\n  \\Sigma(z) =   \\frac{1}{256 \\pi^4 \\mu^4} \\int_0^{\\infty}\\frac{dy}{z-y} \\int_0^y d\\xi (y^2-\\xi^2)^2 = \\frac{1}{480\\pi^4 \\mu^4}\\int_0^{\\infty} \\frac{dy y^5}{z-y}, \\label{sz2f}\n  \\end{eqnarray}\nThe integral in Eq. (\\ref{sz2f}) diverges, so we introduce a high-frequency cut-off $\\Lambda << \\mu$ and restrict the integration over $y$ to the interval $[0, \\Lambda]$. Thus, we obtain\n\\begin{eqnarray}\n\\Sigma(z) = - \\frac{1}{240 \\pi^4 \\mu^4} \\left[ \\Lambda^5\\left(\\frac{1}{5} + \\frac{z}{4 \\Lambda}  + \\frac{z^2}{3\\Lambda^2} + \\frac{z^3}{2\\Lambda^3} + \\frac{z^4}{\\Lambda^4}\\right)+z^5 \\ln \\Lambda -z^5 \\ln(-z)   \\right].\n\\end{eqnarray}\nwhere we approximated  $\\ln(\\Lambda- z) \\simeq \\ln \\Lambda$.\n\nAs in Sec. 3.1.2, the branch point $z = 0$ is logarithmic. Following the same procedure, we evaluate the level-shift and decay functions\n\n\n\\begin{eqnarray}\nF(E) &=& - \\frac{\\Lambda^5}{1200\\pi^4 \\mu^4} \\left(1+ \\frac{5E}{4 \\Lambda} + \\frac{5E^2}{3\\Lambda^2} + \\frac{5E^3}{2\\Lambda^3} + \\frac{5E^4}{\\Lambda^4}\\right) - \\frac{E^5}{240 \\pi^4 \\mu^4}  \\ln\\frac{\\Lambda}{E}\n\\\\\n\\Gamma(E) &=& \\frac{E^5}{120 \\pi^3 \\mu^4}.  \\label{gamma2ef}\n\\end{eqnarray}\nThe beta decay rate is\n\\begin{eqnarray}\n\\Gamma = \\Gamma(\\Omega) =  \\frac{\\Omega^5}{120 \\pi^3 \\mu^4}\n\\end{eqnarray}\n\n\n\\section{Resonant decays}\nIn this section, we consider decays in presence of resonance. To this end, we study  a  variation of the spontaneous emission model of Sec. 3.1.2, in which the 2LS is  within a cavity, consisting of two (infinite) parallel metal plates at distance $L$.  The cavity is perfect, so the field satisfies Dirichlet boundary conditions on the plates.   The model has been studied in Refs. \\cite{AnHu, Cum}, but most of the results presented here are new.\n\n\\subsection{The self-energy function}\n\nIn the directions parallel to the cavity the photon momenta take continuous values. The momentum in the perpendicular direction is an  integer multiple  of the fundamental frequency\n\\begin{eqnarray}\n\\omega_0 = \\frac{\\pi}{L}.\n\\end{eqnarray}\n  Thus, the index $r$ of the bosonic Lee model corresponds to the pair $({\\pmb k}, n)$, where ${\\pmb k}$ is a two-dimensional vector parallel to the plates and $n = 0, 1, 2, \\ldots$. The energies are $\\omega_{{\\pmb k},n} = \\sqrt{{\\pmb k}^2 +n^2 \\omega_0^2}$ and the mode summation corresponds to $\\sum_{n=0}^{\\infty} \\int \\frac{d^2k}{(2\\pi)^2}$.\n\n The coupling constants have the same dependence on energy as in Sec. 3.1.2, but we express them as\n  \\begin{eqnarray}\n  g_{{\\pmb k},n} = \\lambda \\sqrt{\\frac{\\omega_0}{\\omega_{{\\pmb k},n}} },\n  \\end{eqnarray}\n  in terms of the dimensionless constant $\\lambda$.\n\nThen, the self-energy function takes the form\n\\begin{eqnarray}\n\\Sigma(z) = \\frac{\\lambda^2 \\omega_0}{2\\pi} \\sum_{n=0}^{\\infty}\\int_0^{\\infty} \\frac{kdk}{\\omega_{{\\pmb k},n}(z- \\omega_{{\\pmb k},n})}  = - \\frac{\\lambda^2 \\omega_0}{2\\pi}  \\sum_{n=0}^{\\infty}  \\int_{n \\omega_0 -z}^{\\infty} \\frac{dx}{x}, \\label{szcav0}\n\\end{eqnarray}\n where in the last step we set $x = \\omega_{{\\pmb k},n} - z$.\n\n The integral for $\\Sigma(z)$ in Eq. (\\ref{szcav0}) diverges at high energies. We regularize by introducing a high-energy cut-off $\\Lambda$ in the integral over $x$, and a maximum integer $N = \\Lambda\/\\omega_0$ in the summation over $n$.  Then,\n \\begin{eqnarray}\n \\Sigma(z) &=& -\\frac{\\lambda^2 \\omega_0}{2\\pi} \\left[ N \\ln \\frac{\\Lambda}{\\omega_0} - \\sum_{n=0}^{N} \\ln(n - z\/\\omega_0)\\right]   \\nonumber \\\\\n &=& -\\frac{\\lambda^2 \\omega_0}{2\\pi} \\left[\\frac{\\Lambda}{\\omega_0}  \\ln \\frac{\\Lambda}{\\omega_0} - \\ln\\Gamma(N - z\/\\omega_0) +  \\ln\\Gamma( -z\/\\omega_0) \\right],\n \\end{eqnarray}\n where $\\ln\\Gamma(z)$ is the logarithmic gamma function. Since the physical values of $z$ are much smaller than $\\Lambda$,  we use the asymptotic form of the logarithmic gamma function (Stirling's formula)\n \\begin{eqnarray}\n   \\ln\\Gamma(z) \\simeq z \\ln z -z, \\label{stirling}\n \\end{eqnarray}\nto obtain\n\\begin{eqnarray}\n\\Sigma(z) =  - \\frac{\\lambda^2\\Lambda }{2\\pi} -    \\frac{\\lambda^2\\log (\\Lambda\/\\omega_0) }{2\\pi}z-  \\frac{\\lambda^2 \\omega_0}{2\\pi} \\ln\\Gamma( -z\/\\omega_0). \\label{szcav}\n\\end{eqnarray}\n As in Sec. 3.1.2, we  incorporate the constant part of $\\Sigma(z)$ into a frequency redefinition: $\\tilde{\\Omega} =  \\Omega - \\frac{\\lambda^2\\Lambda }{2\\pi}$, so that\n\n\\begin{eqnarray}\n\\Sigma(z) =  - \\frac{\\lambda^2\\log (\\Lambda\/\\omega_0) }{2\\pi}z  - \\frac{\\lambda^2 \\omega_0}{2\\pi} \\ln\\Gamma( -z\/\\omega_0). \\label{szcav2}\n\\end{eqnarray}\n\n\nThe logarithmic gamma function has no poles, but it has infinitely many branch points at all negative integers. The identity $\\Gamma(z+1) = z \\Gamma(z)$ implies that\n\\begin{eqnarray}\n\\ln \\Gamma(-x) = - \\ln(-x) - \\ln(-x+1) - \\ldots - \\ln(-x + [x]) +\\ln\\Gamma[-x+[x]+1], \\label{lngaid}\n\\end{eqnarray}\nfor $x > 0$ and where $[x]$ is the integer part of $x$.\n\nEq. (\\ref{lngaid}) implies that\n $\\Sigma(E^-)$   involves the sum of  $[E\/\\omega_0]+1$ logarithms with negative argument. Each logarithm contributes a term $ \\pi$ to the imaginary part of $\\Sigma(E^-)$, hence,\n\\begin{eqnarray}\n  \\Gamma(E)  =  \\lambda^2 \\omega_0 ([E\/\\omega_0]+1).\n\\end{eqnarray}\n\n The level-shift function $F(E)$ involves a term $ - \\ln(E\/\\omega_0) -   \\ln(E\/\\omega_0 - 1) - \\ln(E\/\\omega_0 - [E\/\\omega_0]) + \\ln \\Gamma( [E\/\\omega_0] + 1 -  E\/\\omega_0)$, which can be written as\n$ \\ln \\left( \\frac{\\Gamma(1+ [E\/\\omega_0] - E\/\\omega_0)   \\Gamma(E\/\\omega_0 - [E\/\\omega_0]) }{ \\Gamma(E\/\\omega_0) }\\right)$. Hence,\n  \\begin{eqnarray}\n  F(E) =   - \\frac{\\lambda^2\\log (\\Lambda\/\\omega_0) }{2\\pi}z + \\frac{\\lambda^2\\omega_0 }{2\\pi} \\ln \\left( \\frac{ \\Gamma(E\/\\omega_0) }{ \\Gamma(E\/\\omega_0 - [E\/\\omega_0])\\Gamma(1+ [E\/\\omega_0] - E\/\\omega_0)  }\\right).\n  \\end{eqnarray}\n  Both $F(E)$ and $\\Gamma(E)$ have finite  discontinuities across the resonances $E = n \\omega_0$, for $n = 1, 2, \\ldots$.\n\n\n\n\\medskip\n\n\\noindent {\\em Large cavity.} For a large cavity ($L \\Omega >>1$), $\\omega_0$ is very small, so we expand the logarithmic gamma function in Eq. (\\ref{szcav2}) using  Stirling's formula. Then,\n\\begin{eqnarray}\n\\Sigma(z) =   \\frac{\\lambda^2  }{2\\pi}\\left[ - ( \\ln \\Lambda - 1)z + z \\ln(-z) \\right] . \\label{szcav2b}\n\\end{eqnarray}\nEq. (\\ref{szcav2b}) has the same functional dependence on $z$ with the self-energy function of Eq. (\\ref{seel6})   that is obtained in absence of a cavity. If the coupling constant and the cut-off parameters are properly redefined, the two expressions coincide.\n\n\\medskip\n\nWe evaluate the persistence amplitude using with Eq. (\\ref{mainampl}). We define $n_c = [\\tilde{\\Omega}\/\\omega_0] $ and $x_0 = \\tilde{\\Omega}\/\\omega_0 - n_c $. Then, we express the integral $\\int_0^{\\infty} dE$   as $\\sum_{n=0}^{\\infty} \\int_{n \\omega_0}^{(n+1)\\omega_0}dE$, where $n = [E\/\\omega_0]$.\n\\begin{eqnarray}\n{\\cal A}_a(t) = \\frac{\\lambda^2  }{2 \\pi} \\sum_{n=0}^{\\infty}(n+1)e^{-i n \\omega_0 t} \\int_0^1 dx \\frac{e^{-i(\\omega_0t)x}}{[x + n - n_c - x_0 - \\frac{\\lambda^2 }{2\\pi} f_n(x)]^2 + \\frac{\\lambda^4 }{4}(n+1)^2 } \\label{offres}\n\\end{eqnarray}\n where we substituted $ E = \\omega_0(n+x)$ and we wrote\n\\begin{eqnarray}\nf_n(x) =   - \\ln(\\Lambda\/\\omega_0) (n +x)  + \\ln \\frac{ \\Gamma(n+x)}{\\Gamma(1-x)\\Gamma(x)}.\n\\end{eqnarray}\nThe function $f_n(x)$ diverges near $x = 0$ and near $x = 1$. Using the expansion,  $\\Gamma(x) \\simeq \\frac{1}{x}  + \\ldots$ near $x = 0$, we obtain\n\\begin{eqnarray}\nf_n(x) &=&   \\ln x  + \\ln (n-1)! -   \\ln(\\Lambda\/\\omega_0) n \\; \\; (n > 0), \\label{fn0}\n\\\\ f_n(1-x) &=& \\ln x + \\ln n! -   \\ln(\\Lambda\/\\omega_0) (n+1), \\label{fn1}\n\\end{eqnarray}\nfor $x << 1$.\n\n\n The integrals in Eq. (\\ref{offres}) are dominated by  values of $x$ for which the denominator is of order $\\lambda^2$.\n Their behavior  depends crucially on whether the system is near resonance or not.\n\n\n\n\n\n\n\\subsection{ Off resonance}\n\nFirst, we assume that the atomic frequency is far from the cavity's resonances. Hence, $x_0$ and $1 - x_0$ are much larger than $O(\\lambda^2)$. The integrals are dominated by their values near solutions to the equation\n\\begin{eqnarray}\nx + n - n_c - x_0 - \\frac{\\lambda^2 }{2\\pi} f_n(x) = 0, \\label{eqoll}\n\\end{eqnarray}\nfor $x\\in [0, 1]$. For $n = n_c$, Eq. (\\ref{eqoll}) admits a perturbative solution at $x = x_0 + \\frac{\\lambda^2 }{2\\pi} f_{n_c}(x_0) + O(\\lambda^4)$.\n\nThere are   other solutions to Eq. (\\ref{eqoll}), near the boundaries  $x = 0$ and $x = 1$ where $f_n(x)$ diverges.\n It is easy to show that these solutions lie at distance  smaller than $e^{-\\frac{2 \\pi x_0}{\\lambda^2}}$ or $e^{-\\frac{2 \\pi (1-x_0)}{\\lambda^2}}$ from the boundaries,  and that their contribution to the integral is negligible.\n\nTherefore, the dominant term to the persistence amplitude corresponds to $n = n_c$. Since the integral is dominated by values near the perturbative solution with a width of order $\\lambda^2$, we can extend the range of integration to $(-\\infty, \\infty)$. But then, we recover the integral (\\ref{ampl3}) of the WWA, with\n\n\\begin{eqnarray}\n\\delta E &=&  - \\frac{\\lambda^2 \\omega_0\\log (\\Lambda\/\\omega_0) }{2\\pi} (n_c  + x_0)  +  \\frac{\\lambda^2\\omega_0 }{2\\pi} \\ln \\left( \\frac{\\Gamma(n_c + x_0) }{\\Gamma(1- x_0 )  \\Gamma(x_0)  }\\right) \\\\\n\\Gamma &=&  \\lambda^2 \\omega_0 (n_c + 1).\n\\end{eqnarray}\n\nAs expected, the persistence amplitude off-resonance does not differ significantly from the persistence amplitude of an atom outside a cavity.\n\n\n\\subsection{Resonance}\nThe condition for resonance is that either $x_0$ or $1 - x_0$ is of order $\\lambda^2$. In the former case,\n$\\Omega_R$ is just above the resonance frequency $n_R \\omega_0$, for $n_R = n_c$; we define the detuning parameter $\\delta = x_0$.\nIn the latter  case, $\\Omega_R$ is just below the resonance frequency $n_R \\omega_0$, for $n_R = n_c + 1$; we define the detuning parameter as $\\delta =   x_0 - 1$.\n\nTwo terms in the series (\\ref{offres}) dominate. The first corresponds to $n = n_R$. In this term, the denominator of the integral is of order $\\lambda^2$ near $x = 0$. Therefore, we can use the approximation (\\ref{fn0}) for $f_n(x)$. We change variables to $y = \\frac{\\lambda^2}{2\\pi}x$ and we extend the limit of integration for $y$ form $\\frac{2\\pi}{\\lambda^2}>> 1$ to $\\infty$. Then, this term equals $e^{-in_R\\omega_0 t} G_-(\\frac{\\Gamma_0t}{\\pi}, -d, (n_R+1)\\pi)$, where\n\n\\begin{eqnarray}\nG_{\\pm}(s, a, b) :=  \\frac{b}{\\pi}  \\int_0^{\\infty} \\frac{dy e^{- i    s y}}{(y \\pm \\ln y - a)^2  + b^2 },  \\label{gsab}\n\\end{eqnarray}\n\\begin{eqnarray}\n d := \\ln \\frac{2 \\pi}{\\lambda^2} - \\frac{2\\pi\\delta}{\\lambda^2} - \\ln (n_R-1)!  +   \\ln(\\Lambda\/\\omega_0) n_R,\n  \\end{eqnarray}\n  and $\\Gamma_0 =   \\lambda^2 \\omega_0$.\n\n The second term corresponds to $n = n_R - 1 $. In this term, the denominator is of order $\\lambda^2$ near $x = 1$. Again, we can use the approximation (\\ref{fn1}) for $f_n(x)$. We change variables to $y = \\frac{2\\pi}{\\lambda^2}(1-x)$ and take the limit of integration for $y$ to $\\infty$. Then, this term equals $e^{-in_R\\omega_0 t} G_+^*(\\frac{\\Gamma_0t}{\\pi}, d, \\frac{1}{2}n_R\\pi)$. Hence,\n \\begin{eqnarray}\n{\\cal A} (t) =  e^{-in_R \\omega_0 t} \\left[ G_-[\\frac{\\Gamma_0t}{\\pi}, -d,  (n_R+1)\\pi ] + G^*_+[\\frac{\\Gamma_0t}{\\pi}, d,  n_R \\pi ]   \\right], \\label{atres}\n\\end{eqnarray}\n\n\n\n\nFor general values of $a$ and $b$ the functions $G_{\\pm}(s, a,b)$ can only evaluated numerically. In general, they decay with increasing $s$, with some oscillations for positive $a$.  Plots of $G_{\\pm}(s, a, b)$ as a function of $s$ for different values of $a$ and $b$ are given in Fig. 3.\n\n\\begin{figure}\n\\includegraphics[height=10cm]{functionG}\n\\caption{The real and the imaginary part of   $G_{\\pm}(s, a, b)$ are plotted as  functions of $s$ for different choices of $a$ and   $b$. }\n\\end{figure}\nFor  $a >>b $  , the contribution of the logarithm to the integral (\\ref{gsab}) is negligible, and the integral is little affected if the range is extended to $(-\\infty, \\infty)$. Then,\n\\begin{eqnarray}\nG_{\\pm}(s, a, b) = \\frac{b}{\\pi}   \\int_{-\\infty}^{\\infty} \\frac{dy e^{-i    s y}}{(y- a )^2  + b^2 } = e^{-b s-ias}\n\\end{eqnarray}\nIn contrast, if $|a|>> b$ with $a <0$, $G_{\\pm}(s, a, b)$ is of order $(|a|\/b)^2$. We readily  verify  that Eq. (\\ref{atres}) recovers the exponential decay form for $ |d| >> \\pi n_R$.\n\nIn all other regimes, the decays are non exponential. This can be seen in  Fig. 4, where the logarithm of the persistence probability   is plotted as a function of $\\Gamma_0t\/\\pi$ for different values of $d$. The graph becomes a straight line for larger values of $d$, signaling exponential decay. Deviations  appear when a tiny fraction of the initial  2LS remains excited. For several values of $d$, the persistence probability is not a decreasing function of $t$. Again, this signifies a failure of the definition (\\ref{decprob2}) for the decay probability.\n\n\\begin{figure}\n\\includegraphics[height=7cm]{persistence}\n\\caption{The logarithm of the persistence probability $\\ln |{\\cal A}(t)|^2$ as a function of $\\Gamma_0t\/\\pi$ for different values of $d$ and for $n_R = 1$.   }\n\\end{figure}\n\nThe behavior of the persistence probability derived here is typical  for decaying systems with energies close to a branch point of the self-energy function. \n In quantum field theory, such branch points appear at energy thresholds, i.e., for energies near  the activation energy $E_0$ of a new decay channel. For example, if the energy of a photon becomes $2m_e$, where $m_e$ is the electrons's mass, the photon decay to a electron-positron pair is possible. In such cases, the persistence amplitude receives two distinct contributions, one from energies slightly beneath and one from energies slightly above the threshold.  For discussions of non-exponential decays due to threshold effects, see, Refs. \\cite{LZMM, RZ93, JMSST, DJN09}.\n\n\n\n\\section{Decay through barrier tunneling}\nThe methodology developed in Sec. 2 applies to decays that originate from a small perturbation in the Hamiltonian. In this section, we consider non-perturbative decays that can be understood in terms of tunneling. Examples of such decays are the alpha emission of nuclei, tunneling ionization of atoms due to an external field, and leakage of particles from a trap.\nWe will consider a simple model of a particle in one dimension that mimics the classic treatment of alpha decay by Gamow and by Gurney and Condon \\cite{Gamow, GuCo}.\n\n \\subsection{Set-up}\n\n\\noindent {\\em Dynamics.} We consider a particle in the half-line ${\\pmb R}^+ = [0, \\infty)$ in presence of  a potential $V(x)$. The potential vanishes outside $ [a, b]$, where $a$ and $b$ are microscopic lengths.\n\nIt is convenient to express the potential in terms of the transmission and reflection coefficients  of the  Schr\\\"odinger operator  $\\hat{H} = \\frac{\\hat{p}^2}{2m}+ V(\\hat{x})$ over the full real line.  $\\hat{H}$    has two generalized eigenstates $f_{k\\pm}(x)$ for each value of energy $E = \\frac{k^2}{2m}$.\n\\begin{eqnarray}\nf_{k+}(x) &=& \\left\\{ \\begin{array}{cc} \\frac{1}{\\sqrt{2\\pi}} (e^{ikx} + R_k e^{-ikx}), & x < a\\\\\n \\frac{1}{\\sqrt{2\\pi}} T_k e^{ikx}& x > b\n \\end{array}\\right. \\nonumber \\\\\n f_{k-}(x) &=& \\left\\{ \\begin{array}{cc} \\frac{1}{\\sqrt{2\\pi}}  T_k e^{-ikx}, & x < a\\\\\n\\frac{1}{\\sqrt{2\\pi}} (e^{ikx} + \\tilde{R}_k e^{ikx})& x > b\n \\end{array}\\right. \\label{f+-}\n\\end{eqnarray}\nwhere   the complex amplitudes $T_k$, $R_k$ and $\\tilde{R}_k$ satisfy\n\\begin{eqnarray}\n|T_k|^2+|R_k|^2 = 1, \\hspace{0.75cm}  |R_k| = |\\tilde{R}_k|,\n  \\hspace{0.75cm}  T^*_k R_k +T_k \\tilde{R}_k^* = 0. \\label{idtyscat}\n\\end{eqnarray}\n$T_k$ is the transmission amplitude, $R_k$ is the reflection amplitude for a right-moving particle and $\\tilde{R}_k$ is the reflection amplitude for a left-moving particle.\n\nWhen the range of $x$ is restricted into the half-line, the generalized eigenfunction $g_k$ of the Schr\\\"odinger operator   is the linear combination of $f_{k+}$ and $f_{k-}$ that satisfies  $g_E(x) = 0$, i.e.,\n\\begin{eqnarray}\ng_k =  \\left( -\\frac{T_k}{1+R_k} f_{k+} + f_{k-}\\right).\n\\end{eqnarray}\nThis implies that\n\\begin{eqnarray}\ng_{k}(x) = \\left\\{ \\begin{array}{cc} -\\frac{2i}{\\sqrt{2\\pi}} \\frac{T_k}{1+R_k} \\sin kx, & x < a\\\\\n \\frac{1}{\\sqrt{2\\pi}} \\left[ e^{- ikx} - e^{iS_k} e^{ikx} \\right] & x > b\n \\end{array}\\right. , \\label{gkx}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\ne^{i S_k} = \\frac{T_k^2}{1+R_k} - \\tilde{R}_k =  \\frac{1+R^*_k}{1+R_k} \\left(\\frac{T_k^2}{|T_k|^2}\\right), \\label{eithe}\n\\end{eqnarray}\nis the reflection amplitude of a left-moving particle. The absolute value of the reflection amplitude is unity, because there is no possibility of transmission to $x < 0$.\n\nThe eigenfunctions $g_k(x)$ are normalized so that $\\int_0^{\\infty} g_k(x)^* g_{k'}(x) = \\delta(k-k')$. We will represent them by kets $|k\\rangle_D$.\n\n\\medskip\n\n\\noindent {\\em Initial state.} We consider an initial state $\\psi_0(x)$ with the following four properties. First, it vanishes  outside $[0, a]$. Second, it belongs to the Hilbert space of square-integrable harmonic functions on ${\\pmb R}^+$ subject to Dirichlet boundary conditions. Hence, it can be expressed as\n\\begin{eqnarray}\n\\psi_0(x) = \\sqrt{\\frac{2}{\\pi}} \\int_0^{\\infty} \\sin(kx) \\tilde{\\psi}_0(k). \\label{Dirich}\n\\end{eqnarray}\nThe function $\\tilde{\\psi}_0(k)$ is defined for $k > 0$. Extending to $k <0$ by $\\tilde{\\psi}_0(-k) = -\\tilde{\\psi}(k)$, we can write Eq. (\\ref{Dirich}) as $\\psi_0(x) = - \\frac{i}{\\sqrt{2\\pi}} \\int_{-\\infty}^{\\infty} dk e^{ikx} \\tilde{\\psi}_0(k)$. By Eq. (\\ref{gkx}),\n\\begin{eqnarray}\n{}_D\\langle k|\\psi_0\\rangle = i \\frac{T^*_k}{1+R^*_k}   \\tilde{\\psi}_0(k).\n\\end{eqnarray}\nThird, we assume that $\\psi_0(x)$ is real-valued. For example, $\\psi_0$ may be an eigenstate of a  Schr\\\"odinger operator $\\hat{H}'$ with a different potential $U(x)$.  The physical interpretation of this condition is that we prepare the system in an eigenstate of $\\hat{H}'$ and at time $t = 0$ we change the potential to $V(x)$, for example, by switching on an external electric field. Fourth,  we assume that   $\\psi_0$  has a sharp energy distribution with respect to the Hamiltonian $\\hat{H}$: the energy spread for $\\psi_0$ is much smaller than the mean energy.\n\n\n\n \\medskip\n Given an initial state $\\psi_0$, we find the state at time $t$\n \\begin{eqnarray}\n \\psi_t(x) = \\int_0^{\\infty} dk {}_D\\langle k|\\psi_0\\rangle  g_k(x) e^{ - i\\frac{k^2}{2m}t}. \\label{evolvt}\n \\end{eqnarray}\n The  definition  (\\ref{decprob2}) of the decay probability in terms of the persistence probability  is not appropriate for this problem, because the initial configuration of the system does not correspond to an one-dimensional subspace, i.e., the particle may remain confined by the potential without remaining in its initial state. A better choice is perhaps to define the decays probability as $p(t) = - \\frac{d}{dt} W(t)$, where\n$W(t) = \\int_0^a dx|\\psi_t(x)|^2$ is the  {\\em non-escape probability}, i.e., the probability that the particle is found in $[0, a]$ at time $t$---see, Ref. \\cite{CMM95} for the relation between persistence and non-escape probability. We note that the definition of the non-escape probability  is   arbitrary:\n we could have equally well  taken the integration range to $[0, b]$.\n\n The main drawback of both the non-escape and the persistence probability  is that they  lack a clear operational interpretation.\n We do not carry out measurements of particles at  microscopic scales inside the confining potential, rather we measure the number of particles that have tunneled from the barrier, as recorded by a detector  far from the tunneling region. Furthermore, the persistence probability and the non-escape probability are not always  increasing functions of $t$ \\cite{Peshk}. Hence, they may lead to negative values for $p(t)$.\n\nHere, we will proceed  by calculating the probability flux far from the potential region, i.e., we evaluate\n\\begin{eqnarray}\nJ(x, t) = \\frac{1}{m}\\mbox{Im} \\left[ \\psi^*_t(x) \\partial_x \\psi_t( x)\\right],\n\\end{eqnarray}\nfor $x >> b$.\n\n Eqs. (\\ref{gkx}) and (\\ref{evolvt}) imply that\n\\begin{eqnarray}\n\\psi_t(x)  = -\\frac{i}{\\sqrt{2\\pi}} \\int _0^{\\infty} dk  \\frac{T^*_k}{1+R^*_k}  \\left[ e^{iS_k +ikx}-  e^{-ikx}\\right] e^{-i\\frac{k^2}{2m}t}  \\tilde{\\psi}_0(k). \\label{psitxx}\n\\end{eqnarray}\n\nFor $ x>> b$, there is no stationary phase in the integral involving $e^{-ikx -i\\frac{k^2}{2m}t}$. Hence, its contribution to $\\psi_t(x)$ is much smaller than that of  the integral involving $e^{ikx -i\\frac{k^2}{2m}t}$, which has a stationary phase. By Eq.  (\\ref{eithe}),\n\\begin{eqnarray}\n\\psi_t(x) = -\\frac{i}{\\sqrt{2\\pi}} \\int _0^{\\infty} dk \\frac{T_k}{1+R_k}  e^{ikx -i\\frac{k^2}{2m}t } \\tilde{\\psi}_0(k). \\label{psitx}\n\\end{eqnarray}\n  The integral (\\ref{psitx}) contains only out-coming waves, hence, there is  no backflow in the probability current. Note that, in general, $J(x, t)$ can take negative values close to the barrier \\cite{Winter}, and hence, it is not reliable for the description of near-field experiments, i.e., when a particle detector is placed at a microscopic distance from the tunneling region.\n\nWe expand $(1+R_k)^{-1} = \\sum_{n=0}^{\\infty}(-R_k)^n$, to write Eq. (\\ref{psitx}) as\n\\begin{eqnarray}\n\\psi_t(x) = -\\frac{i}{\\sqrt{2\\pi}}\\sum_{n=0}^{\\infty}\\int _0^{\\infty} dk T_k(-R_k)^n  e^{ikx -i\\frac{k^2}{2m}t } \\tilde{\\psi}_0(k). \\label{psitx1b}\n\\end{eqnarray}\n\n\\subsection{Exponential decay }\nEq. (\\ref{psitx}) is accurate for the asymptotic behavior of the wave-function at $x >> b$. To proceed further, we exploit the fact that $\\psi_0(k)$ is strongly peaked about a specific value $k_0$, and we evaluate Eq. (\\ref{psitx}) in the saddle-point approximation. To this end, we write\n  $R_k = -|R_k|e^{i\\phi_k}$ and $T_k = |T_k|e^{i\\chi_k}$, so that Eq. (\\ref{psitx1b}) becomes\n\\begin{eqnarray}\n\\psi_t(x) =  - \\frac{i}{\\sqrt{2\\pi}}  \\sum_{n=0}^{\\infty}  \\int _0^{\\infty}dk |T_kR_k^n|  e^{ikx -i\\frac{k^2}{2m}t + i \\chi_k + i n \\phi_k} \\tilde{\\psi}_0(k). \\label{psitx2}\n\\end{eqnarray}\nThen, we extend the range of integration of $k$ to $(-\\infty, \\infty)$ setting $\\tilde{\\psi}_0(-k) =  - \\tilde{\\psi}_0(k)$ for negative $k$. The integral is not affected, because the additional terms involve a term $e^{-i|k|x -i\\frac{k^2}{2m}t}$, with no stationary phase. Then, we  approximate $|T_k| \\simeq |T_{k_0}|, |R_k| \\simeq |R_{k_0}|$, $k^2 \\simeq k_0^2 + 2 k_0 (k-k_0)$, $\\phi_k \\simeq \\phi_{k_0}+ \\phi'_{k_0} (k-k_0)$, $\\chi_k \\simeq \\chi_{k_0}+ \\chi'_{k_0} (k-k_0)$. The resulting integral is simply the inverse Fourier transform of $\\tilde{\\psi}_0(k)$. Hence,\n\\begin{eqnarray}\n\\psi_t(x) =  T_{k_0}  e^{ik_0x -i\\frac{k_0^2}{2m}t } \\sum_{n=0}^{\\infty} (-R_{k_0})^n \\psi_0(x - \\frac{k_0t}{m} + \\chi'_{k_0} + n \\phi'_{k_0}) \\label{psitx3}\n\\end{eqnarray}\n\n\nEq. (\\ref{psitx3}) has a natural interpretation in terms of classical concepts. The particle makes successive attempts to cross the barrier at $x = a$. On failure, it is reflected back,  it is reflected again at $x = 0$, and then it makes a new attempt. The $n$-th term in the sum of Eq. (\\ref{psitx3}) is the amplitude associated to a particle that succeeded in crossing the barrier at its\n $(n+1)$-th attempt: it is proportional to $T_{p_0}$ (one success) and to $R^n_{p_0}$ (after $n$ failures).\n\nSince $\\psi_0(x)$ has support only on $[0, a]$, $\\psi_t(x)$ vanishes for $t < t_0 := m (x -a + \\chi_{k_0}')\/p_0$. The time-scale $t_0$ has an obvious classical interpretation: it is the time it takes a particle inside the barrier region to traverse the distance to point $x$. The term $\\chi_{k_0}$ corresponds to the Wigner-Bohm time delay due to the particle crossing the\nclassically forbidden region \\cite{BohmWig}. We rewrite Eq. (\\ref{psitx3}) as\n\\begin{eqnarray}\n\\psi_t(x) =  T_{k_0}  e^{ik_0x -i\\frac{k_0^2}{2m}t } \\theta(t-t_0)\\sum_{n=0}^{\\infty} (-R_{k_0})^n \\psi_0\\left(a - \\frac{k_0(t - t_0 )}{m} + n\\Delta x \\right)  \\label{psitx4}\n\\end{eqnarray}\nwhere we defined $\\Delta x = \\phi'_{k_0}$ the position-shift between successive terms in the series (\\ref{psitx4}). If $|\\Delta x| > a$, the partial amplitudes at different $n$ do not overlap. Hence, there is no quantum interference between different attempts of the particle to cross the barrier. For $|\\Delta x| < a$, there is  quantum interference between $M = [a\/|\\Delta x|]$ successive attempts to cross the barrier.\n\nSince we assumed the initial state $\\psi_0$ to be almost monochromatic at energy $\\frac{k_0^2}{2m}$, the dominant contribution to the current is $ \\frac{k_0}{m} |\\psi_t(x)|^2$. Hence,\n\\begin{eqnarray}\nJ(t,x) &=& \\frac{k_0}{m} |T_{k_0}|^2  \\theta(t-t_0) \\sum_{n=0}^{\\infty} \\sum_{\\ell=0}^{\\infty}(-R_{k_0})^n(-R_{ k_0}^{*})^{\\ell}\\nonumber \\\\\n&\\times& \\psi_0(a- \\frac{k_0(t - t_0 )}{m} + \\ell \\Delta x )\\psi_0(a- \\frac{k_0(t - t_0 )}{m} + n \\Delta x ) \\label{jxt1}\n\\end{eqnarray}\nTerms in the summation with $|n- \\ell| > M$ vanish because the corresponding wave functions do not overlap. Then, we write\n\\begin{eqnarray}\nJ(t,x) = \\frac{k_0}{m} |T_{k_0}|^2  \\theta(t-t_0) \\sum_{N=0}^{\\infty}|R_{k_0}|^N \\rho\\left(\\frac{k_0(t - t_0 )}{m} - \\frac{1}{2} N\\Delta x  \\right), \\label{jtx3}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\rho(x) = \\sum_{j=-M}^M \\psi_0(a - x - \\frac{j}{2} \\Delta x) \\psi_0(a - x +\\frac{j}{2} \\Delta x) e^{i j\\phi_{k_0}}.\n\\end{eqnarray}\nThe function $\\rho(x)$ is localized within a width of order $a$.\n\nIn order to connect Eq. (\\ref{jtx3}) with experiments, we have to treat both $x$ and $t$ as macroscopic variables. This means that they can be measured with an accuracy of order $\\sigma_X$ and $\\sigma_T$, respectively, that is much larger than the microscopic scales that characterize the system. Hence, $\\sigma_X >> a$ and $\\sigma_T >> ma\/k_0$. At such scales, the width of $\\rho(x)$ is negligible, and we can substitute it with a delta function,\n\\begin{eqnarray}\n\\rho(x) \\simeq \\alpha \\delta(x), \\label{Markovtunel}\n\\end{eqnarray}\nwhere $\\alpha = \\int_{-\\infty}^{\\infty}dx \\rho(x)$ is a number close to unity. In particular,  $\\alpha =1 $ for $M = 0$. In this regime, we can also approximate the sum over $N$ with an integral, so that\n\\begin{eqnarray}\nJ(t,x) &=& \\alpha \\frac{k_0}{m} |T_{k_0}|^2  \\theta(t-t_0) \\int_0^{\\infty}dN |R_{k_0}|^N \\delta\\left(\\frac{k_0(t - t_0 )}{m} - \\frac{1}{2} N\\Delta x  \\right)\\nonumber \\\\  &=& \\alpha \\frac{|T_{k_0}|^2}{\\Delta t}  \\theta(t-t_0) e^{\\log |R_{p_0}|^2 \\frac{(t-t_0)}{\\Delta t}} \\label{jx5}\n\\end{eqnarray}\nwhere  $\\Delta t = m \\Delta x\/k_0$ has the classical interpretation as the time between two successive attempts of the particle to cross the barrier. For $|T_{k_0}| << 1$, $\\log|R_{p_0}|^2 \\simeq -|T_{p_0}|^2$. Then, Eq. (\\ref{jx5}) describes exponential decay,\n\n\\begin{eqnarray}\nJ(t,x) =  \\alpha \\Gamma  e^{-\\Gamma  (t-t_0) } \\theta (t-t_0), \\label{expdec}\n\\end{eqnarray}\n with a decay constant\n\\begin{eqnarray}\n\\Gamma = \\frac{ |T_{k_0}|^2}{\\Delta t}. \\label{Gammat}\n\\end{eqnarray}\nthat does not depend on the detailed properties of the initial state.\n\n\nNote that  exponential decay fails at early times; the derivation of  Eq. (\\ref{expdec}) requires that $|t - t_0 |>> \\Delta t$.\n\n\nIn deriving the exponential decay law, we employed the saddle point approximation. This it is reasonably accurate for   $\\cap$-shaped potentials. In general, it does not\napply  to potentials with multiple transmission and reflection points, like the double well potential \\cite{Matsu}. Such potentials may trap the particle in an intermediate region, and they require an analysis of the escape from this region. The  escape satisfies an exponential decay law, except for energies near resonance \\cite{AnSav13}.\n\nThe exponential decay law also fails at very long times, when wave-function dispersion becomes important. To see this, we change variables to  $y = \\frac{k^2t}{2m}$ in Eq. (\\ref{psitx}) for $\\psi_t(x)$. The dominant term at $t \\rightarrow \\infty$ is\n\\begin{eqnarray}\n\\psi_t(x) =  - i \\sqrt{\\frac{m}{4\\pi t}} A_0  \\int_0^{\\infty} \\frac{dy}{y}  e^{-iy} \\tilde{\\psi}_0(\\sqrt{2my\/t}). \\label{asymptunel}\n\\end{eqnarray}\nwhere $A_0 $ stands for $\\lim_{k\\rightarrow \\infty} T_k\/(1+R_k)$. In general, $A_0 \\neq 0$, as can be readily checked in elementary systems. Hence, the asymptotic behavior of $\\psi_t(x)$ depends on  the infrared behavior of $\\tilde{\\psi}_0$. For  a power law dependence, $\\psi_0 \\sim k^n$ near $k = 0$, Eq. (\\ref{asymptunel}) gives $\\psi_t(x) \\sim t^{-\\frac{n+1}{2}}$. It follows that $J(t, x) \\sim t^{-(n+1)}$, i.e., the flux decays  with an inverse power law. We note here that the asymptotic regime captures some of the information of the initial state \\cite{MDCG}.\n\n\nThe key property in deriving the exponential decay law is the lack of interference between  different attempts of the particle to cross the barrier. Let us assume that the maximum number of successive attempts that interfere in the probability amplitude is $M$. The decay  time scale $\\Gamma^{-1}$ corresponds to $|T_{k_0}|^{-2}$  attempts to cross the barrier. As long as\n\\begin{eqnarray}\n|T_{k_0}|^{-2} >> M, \\label{tkm}\n\\end{eqnarray}\nthe effects of interference are negligible. This also implies that the particle has very short `memory' about its past attempts to cross the barrier. Hence, the memory time-scale  is much shorter than the decay time-scale. This feature is known as the {\\em Markov property}.\n\nIn absence of quantum interferences and memory effects,  decays due to tunneling are indistinguishable from classical probabilistic processes that can be described using elementary arguments.  Consider a classical particle that attempts to cross a barrier with probability $w << 1$ of success\\footnote{The non-zero probability to cross the barrier needs not be quantum mechanical in origin. The particle may interact with a stochastic environment, such as a thermal bath. Then, the crossing of the  barrier may be due to a random force.}. After $N$ attempts, the survival probability is $(1-w)^N \\simeq e^{-Nw}$. If every attempt takes time $\\Delta t$, then for $N >> 1$, the system is described by an exponential decay law with constant $\\Gamma = w \/\\Delta t$, in full agreement with Eq. (\\ref{Gammat}).\n\n\n\n\n\\subsection{Alternative description of tunneling decays}\nWe can understand the emergence of exponential decay in tunneling using a different argument that does not rely on the saddle-point approximation. Instead, we use the analyticity properties of the time-evolution operator. For the full development of such methods, see, Ref. \\cite{RNewt}, and for applications to tunneling decays, see, Refs. \\cite{CP75, CMM95, CCM09}.\n\nAssume that we can analytically extend $\\tilde{\\psi}_0$ to the fourth quadrant of the complex plane. Then, we can write $\\psi_t(x) = K(t, x) + I_N(t,x)$, where\n\n\\begin{eqnarray}\nK(t, x) = \\frac{i}{\\sqrt{2\\pi}}  \\oint_C dz  \\frac{T_z}{1+R_z}  e^{izx -i\\frac{z^2}{2m}t } \\tilde{\\psi}_0(z), \\label{ktn}\n\\end{eqnarray}\nis a line integral along the contour $C$ of Fig. 5, and $I_N(t)$ is the integral across the line segment $N$ of $C$.  By Cauchy's theorem, we can evaluate $K(t,x)$  in terms of the poles of the integrand in the interior of $C$.\n\n\\begin{figure}\n\\includegraphics[height=7cm]{curven}\n\\caption{The integration contour of the line-integral (\\ref{ktn}). The contour is traversed clockwise. }\n\\end{figure}\nLet us denote by $z_n = q_n -i \\gamma_n$ the poles of $\\frac{T_z}{1+R_z} $ in  the interior of $C$. The integer $n$ labels the poles, in the interior of $C$, $q_n$ and $\\gamma_n$ are positive. Then,\n\\begin{eqnarray}\n\\psi_t(x) = \\sum_n c_n \\tilde{\\psi}_0(q_n -i \\gamma_n) e^{iq_n x - i\\frac{q_n^2-\\gamma_n^2}{2m}t - \\frac{q_n\\gamma_n}{m}(t - \\frac{mx}{q_n})} + I_N(t),\n\\end{eqnarray}\nfor some complex constants $c_n$. Each term in the sum, is suppressed by an exponential factor $\\exp[\\frac{q_n\\gamma_n}{m}(t - \\frac{x}{q_n})]$, for $t > \\frac{x}{q_n}$. For an almost monochromatic state $\\tilde{\\psi}_0$ with energy $k_0$, only a small number of poles with $q_n$ near $k_0$ contribute. Furthermore, the contribution   $I_N$ to $\\psi_t(x)$ drops exponentially for $t > mx\/k_0$, i.e., after the earliest possible time of detection.\n\nFor simplicity, let us assume that the contribution of only one pole at $n = n_0$ is significant, and that $q_{n_0} \\simeq k_0$. Then, for $t > mx\/k_0$,\n \\begin{eqnarray}\n \\psi_t(x) \\sim   e^{ik_0 x - i\\frac{k_0^2-\\gamma_{n_0}^2}{2m}t - \\frac{k_0\\gamma_{n_0}}{m}|t - \\frac{mx}{k_0}|}.\n \\end{eqnarray}\n  Therefore, the flux $J(t, x)$ is proportional to $e^{-2 \\frac{k_0\\gamma_{n_0}}{m} |t - \\frac{mx}{k_0}|}$. The decay constant is\n  \\begin{eqnarray}\n  \\Gamma = \\frac{2k_0\\gamma_{n_0}}{m}, \\label{decaytb}\n  \\end{eqnarray}\n   and it is determined {\\em solely} by the pole of $\\frac{T_z}{1+R_z} $ near $z = k_0$. No information about the initial state other than its energy is required.\n\nThe above analysis also provides a criterion for the breakdown of exponential decay. If the initial state allows for the contribution of different poles $z_n$, such that there is a significant variation in the values of $\\gamma_n$, then corrections to exponential decay, or even its breakdown are possible.\n\nFor example, consider an initial state $ \\psi_0 = a_1 \\psi_1 + a_2 \\psi_2$ that is a superposition of  two almost monochromatic states $\\psi_1$ and $\\psi_2$ with energies $\\frac{k_1^2}{2m}$ and $\\frac{k_2^2}{2m}$. Furthermore, assume that $k_1$ is close to   one pole of $T_z\/(1+R_z)$ at $n = n_1$, and $k_2$ close to  another pole of $T_z\/(1+R_z)$ at $n = n_2$, and that there is no overlap. Then, for $t > \\max \\{ mx\/k_1, mx\/k_2 \\}$,\n\\begin{eqnarray}\n \\psi_t(x) \\sim c_1   e^{ik_1 x - i\\frac{k_1^2-\\gamma_{n_1}^2}{2m}t - \\frac{k_0\\gamma_{n_1}}{m}|t - \\frac{x}{k_1}|} +\n  c_2   e^{ik_2 x - i\\frac{k_2^2-\\gamma_{n_2}^2}{2m}t - \\frac{k_0\\gamma_{n_2}}{m}|t - \\frac{x}{k_2}|},\n\\end{eqnarray}\nfor some constants $c_1$ and $c_2$.\n\n\nThe dominant contribution to the current is\n\\begin{eqnarray}\nJ(t, x)  &=& |c_1|^2 k_1 e^{- \\Gamma_1 |t - \\frac{mx}{k_1}|} + |c_2|^2 k_2 e^{- \\Gamma_2 |t - \\frac{mx}{k_2}|}\n\\nonumber \\\\\n&+& (k_1 + k_2) e^{- \\frac{1}{2} \\Gamma_1 |t - \\frac{mx}{k_1}| - \\frac{1}{2} \\Gamma_2 |t - \\frac{mx}{k_2}| } \\mbox{Re} \\left[ c_1c_2^* e^{i \\theta(t, x)}\\right],\n\\end{eqnarray}\nwhere   $\\Gamma_i = \\frac{2k_i\\gamma_{n_i}}{m}$, and the interference phase is\n\\begin{eqnarray}\n\\theta(t, x) = (k_1 - k_2)x  - \\frac{k_1^2 -k_2^2}{2m} t + \\frac{\\gamma_{n_1}^2 - \\gamma_{n_2}^2}{2m}t.\n\\end{eqnarray}\nThe flux is characterized by an exponential decay with a periodic modulation due to the energy difference between the interfering states. This is the well-known phenomenon of {\\em quantum beats}.\n\n\n\\section{Detection probabilities}\nIn the previous sections, we employed two methods for constructing the decay probability, namely, persistence probabilities and probabilities currents. Both methods work fine for exponential decays, where the decay probability is determined by a single parameter $\\Gamma$. Outside exponential decay they have a restricted domain of validity. The key problem is that they are not guaranteed to define positive-definite probabilities.  This is due to the fact that they do not express probabilities in terms of measurement outcomes for concrete observables, These probabilities are  guaranteed to be positive by the rules of quantum theory.\n\nA rigorous  description of decays requires a consideration of the explicit measurement scheme through which the decay products are detected, and the construction of appropriate measurement observables. The latter correspond to\n  positive operators $\\hat{\\Pi}(t)$, in which the detection time $t$ appears as a random variable. Then, given an initial state $\\hat{\\rho}_0$, the detection probability $p(t)$ is determined by $Tr\\left[\\hat{\\rho}_0 \\hat{\\Pi}(t)\\right]$.  A scheme for constructing temporal observables of this type has been developed in \\cite{AnSav}. Here, we will present an elementary example of such observables that generalizes the well-established photodetection model by Glauber \\cite{Glauber}.\n\n\nSuppose that one of the decay products is a particle that is described by  quantum field  operators $\\hat{\\phi}({\\pmb x})$ and Hamiltonian $\\hat{H}$. The field operators are split into a positive frequency part $\\hat{\\phi}^{(+)}({\\pmb x})$ that contains annihilation operators and a negative frequency part $\\hat{\\phi}^{(-)}({\\pmb x})$ that contains creation operators. Consider an elementary apparatus located at a point ${\\pmb x}$ that gives a detection signal at time $t$ after having absorbed the incoming particle. The amplitude associated to this process is then proportional to $\\hat{\\phi}^{(+)}({\\pmb x})|\\psi_t\\rangle$, where  $|\\psi_t \\rangle$  is the state of the quantum field at time $t$. The probability of detection is the determined by the modulus square of this amplitude. It is given by\n  Glauber's formula\n\\begin{eqnarray}\nP(t, {\\pmb x})  = C  \\langle \\psi_t| \\hat{\\phi}^{(-)}({\\pmb x})\\hat{\\phi}^{(+)}({\\pmb x})|\\psi_t\\rangle, \\label{Glauber}\n\\end{eqnarray}\nwhere   $C$ is a normalization constant.  We do not obtain normalized probabilities, because there is a non-zero probability that the particle will not be detected and this probability depends on the field-state.\n\nEq. (\\ref{Glauber}) was first proposed by Glauber for photodetection. In Glauber's theory,  the role of $\\hat{\\phi}$ is played by the electric field, and the absorption interaction corresponds to the dipole coupling between the electromagnetic field and a macroscopic detectors.    Glauber's formula is a special case of a larger class of particle detection observables that can be defined in quantum fields \\cite{AnSav}.\n\n\nWe will apply Glauber's formula to the bosonic Lee model. The field operators associated to the bosonic creation and annihilation operators are\n\\begin{eqnarray}\n\\hat{\\phi}^{(+)}({\\pmb x}) = \\sum_r \\hat{a}_r \\chi_r({\\pmb x}) \\hspace{1cm}  \\hat{\\phi}^{(-)}({\\pmb x}) = \\sum_r \\hat{a}_r \\chi^*_r({\\pmb x}) \\label{scalar}\n\\end{eqnarray}\nwhere $\\chi_r(x)$ are eigenfunctions of the single-particle Hamiltonian. For particles in three dimensions,  $r$ corresponds to the three-momentum ${\\pmb k}$, and\n\\begin{eqnarray}\n\\chi_{\\pmb k}({\\pmb x}) = \\frac{1}{\\sqrt{2 \\omega_{\\pmb k}} }e^{i{\\pmb k} \\cdot {\\pmb x}}. \\label{chik}\n\\end{eqnarray}\n\nBy Eq. (\\ref{decay555}),\n\\begin{eqnarray}\n\\hat{\\phi}^{(+)}({\\pmb x}) \\frac{1}{z - \\hat{H}} |A'\\rangle = \\frac{V(z; {\\pmb x})}{z - \\Omega - \\Sigma(z)} |g\\rangle \\otimes  |0\\rangle,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nV_ {\\pmb x}(z)  = \\sum_r \\frac{g_r \\chi_r({\\pmb x})  }{z- \\omega_r}. \\label{Vz}\n\\end{eqnarray}\nThen, Eq. (\\ref{Glauber}) gives\n\\begin{eqnarray}\nP(t, {\\pmb x})  = C  |{\\cal B}(t, {\\pmb x})|^2, \\label{Glauber2}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{\\cal B}(t, {\\pmb x}) =   \\lim_{\\epsilon \\rightarrow 0} \\int_{-\\infty+i \\epsilon }^{\\infty+i \\epsilon } \\frac{dE V_ {\\pmb x}(E)  e^{-iEt}}{[E - \\Omega - \\Sigma_a(E) }.\n\\end{eqnarray}\nFollowing the same steps that lead to  Eq. (\\ref{mainampl}), we find that\n\\begin{eqnarray}\n{\\cal B}(t,  {\\pmb x}) =\n   \\int_{0 }^{\\infty } \\frac{dE e^{-iEt}}{2\\pi}  \\frac{ \\frac{1}{2}\\Gamma(E) [V_{\\pmb x}^+(E) +V_{\\pmb x}^-(E) ] + i [E - \\Omega - F(E)][V_{\\pmb x}^+(E) - V_{\\pmb x}^-(E) ]  }{ [E - \\Omega - F(E)]^2 + \\frac{1}{4}[\\Gamma(E)]^2} \\label{mainampl44}\n\\end{eqnarray}\n where\n \\begin{eqnarray}\n V_{\\pmb x}^{\\pm}(E, {\\pmb x}) :=   \\lim_{\\eta \\rightarrow 0 } V_{\\pmb x}(E \\pm i \\eta).\n \\end{eqnarray}\n\nWe evaluate the amplitude (\\ref{mainampl44})  in the WWA, in which the dominant contribution to the integral comes\n from values of $E$ near $\\Omega$. Hence,\n\\begin{eqnarray}\n{\\cal B}(t,  {\\pmb x}) &\\simeq&\n  \\frac{1}{2 \\pi }  \\int_{-\\infty }^{\\infty } dE e^{-iEt}  \\frac{ \\Gamma  [V_{\\pmb x}^+(\\Omega) +V_{\\pmb x}^-(\\Omega) ]  + i [E - \\Omega - \\delta E][V_{\\pmb x}^+(\\Omega)  - V_{\\pmb x}^-(\\Omega) ]   }{ [E - \\Omega - \\delta E]^2 + \\frac{1}{4}\\Gamma^2} \\nonumber \\\\\n&=& \\frac{1}{2} e^{-i (\\Omega t + \\delta E)t - \\frac{1}{2}\\Gamma t}   V_+(\\Omega; {\\pmb x})   , \\label{mainampl44b}\n\\end{eqnarray}\nwhere we set $\\Gamma = \\Gamma(\\Omega)$ and $\\delta E = F(\\Omega)$. Hence, we obtain\n\\begin{eqnarray}\nP(t, {\\pmb x})  = \\frac{1}{4} C|V_{\\pmb x}^+(\\Omega) |^2 e^{-\\Gamma t}. \\label{decayGl}\n\\end{eqnarray}\nThe constant $C$ can be determined by normalizing over all particle detection events, i.e., by the requirement that\n\\begin{eqnarray}\n\\int_0^{\\infty} dt \\oint_{S} d^2n  P(t, {\\pmb x}) =1  , \\label{normalk}\n\\end{eqnarray}\nwhere $S$ is a two-sphere at distance $r$ from the location of the 2LS, ${\\pmb n}$ is a unit vector such that ${\\pmb x} = r {\\pmb n}$ on $S$.\n\nWe conclude  that the WWA guarantees exponential decay with constant $\\Gamma$, {\\em irrespective of the method used}. However, outside the exponential decay regime the probability density (\\ref{Glauber2}) differs significantly from Eq. (\\ref{decprob2}). In particular, it is guaranteed to be always positive.\n\n\n\n\\medskip\n\nAs an example, we revisit the photoemission model of Sec. 3.1.2. We employ Eqs. (\\ref{chik}) and (\\ref{Vz}), to obtain\n\\begin{eqnarray}\nV_{\\pmb x}(z) = \\frac{\\lambda}{8\\sqrt{2} \\pi^3} \\int \\frac{d^3k}{|{\\pmb k}|(z - |{\\pmb k}|)} e^{i {\\pmb k} \\cdot {\\pmb x}}.\n\\end{eqnarray}\nWe introduce spherical coordinates $(k, \\theta, \\phi)$ for ${\\pmb k}$, so that\n\\begin{eqnarray}\nV_{\\pmb x}(z) = \\frac{\\lambda}{4\\sqrt{2} \\pi^2} \\int_0^{\\infty} \\frac{kdk}{z- k} \\int_0^{\\pi} d\\theta \\sin \\theta e^{ikr \\cos \\theta} =\n \\frac{\\lambda}{2\\sqrt{2} \\pi^2 r} \\int_0^{\\infty} \\frac{dk \\sin(kr)}{z-k}, \\label{vzr}\n\\end{eqnarray}\ni.e., $V_{\\pmb x}(z)$ depends only on the radial coordinate $r = |{\\pmb x}|$.\n\nWe evaluate the integral (\\ref{vzr}) to\n\\begin{eqnarray}\nV_r(z) =  \\frac{\\lambda}{2\\sqrt{2} \\pi^2r}  \\left[ [\\gamma + \\ln(-rz) + \\mbox{Cin}(rz)] \\sin rz - \\mbox{si}(rz)\\cos rz\\right],\n \\end{eqnarray}\nwhere the functions\n $\\mbox{Cin}$ and $\\mbox{si}$ are    defined as\n\\begin{eqnarray}\n\\mbox{Cin}(z) = \\int_0^z dt \\frac{1-\\cos t}{t} \\hspace{1cm} \\mbox{si}(z) = \\int_z^{\\infty} dt \\frac{\\sin t}{t}, \\label{CiSi}\n\\end{eqnarray}\nand $\\gamma$ is the Euler-Mascheroni constant \\cite{ASt}.\n\nWe straightforwardly evaluate\n\\begin{eqnarray}\nV_r^{\\pm}(E;r) = \\frac{\\lambda}{2\\sqrt{2} \\pi^2r}  \\left[ [\\gamma + \\ln(rE) + \\mbox{Cin}(rE)] \\sin rE - \\mbox{si}(rE)\\cos rE \\mp i \\pi \\sin(rE)\\right].\n\\end{eqnarray}\n\nWe assume that the detectors are located at macroscopic distance from the decaying atom, so that $\\tilde{\\Omega}r >> 1$. Then, the terms involving the trigonometric integrals vanish, and the imaginary part of $V_+(\\Omega, r)$  dominates. Eq. (\\ref{decayGl}) gives\n\\begin{eqnarray}\nP(t, r) =  \\frac{\\lambda^2 \\sin^2(\\tilde{ \\Omega} r)}{32 \\pi^2 r^2} C e^{-\\Gamma t}.\n\\end{eqnarray}\n The sinusoidal dependence  on $r$ disappears if we average $P(t, r)$ over a thin shell of width $d >> \\tilde{\\Omega}^{-1}$ at distance $r$, since $\\langle\\sin^2(\\tilde{\\Omega}r)\\rangle = \\frac{1}{2}$. Then, the normalization condition (\\ref{normalk}) is satisfied for $C = (16 \\tilde{\\Omega})^{-1} $.\n\n  Note that outside the exponential decay regime, the probability density (\\ref{Glauber2}) leads to different predictions from the persistence probability method. The latter predicts an asymptotic probability density decaying with $t^{-5}$. Eq. (\\ref{mainampl44}) leads to an asymptotic decay of ${\\cal B}$ with $t^{-2}$, hence,  the probability density (\\ref{Glauber2})\ndecays as $t^{-4}$.\n\n\n\\section{Conclusions}\nWe presented an overview of the quantum description of decay processes. We showed that the emergence of the exponential decay law is explained in terms of a scale separation. In perturbative decays, the scale separation refers to energy: exponential decays emerge when the released energy associated to the decay  is much larger the energy of the interaction, as described by the self-energy function. In non-perturbative decays, the scale separation refers to time: exponential decay emerges when the decay time-scale is much larger than the time-scale of coherence between different attempts of the particle to cross the barrier.\n\nExponential decay may be extremely common, but it is not universal. It is not valid at very early and very late times, and in specific systems, it is not relevant at all. There is good experimental evidence for non-exponential decays, some of which pose persistent theoretical puzzles \\cite{GSI}. Our increasing access and control of multi-partite\/multi-particle systems is expected to uncover further unconventional types of decay---for example,  memory effects due to interaction with an environment \\cite{nonMark, BKEC},  effects due to the entanglement of the initial state \\cite{AnHu2, CVS17, C18}, or effects from particle statistics in many-particle systems \\cite{ADCM, TS11, CL11, DC11, MG11}. Furthermore, studies of    decay dynamics in many-particle quantum systems have demonstrated the need of  a genuinely many-particle characterization \\cite{ PSC12, HZHB13, DC16}, i.e., going beyond\ndescription in terms of single-particle observables.  We believe that a significant upgrade of traditional methods for quantum decays will be needed, in order to address such challenges.\n\n\\section*{Acknowledgements}\nResearch was supported by Grant No. E611 from the Research Committee of the University of Patras via the \"K. Karatheodoris\" program.\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\begin{subfigure}{0.15\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic1.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:motivation:A}\n\t\\end{subfigure}\n\t\\quad \\quad \\quad \\quad\n\t\\begin{subfigure}{0.17\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic2.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:motivation:B}\t\n\t\\end{subfigure}\n\t\\quad \\quad \\quad \\quad\n\t\\begin{subfigure}{0.16\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:motivation:C}\n\t\\end{subfigure}\n\t\\quad \\quad \\quad \\quad\n\t\\begin{subfigure}{0.15\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic4.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:motivation:D}\n\t\\end{subfigure}\n\t\\\\ \n\t\\begin{subfigure}{0.19\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic5.pdf}\n\t\t\\caption{Tied filtering}\n\t\t\\label{fig:motivation:E}\n\t\\end{subfigure}\n\t\\quad \\quad\n\t\\begin{subfigure}{0.23\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic6.pdf}\n\t\t\\caption{Ranking filtering}\n\t\t\\label{fig:motivation:F}\n\t\\end{subfigure}\n\t\\quad \\quad\n\t\\begin{subfigure}{0.20\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic7.pdf}\n\t\t\\caption{Gaussian-induced filtering}\n\t\t\\label{fig:motivation:G}\n\t\\end{subfigure}\n\t\\quad \\quad\n\t\\begin{subfigure}{0.17\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic8.pdf}\n\t\t\\caption{$3\\times 3$ regular filtering}\n\t\t\\label{fig:motivation:H}\n\t\\end{subfigure}\n\t\\caption{Different filters operations on graph vertices. Examples of one-hop subgraphs are given in (a)-(d), where $v_0$ is the reference vertex and each vertex is assigned to a signal. The tied filtering (e) summarizes all neighbor vertices, and generates the same responses to (a) and (b) under the filter $f$, {i.e.}} \\def\\etal{{et.al}, $f(\\sum \\widetilde{w_{0i}}x_i)=f(1.9)$, although the two graphs are completely different in structures. The ranking filtering (f) sorts\/prunes neighbor vertices and then performs different filtering on them. It might result into the same responses $f_1(1)+f_2(3)+f_3(1)+f_4(4)$ to different graphs such as (b) and (c), where the digits in red boxes denote the ranked indices and the vertex of dashed box in (b) is pruned. Moreover, the vertex ranking is uncertain\/non-unique for equal connections in (d).To address these problems, we derive edge-induced GMM to coordinate subgraphs as shown in (g). Each of Gaussian model can be viewed as one variation component (or direction) of subgraph. Like the standard convolution (h), the Gaussian encoding is sensitive to different subgraphs, \\eg, (a)-(d) will have different responses. Note $f, f_i$ are linear filters, and the non-linear activation functions are put on their responses.\n\t}\n\t\\label{fig:motivation}\n\\end{figure*}\n\nAs witnessed by the widespread applications, graph is one of the most successful models to conduct structured and semi-structured data, ranging from text~\\cite{defferrard2016convolutional}, bioinformatics~\\cite{yanardag2015deep,niepert2016learning,song2018eeg} and social network~\\cite{gomez2017dynamics,orsini2017shift} to images\/videos~\\cite{marino2016more,cui2018context,cui2017spectral}. Among these applications, learning robust representations from structured graphs becomes the main topic. To this end, various methods have come forth in recent years. Graph kernels~\\cite{yanardag2015deep} and recurrent neural networks (RNNs)~\\cite{scarselli2009graph} are the most representative ones. Graph kernels usually take the classic R-convolution strategy~\\cite{haussler1999convolution} to recursively decompose graphs into atomic sub-structures and then define local similarities between them. RNNs based methods sequentially traverse neighbors with tied parameters in depth. With the increase of graph size, graph kernels would suffer diagonal dominance of kernels~\\cite{scholkopf2002kernel} while RNNs would have the explosive number of combinatorial paths in the recursive stage.\n\nRecently convolutional neural networks (CNNs)~\\cite{lecun2015deep} have achieved breakthrough progresses on representing grid-shaped image\/video data. In contrast, graphs are with irregular structures and fully coordinate-free on vertices and edges. The vertices\/edges are not strictly ordered, and can not be explicitly matched between two graphs. To generalize the idea of CNNs onto graphs, we need to solve this problem therein that the same responses should be produced for those homomorphic graphs\/subgraphs when performing convolutional filtering. To this end, recent graph convolution methods~\\cite{defferrard2016convolutional,atwood2016diffusion,hamilton2017inductive} attempted to aggregate neighbor vertices as shown in Fig.~\\ref{fig:motivation:E}. This kind of methods actually employ a fuzzy filtering ({i.e.}} \\def\\etal{{et.al}, a tied\/shared filter) on neighbor vertices because only first-order statistics (mean) is used. Two examples are shown in Fig.~\\ref{fig:motivation:A} and Fig.~\\ref{fig:motivation:B}. Although they have different structures, the responses on them are fully equal. Oppositely, Niepert \\etal~\\cite{niepert2016learning} ranked neighbor vertices according to weights of edges, and then used different filters on these sorted vertices, as shown in Fig.~\\ref{fig:motivation:F}. However, this rigid ranking method will suffer some limitations: i) probably consistent responses to different structures (\\eg, Fig.~\\ref{fig:motivation:B} and Fig.~\\ref{fig:motivation:C}) because weights of edges are out of consideration after ranking; ii) information loss of node pruning for a fixed-size receptive field as shown in Fig.~\\ref{fig:motivation:B}; and iii) ranking ambiguity for equal connections as shown in Fig.~\\ref{fig:motivation:D}; and iv) ranking sensitivity to (slightly) changes of edge weights\/connections.\n\nIn this paper we propose a Gaussian-induced graph convolution framework to learn graph representation. For a coordinate-free subgraph region, we design an \\textit{edge-induced} Gaussian mixture model (EI-GMM) to implicitly coordinate the vertices therein. Specifically, the edges are used to regularize Gaussian models such that variations of subgraph can be well-encoded. In analogy to the standard convolutional kernel as shown in Fig.~\\ref{fig:motivation:H}, EI-GMM can be viewed as a coordinate normalization by projecting variations of subgraph into several Gaussian components. For example, the four subgraphs w.r.t. Fig.~\\ref{fig:motivation:A}$\\sim$\\ref{fig:motivation:D} will have different representations\\footnote{Suppose three Gaussian models are $\\mcN(0,1), \\mcN(0,2)$ and $\\mcN(0,3)$, then we can compute the responses on (a)-(d) respectively as $f_1([0.49, -0.93])+f_2([0.17, -0.65])+f_3([0.07, -0.44])$, $f_1([0.35, -0.73])+f_2([0.15, -0.58])+f_3([0.10, -0.64]$, $f_1([0.35, -0.71])+f_2([0.15, -0.39])+f_3([0.10, -0.43])$, $f_1([0.46, -0.99])+f_2([0.18, -0.62])+f_3([0.08, -0.42])$. Please refer to incoming section.} through our Gaussian encoding in Fig.~\\ref{fig:motivation:G}. To make the network inference forward, we transform Gaussian components of each subgraph into the gradient space of multivariate Gaussian parameters, instead of employing the sophisticated EM algorithm. Then the filters (or transform functions) are performed on different Gaussian components like latticed kernels on different directions in Fig.~\\ref{fig:motivation:H}. Further, we derive a \\textit{vertex-induced} Gaussian mixture model (VI-GMM) to favor dynamic coarsening of graph. We also theoretically analyze the approximate equivalency of VI-GMM to weighted graph cut~\\cite{dhillon2007weighted}. Finally, EI-GMM and VI-GMM can be alternately stacked into an end-to-end optimization network.\n\nIn summary, our main contributions are four folds: i) propose an end-to-end Gaussian-induced convolutional neural network for graph representation; ii) propose edge-induced GMM to encode variations of different subgraphs; iii) derive vertex-induced GMM to perform dynamic coarsening of graphs, which is an approximation to the weighted graph cut; iv) verify the effectiveness of our method and report state-of-the-art results on several graph datasets.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.83\\textwidth]{pic9.pdf}\n\t\\caption{The GIC network architecture. The GIC main contains two module: convolution layer (EI-GMM) and coarsening layer (VI-GMM). The GIC stacks several convolution and coarsening layers alternatively and iteratively. More details can be found in incoming section.}\n\t\\label{fig:network:A}\n\\end{figure*}\n\n\n\\section{Related Work}\n\nGraph CNNs mainly fall into two categories: spectral and spatial methods. Spectral methods~\\cite{bruna2013spectral,scarselli2009graph,henaff2015deep,such2017robust,li2018action,li2018spatio} construct a series of spectral filters by decomposing graph Laplacian, which often suffers high-computational burden. To address this problem, the fast local spectral filtering method~\\cite{defferrard2016convolutional} parameterizes the frequency responses as a Chebyshev polynomial approximation. However, as shown in Fig.~\\ref{fig:motivation:B}, after summarizing all nodes, this method will discard topology structures of a local receptive field. This kind of methods usually require equal sizes of graphs like the same sizes of images for CNNs~\\cite{kipf2016semi}. Spatial methods attempt to define spatial structures of adjacent vertices and then perform filtering on structured graphs. Diffusion CNNs~\\cite{atwood2016diffusion} scans a diffusion process across each node. PATCHY-SAN~\\cite{niepert2016learning} linearizes neighbors by sorting weights of edges and deriving convolutional filtering on graphs, as shown in Fig.~\\ref{fig:motivation:C}. As an alternative, random walks based approach is also used to define the neighborhoods~\\cite{perozzi2014deepwalk}. For the linearized neighbors, RNNs~\\cite{li2015gated} could be used to model the structured sequences. Similarly, NgramCNN~\\cite{luo2017deep} serializes each graph by introducing the concept of $n$-gram block. \nGAT~\\cite{velickovic2017graph} attempts to weight edges through the attention mechanism. WSC~\\cite{jiang2018walk} attempts to aggregate walk fields defined by random walks into Gaussian mixture models. Zhao~\\cite{zhao2018work} attempts to define a standard network with different graph convolutions. Besides, some variants~\\cite{hamilton2017inductive,duran2017learning,zhang2018tensor} employ the aggregation or propagation of local neighbor nodes. Different from these tied filtering or ranking filtering methods, we use Gaussian models to encode local variations of graph. Also different from the recent mixture models~\\cite{monti2017geometric}, which uses GMM to only learn the importance of adjacent nodes, our method uses weighted GMM to encode the distributions of local graph structures.\n\n\n\\section{The GIC Network}\n\n\\subsection{Attribute Graph}\n\nHere we consider an undirected attribute graph $\\mcG=(\\mcV,\\A,\\X)$ of $m$ vertices (or nodes), where $\\mcV=\\{v_i\\}_{i=1}^{m}$ is the set of vertices, $\\A$ is a (weighted) adjacency matrix, and $\\X$ is a matrix of graph attributes (or signals). The adjacency matrix $\\A\\in\\mbR^{m\\times m}$ records the connections between vertices. If $v_i, v_j$ are not connected, then $A(v_i,v_j)=0$, otherwise $A(v_i,v_j)\\neq 0$. We sometimes abbreviate $A(v_i,v_j)$ as $A_{ij}$. The attribute matrix $\\X\\in\\mbR^{m\\times d}$ is associated with the vertex set $\\mcV$, whose $i$-th row $\\X_{i}$ (or $\\X_{v_i}$) denotes a $d$-dimension attribute of the $i$-th node ({i.e.}} \\def\\etal{{et.al}, $v_i$).\n\nThe graph Laplacian matrix $\\L$ is defined as $\\L = \\D-\\A$, where $\\D\\in\\mbR^{m\\times m}$ is the diagonal degree matrix with $D_{ii}=\\sum_{j}A_{ij}$. The normalized version is written as $\\L^{norm} = \\D^{-1\/2}\\L\\D^{-1\/2}= \\I-\\D^{-1\/2}\\A\\D^{-1\/2}$.\nwhere $\\I$ is the identity matrix. Unless otherwise specified, we use the latter. We give the definition of subgraph used in the following.\n\\begin{defn} \\label{def:graph}\n\tGiven an attribute graph $\\mcG=(\\mcV,\\A,\\X)$, the attribute graph $\\mcG'=(\\mcV',\\A',\\X')$ is a subgraph of $\\mcG$, denoted $\\mcG'\\subseteq\\mcG$, if (i) $\\mcV'\\subseteq\\mcV$, (ii) $\\A'$ is the submatrix of $\\A$ w.r.t. the subset $\\mcV'$, and (iii) $\\X'=\\X_{\\mcV'}$.\n\\end{defn}\n\n\\subsection{Overview}\n\nThe GIC network architecture is shown in Fig.~\\ref{fig:network:A}. Given an attribute graph $\\mcG^{(0)}=(\\mcV^{(0)},\\A^{(0)},\\X^{(0)})$, where the superscript denotes the layer number, we construct multi-scale receptive fields for each vertex based on the adjacency matrix $\\A^{(0)}$. Each receptive field records $k$-hop neighborhood relationships around the reference vertex, and forms a local centralized subgraph. To encode the centralized subgraph, we project it into edge-induced Gaussian models, each of which defines one variation ``direction\" of the subgraph. We perform different filtering operations on different Gaussian components and aggregate all responses as the convolutional output. After the convolutional filtering, the input graph $\\mcG^{(0)}$ is transformed into a new graph $\\mcG^{(1)}=(\\mcV^{(1)},\\A^{(1)},\\X^{(1)})$, where $\\mcV^{(1)}=\\mcV^{(0)}$ and $\\A^{(1)}=\\A^{(0)}$. To further abstract graphs, we next stack a coarsening layer on the graph $\\mcG^{(1)}$. The proposed vertex-induced GMM is used to downsample the graph $\\mcG^{(1)}$ into the low-resolution graph $\\mcG^{(2)}=(\\mcV^{(2)},\\A^{(2)},\\X^{(2)})$. Taking the convolution and coarsening modules, we may alternately stack them into a multi-layer GIC network, With the increase of layers, the receptive field size of filters will become larger, so the higher layer can extract more global graph information. In the supervised case of graph classification, we finally concatenate with a fully connected layer followed by a softmax loss layer.\n\n\\subsection{Multi-Scale Receptive Fields} \n\nIn the standard CNN, receptive fields may be conveniently defined as latticed spatial regions. Thus convolution kernels on grid-shaped structures are accessible. However, the construction of convolutional kernels on graphs are intractable due to coordinate-free graphs, \\eg, unordered vertices, unfixed number of adjacent edges\/vertices. To address this problem, we resort to the adjacent matrix $\\A$, which expresses connections between vertices. Since $\\A^k$ exactly records the $k$-step reachable vertices, we may construct a $k$-neighbor receptive field by using the $k$-order polynomial of $\\A$, denoted as $\\psi_k(\\A)$. Taking the simplest case, $\\psi_k(\\A)=\\A^k$ reflects the $k$-hop neighborhood relationships. In order to remove the scale effect, we may normalize $\\psi_k(\\A)$ as $\\psi_k(\\A) \\text{diag} (\\psi_k(\\A)\\1)^{-1}$, which describes the reachable possibility in a $k$-hop walking. Formally, we define the $k$-th scale receptive field as a subgraph.\n\\begin{defn}\\label{def:receptive}\n\tThe $k$-th scale receptive field around a reference vertex $v_i$ is a subgraph $\\mcG_{v_i}^k=(\\mcV',\\A',\\X')$ of the k-order graph $(\\mcV, \\tbA=\\psi_k(\\A), \\X)$, where $\\mcV'=\\{v_j|\\wtA_{ij}\\neq 0\\}\\cup\\{v_i\\}$, $\\A'$ is the submatrix of $\\tbA$ w.r.t. $\\mcV'$, and $\\X'=\\X_{\\mcV'}$.\n\\end{defn}\n\n\\subsection{Convolution: Edge-Induced GMM}\n\nGiven a reference vertex $v_i$, we can construct the centralized subgraph $\\mcG_{v_i}^k$ of the $k$-th scale.\nTo coordinate the subgraph, we introduce Gaussian mixture models (GMMs), each of which may be understood as one principal direction of its variations. To encode the variations accurately, we jointly formulate attributes of vertices and connections of edges into Gaussian models. The edge weight $A'(v_i,v_j)$ indicates the relevance of $v_j$ to the central vertex $v_i$. The higher weight is, the stronger impact on $v_i$ is. So the weights can be incorporated into a Gaussian model by observing $A'(v_i,v_j)$ times. As the likelihood function, it is equivalent to raise the power $A'(v_i,v_j)$ on Gaussian function, which is proportional to $\\mcN(\\X'_{v_j}, \\bmu, \\frac{1}{A'(v_i,v_j)}\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta})$. Formally,  we estimate the probability density of the subgraph $\\mcG_{v_i}^k$ from the $C_1$-component GMM,\n\\begin{align}\np_{v_i}(\\X'_{v_j};\\bTheta_1,A'_{ij}) &= \\sum_{c=1}^{C_1}\\pi_c\\mcN(\\X'_{v_j}; \\bmu_c, \\frac{1}{A'_{ij}}\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}_c), \\nonumber \\\\\n&\\st \\pi_c>0, \\sum_{c=1}^{C_1}\\pi_c=1, \\label{eqn:GMM_edge}\n\\end{align}\nwhere $\\bTheta_1=\\{\\pi_1,\\cdots,\\pi_{C_1}, \\bmu_1,\\cdots,\\bmu_{C_1},\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}_1,\\cdots,\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}_{C_1}\\}$ are the mixture parameters, $\\{\\pi_c\\}$ are the mixture coefficients, $\\{\\bmu_c, \\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}_c\\}$ are the parameters of the $k$-th component, and $A'_{ij}>0$ \\footnote{In practice, we normalize $\\A'$ into a non-negative matrix.}. Intuitively, edge weight $A'_{ij}$ is, the stronger impact of the node $v_j$ w.r.t. the reference vertex $v_i$ is. We will refer to the model in Eqn.~(\\ref{eqn:GMM_edge}) as the \\textit{edge-induced Gaussian mixture model} (EI-GMM).\n\nIn what follows, we assume all attributes of nodes are independent on each other, which is often used in signal processing. That means, the covariance matrix $\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}_c$ is diagonal, so we denote it as $\\text{diag}(\\mathbf{\\sigma}_c^2)$. To avoid the explicit constraints for $\\pi_c$ in Eqn.~(\\ref{eqn:GMM_edge}), we adopt the soft-max normalization with the re-parameterization variable $\\alpha_c$, {i.e.}} \\def\\etal{{et.al}, $\\pi_c = {\\exp(\\alpha_c)}\/{\\sum_{k=1}^{C_1}\\exp(\\alpha_k)}$. Thus, the entire subgraph log-likelihood can be written as\n\\begin{align}\n\\zeta(\\mcG_{v_i}^k) &= \\sum_{j=1}^m\\ln p_{v_i}(\\X'_{v_j};\\bTheta_1,\\A') \\nonumber \\\\\n&= \\sum_{j=1}^m\\ln\\sum_{c=1}^{C_1}\\pi_c\\mcN(\\X'_{v_j}; \\bmu_c, \\frac{1}{A'_{ij}}\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}_c),\\label{eqn:GMM_log}\n\\end{align}\nTo infer forward, instead of the expectation-maximization (EM) algorithm, we use the gradients of the subgraph with regard to the parameters of the EI-GMM model $\\bTheta_1$, motivated by the recent Fisher vector work~\\cite{sanchez2013image}, which has been proven to be effective in representation.\n\nFor a convenient calculation, we simplify the notations, $\\mcN_{jc} = \\mcN(\\X'_{v_j}, \\bmu_c, \\frac{1}{A'_{ij}}\\mathbf{\\sigma}_c^2)$ and $ Q_{jc}=\\frac{\\pi_c\\mcN_{jc} }{\\sum_{k=1}^{C_1}\\pi_k\\mcN_{jk}}$, then we can derive the gradients of model parameters from Eqn.~(\\ref{eqn:GMM_log}) as follows\n\\begin{align}\n&\\frac{\\partial \\zeta(\\mcG_{v_i}^k)}{\\partial\\bmu_c}  \\!\\!=\\!\\! \\sum_{j=1}^m\\frac{A'_{ij}Q_{jc}(\\X'_{v_j}-\\bmu_c)}{\\mathbf{\\sigma}_c^2}, \\quad \\nonumber \\\\\n&\\frac{\\partial \\zeta(\\mcG_{v_i}^k)}{\\partial\\mathbf{\\sigma}_c}  \\!\\!=\\!\\! \\sum_{j=1}^m \\frac{Q_{jc}(A'_{ij}(\\X'_{v_j}-\\bmu_c)^2-\\mathbf{\\sigma}_c^2)}{\\mathbf{\\sigma}_c^3},\n\\end{align} \nwhere the division of vectors means a term-by-term operation. Note we do not use $\\partial \\zeta(\\mcG_{v_i}^k) \/ \\partial\\alpha_c$ due to no improvement in our experience. The gradients describe the contribution of the corresponding parameters to the generative process. The subgraph variations are adaptively allocated to $C_1$ Gaussian models. Finally, we ensemble all gradients w.r.t. Gaussian model ({i.e.}} \\def\\etal{{et.al}, directions of graph) to analogize the collection of local square receptive field on image. Formally, for the $k$-scale receptive field $\\mcG_{v_i}^k$ around the vertex $v_i$, the attributes produced from Gaussian models are filtered respectively and then concatenated,\n\\begin{align}\nF(\\mcG_{v_i}^k,\\bTheta_1,f) &= \\text{ReLU}(\\sum_{c=1}^{C_1} f_i(\\text{Cat}[\\frac{\\partial \\zeta(\\mcG_{v_i}^k)}{\\partial\\bmu_c},\\frac{\\partial \\zeta(\\mcG_{v_i}^k)}{\\partial\\mathbf{\\sigma}_c}]), \\label{eqn:gau_fea}\n\\end{align}\nwhere $\\text{Cat}[\\cdot, \\cdot]$ is a concatenation operator, $f_i$ is a linear filtering function ({i.e.}} \\def\\etal{{et.al}, a convolution function) and ReLU is the rectified linear unit. Therefore we can produce the feature vectors that have same dimensionality depending on the number of Gaussian models for different subgraphs. If the soft assignment distribution $ Q_{jc} $ is sharply peaked on a single value of one certain Gaussian for the vertex $v_{j}$, the vertex will be only projected onto one Gaussian direction.\n\n\\subsection{Coarsening: Vertex-Induced GMM}\n\nLike the standard pooling in CNNs, we need to downsample graphs so as to abstract them as well as reduce the computational cost. However, the pooling on images are tailored for latticed structures, and cannot be used for irregular graphs. One solution is to use some clustering algorithms to partition vertices to several clusters, and then produce a new vertex from each cluster. However, we expect that two vertices should not fall into the same cluster with a larger possibility if there is a high transfer difficulty between them. To this end, we derive vertex-induced Gaussian mixture models (VI-GMM) to weight each vertex. To utilize the edge information,  we construct a latent observation $\\phi(v_i)$ w.r.t. each vertex $v_i$ from the graph Laplacian (or adjacent matrix if semi-positive definite), {i.e.}} \\def\\etal{{et.al}, the kernel calculation $\\langle\\phi(v_i),\\phi(v_j)\\rangle=L_{ij}$. Moreover, for each vertex $v_i$, we define an influence factor $w_i$ for Gaussian models. Formally, given $C_2$ Gaussian models, VI-GMM is written as\n\\begin{align}\np(\\phi(v_i);\\bTheta_2,w_i) &= \\sum_{c=1}^{C_2}\\pi_c\\mcN(\\phi(v_i); \\bmu_c, \\frac{1}{w_i}\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}_c), \\nonumber \\\\\n&\\st  w_i=h(\\X_{v_i})>0,\n\\end{align}\nwhere $h$ is a mapping function to be learnt. To reduce the computation cost of matrix inverse on $\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}$, we specify it as an identity matrix. Then we have\n\\begin{align}\np(\\phi(v_i);\\bTheta_2,w_i) =  \\sum_{c=1}^{C_2}\\frac{\\pi_c}{(\\frac{2\\pi}{w_i})^{d\/2}}\\exp^{-\\frac{w_i}{2}\\|\\phi(v_i)-\\bmu_c\\|^2}, \n\\end{align}\nGiven a graph with $m$ vertices, the objective is to maximize the following log-likelihood:\n\\begin{align}\n\\argmax_{\\bTheta_2} \\zeta(\\bTheta_2) = \\sum_{i=1}^m \\ln\\sum_{c=1}^{C_2}\\pi_c\\mcN(\\phi(v_i); \\bmu_c, \\frac{1}{w_i}\\I)). \\label{eqn:likelihood}\n\\end{align}\n\nTo solve above model in Eqn.~(\\ref{eqn:likelihood}), we use the iterative expectation maximization algorithm, which has closed-form solution at each step. Meanwhile, the algorithm may automatically conduct the required constraints. The graphical clustering process is summarized as follows:\n\n(1) {E-Step}: the posteriors, {i.e.}} \\def\\etal{{et.al}, the $i$-th vertex for the $c$-th cluster, are updated with\n$p_{ic} = \\frac{\\pi_c p(\\phi(v_i);\\btheta_c, w_i)}{\\sum_{k=1}^C \\pi_k p(\\phi(v_i);\\btheta_k, w_i)}$,\nwhere $\\btheta_c$ is the $c$-th Gaussian parameters, and $\\bTheta_2=\\{\\btheta_1,\\cdots,\\btheta_{C_2}\\}$.\n\n(2) {M-Step}: we optimize Gaussian parameters $\\pi, \\bmu$. The parameter estimatation is given by $\n\\pi_c = \\frac{1}{m}\\sum_{i=1}^m r_{ic},\n\\bmu_c =\\frac{\\sum_{v_i\\in \\mcG_c}w_i\\phi(v_i)}{\\sum_{v_i\\in \\mcG_c}w_i}$.\n$\\pi_c$ indicates the energy summation of all vertices assigned to the cluster $c$, and $\\bmu_c$ may be understood as a doubly weighted ($w_i, r_{ic}$) average on the cluster $c$.\n\nAfter several iterations of the two steps, we perform hard quantification. The $i$-th vertex is assigned as the class with the maximum possibility, formally, $r_{ic} = 1$ if $c=\\argmax_{k} p_{ik}$, otherwise 0. Thus we can obtain the cluster matrix $\\P\\in\\{0,1\\}^{m\\times C_2}$, where $P_{ic}=1$ if the $i$-th vertex falls into the cluster $c$. During coarsening, we take maximal responses of each cluster as the attributes of new vertex, and derive a new adjacency matrix by using $\\P^\\intercal} \\def\\st{\\text{s.t.~}\\A\\P$.\n\nIt is worth noting that we need not compute the concrete $\\phi$ during the clustering process. The main calculation $\\|\\phi(v_i)-\\bmu_c\\|^2$ in EM can be reduced to the kernel version:\n$K_{ii}-\\frac{2\\sum_{v_j\\in\\mcG_c}w_jK_{ij}}{\\sum_{v_j\\in\\mcG_c}w_j}\n+\\frac{\\sum_{v_j,v_k\\in\\mcG_c}w_jw_k K_{jk}}{(\\sum_{v_j\\in\\mcG_c}w_j)^2}$,\nwhere $K_{ij} = \\langle\\phi(v_i),\\phi(v_j)\\rangle$.\nIn practice, we can use the graph Laplacian $\\L$ as the kernel. In this case, we can easily reach the following proposition, which is relevant to graph cut~\\cite{dhillon2007weighted}.\n\\begin{prop}\n\tIn EM, if the kernel matrix takes the weight-regularized graph Laplacian, {i.e.}} \\def\\etal{{et.al}, $\\mcK= \\text{diag}(\\w)\\L \\text{diag}(\\w)$, then VI-GMM is equal to an approximate optimization of graph cut, {i.e.}} \\def\\etal{{et.al}, $\n\t\\min \\sum_{c=1}^C\\frac{\\text{links}(\\mcV_c, \\mcV\\backslash \\mcV_c)}{w(\\mcV_c)}$,\n\twhere $\\text{links}(\\mcA, \\mcB)=\\sum_{v_i\\in\\mcA,v_j\\in\\mcB} A_{ij}$, and $w(\\mcV_c)=\\sum_{j\\in\\mcV_c}w_j$.\n\\end{prop}\n\n\n\\section{Experiments}\n\n\\subsection{Graph Classification}\n\nFor graph classification, each graph is annotated with one label. We use two types of datasets: Bioinformatics and Network datasets. The former contains MUTAG~\\cite{debnath1991structure}, PTC~\\cite{toivonen2003statistical}, NCI1 and NCI109~\\cite{wale2008comparison}, ENZYMES~\\cite{borgwardt2005protein} and PROTEINS~\\cite{borgwardt2005protein}. The latter has COLLAB~\\cite{leskovec2005graphs}, REDDIT-BINARY, REDDIT-MULTI-5K, REDDIT-MULTI-12K, IMDB-BINARY and IMDB-MULTI.\n\n\\begin{table*}[!t]\n\t\\centering\n\t\\caption{Comparisons with state-of-the-art methods.}\n\t\\begin{sc}\n\t\\scalebox{0.85}{\n\t\t\\begin{tabular}{|l| c c c| c c | c c c |c |c |c |c|}\n\t\t\t\\toprule\n\t\t\tDataset \n\t\t\t&PSCN  &DCNN  &NgramCNN   &FB  &DyF   &WL   &GK  &DGK  &RW   &SAEN  & GIC \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{MUTAG}\n\t\t\t&92.63   &66.98  &\\textbf{94.99}  &84.66   &88.00    &78.3   &81.66     &82.66      &83.72     &84.99 &94.44   \\\\\t\t\n\t\t\t& $\\pm$4.21  &--    &\\textbf{$\\pm$5.63} &$\\pm$2.01 & $\\pm$2.37  & $\\pm$1.9 & $\\pm$2.11  & $\\pm$1.45  & $\\pm$1.50   & $\\pm$1.82  \t&$\\pm$4.30      \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{PTC}\n\t\t\t&60.00   &56.60  &68.57  &55.58   &57.15     &--     &57.26     &57.32   &57.85    &57.04  &\\textbf{77.64}   \\\\\t\t\t\n\t\t\t& $\\pm$4.82  &--   &$\\pm$1.72  &2.30   & $\\pm$1.47  & --  &  $\\pm$1.41  &  $\\pm$1.13  & $\\pm$1.30\n\t\t\t&  $\\pm$ 1.30  & \\textbf{$\\pm$ 6.98}      \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{NCI1}\n\t\t\t&78.59   &62.61     &--  &62.90   &68.27  &83.1   &62.28   &62.48   &48.15    &77.80  &\\textbf{84.08}    \\\\\t\t\t\n\t\t\t& $\\pm$1.89  &--   &--  &$\\pm$0.96  & $\\pm$0.34  & $\\pm$0.2  & $\\pm$0.29  &  $\\pm$0.25  &  $\\pm$0.50  & $\\pm$ 0.42  & \\textbf{$\\pm$1.77}     \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{NCI109}\n\t\t\t& --   &62.86    &--  &62.43  & 66.72   & \\textbf{85.2}    & 62.60   & 62.69   & 49.75 \t & --  & 82.86    \\\\\t\t\t\n\t\t\t& --   &--   &--  &$\\pm$1.13    & $\\pm$ 0.20  & \\textbf{$\\pm$ 0.2}  & $\\pm$ 0.19  &  $\\pm$ 0.23  &  $\\pm$ 0.60\t& --  & $\\pm$ 2.37     \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{ENZYMES}\n\t\t\t& --    &18.10    &--  &29.00   & 33.21   & 53.4   & 26.61   & 27.08   & 24.16   & --   & \\textbf{62.50}  \\\\\t\t\t\n\t\t\t& --    &--    &--  &$\\pm$1.16  & $\\pm$ 1.20  & $\\pm$ 1.4   &  $\\pm$ 0.99  & $\\pm$ 0.79  &  $\\pm$ 1.64\n\t\t\t&--  & \\textbf{$\\pm$ 5.12}     \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{PROTEINS}\n\t\t\t& 75.89  &--   &75.96  &69.97 & 75.04   & 73.7  & 71.67   & 71.68  & 74.22    & 75.31  & \\textbf{77.65}     \\\\\t\t\t\n\t\t\t& $\\pm$ 2.76  &--   &$\\pm$2.98  &$\\pm$1.34  & $\\pm$ 0.65  &  $\\pm$ 0.5  & $\\pm$ 0.55  &  $\\pm$ 0.50  & $\\pm$ 0.42\t&  $\\pm$ 0.70 & \\textbf{$\\pm$ 3.21}      \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{COLLAB}\n\t\t\t& 72.60  &--    &--  &76.35 & 80.61 & -- & 72.84 & 73.09 & 69.01 & 75.63 & \\textbf{81.24}     \\\\\t\t\t\n\t\t\t& $\\pm$ 2.15  &--   &--  &1.64  & $\\pm$ 1.60  & --  & $\\pm$ 0.28  & $\\pm$ 0.25  & $\\pm$ 0.09\n\t\t\t& $\\pm$ 0.31  & \\textbf{$\\pm$ 1.44}      \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{REDDIT-B}\n\t\t\t& 86.30  &--     &--  &88.98 &\\textbf{89.51}  &75.3 & 77.34 & 78.04 & 67.63  & 86.08 & 88.45   \\\\\t\t\t\n\t\t\t& $\\pm$ 1.58  &--   &--  &$\\pm$2.26  & \\textbf{$\\pm$ 1.96}  & $\\pm$ 0.3  & $\\pm$ 0.18  & $\\pm$ 0.39  & $\\pm$ 1.01\n\t\t\t& $\\pm$ 0.53  & $\\pm$ 1.60     \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{REDDIT-5K}\n\t\t\t& 49.10  &--      &--  &50.83   & 50.31 & --  & 41.01 & 41.27 & --  &\\textbf{52.24} &51.58 \\\\\t\t\n\t\t\t& $\\pm$ 0.70  &--    &--  &1.83  &$\\pm$ 1.92  & --  & $\\pm$ 0.17  & $\\pm$ 0.18  & --\n\t\t\t& \\textbf{$\\pm$ 0.38}  & $\\pm$ 1.68   \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{REDDIT-12K}\n\t\t\t& 41.32  &--      &--  &42.37  & 40.30 & --  & 31.82 & 32.22 & --  & \\textbf{46.72} & 42.98 \\\\\t\t\t\n\t\t\t& $\\pm$ 0.42  &--    &--  &1.27   & $\\pm$ 1.41  & -- & $\\pm$ 0.08  & $\\pm$ 0.10  & --\n\t\t\t& \\textbf{$\\pm$ 0.23} & $\\pm$ 0.87     \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{IMDB-B}\n\t\t\t& 71.00   &--    &71.66  &72.02 & 72.87 & 72.4 & 65.87 & 66.96 & 64.54 \t& 71.26 & \\textbf{76.70}     \\\\\t\t\t\n\t\t\t& $\\pm$ 2.29  &--    &$\\pm$2.71  &$\\pm$4.71  & $\\pm$ 4.05  & $\\pm$ 0.5  & $\\pm$ 0.98  & $\\pm$ 0.56  & $\\pm$ 1.22\n\t\t\t& $\\pm$ 0.74  & \\textbf{$\\pm$ 3.25}      \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{IMDB-M}\n\t\t\t& 45.23 &--     &50.66  &47.34  & 48.12 & -- & 43.89 & 44.55 & 34.54 & 49.11 & \\textbf{51.66}      \\\\\t\t\t\n\t\t\t& $\\pm$ 2.84  &--  &$\\pm$4.10  &3.56  & $\\pm$ 3.56  & -- & $\\pm$ 0.38  & $\\pm$ 0.52  & $\\pm$ 0.76\n\t\t\t& $\\pm$ 0.64  & \\textbf{$\\pm$ 3.40}      \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n\t\\end{sc}\n\t\\label{table:state-of-the-art}\n\\end{table*}\n\\begin{table}[!t]\n\t\\centering\n\t\\caption{Node label prediction on Reddit and PPI data (micro-averaged F1 score).}\n\t\\label{table:multi-label}\n\t\\begin{sc}\t\n\t\t\\scalebox{0.9}{\n\t\t\t\\begin{tabular}{l c c}\n\t\t\t\t\\toprule\n\t\t\t\tDataset      & Reddit  & PPI   \\\\\n\t\t\t\t\\midrule\t\n\t\t\t\tRandom       & 0.042   & 0.396                     \\\\\n\t\t\t\tRaw features & 0.585   & 0.422                       \\\\\n\t\t\t\tDeep walk    & 0.324   & --                       \\\\\n\t\t\t\tDeep walk + features  & 0.691  & --               \\\\\n\t\t\t\tNode2Vec + regression  & 0.934 & -- \\\\\n\t\t\t\tGraphSAGE-GCN  & 0.930   & 0.500                       \\\\\t\t\t\n\t\t\t\tGraphSAGE-mean  & 0.950   & 0.598                       \\\\\n\t\t\t\tGraphSAGE-LSTM  & \\textbf{0.954}   & 0.612                       \\\\\n\t\t\t\tGIC    & 0.952   & \\textbf{0.661}                     \\\\\n\t\t\t\t\\bottomrule\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{sc}\n\\end{table}\n\n\n\\subsubsection{Experiment Settings} \n\nWe verify our GIC on the above bioinformatics and social network datasets. In default, GIC mainly consists of three graph convolution layers, each of which is followed by a graph coarsening layer, and one fully connected layer with a final softmax layer as shown in Fig~\\ref{fig:network:A}. Its configuration can simply be set as C(64)-P(0.25)-C(128)-P(0.25)-C(256)-P-FC(256), where C, P and FC denote the convolution, coarsening and fully connected layers respectively. The choices of hyperparameters are mainly inspired from the classic VGG net. For example, the coarsening factor is 0.25 (w.r.t. 0.5$\\times$0.5 in VGG), the attribute dimensions at three conv. layers are 64-128-256 (w.r.t. the channel numbers of conv1-3 in VGG). The scale of respective field and the number of Gaussian components are both set to 7. We train GIC network with stochastic gradient descent for roughly 300 epochs with a batch size of 100, where the learning rate is 0.1 and the momentum is 0.95. \n\nIn the bioinformatics datasets, we exploit labels and degrees of the vertices to generate initial attributes of each vertex. In the social network datasets, we use degrees of vertices. We closely follow the experimental setup in PSCN~\\cite{niepert2016learning}. We perform 10-fold cross-validation, 9-fold for training and 1-fold for testing. The experiments are repeated 10 times and the average accuracies are reported.\n\n\n\\subsubsection{Comparisons with the State-of-the-arts}\n\nWe compare our GIC with several state-of-the-arts, which contain graph convolution networks (PSCN~\\cite{niepert2016learning}, DCNN~\\cite{atwood2016diffusion}, NgramCNN~\\cite{luo2017deep}), neural networks (SAEN~\\cite{orsini2017shift}), feature based algorithms (DyF~\\cite{gomez2017dynamics}, FB~\\cite{bruna2013spectral}), random walks based methods (RW~\\cite{gartner2003graph}), graph kernel approaches (GK~\\cite{shervashidze2009efficient}, DGK~\\cite{yanardag2015deep}, WL~\\cite{morris2017glocalized}). We present the comparisons with the state-of-the-arts, as shown in Table~\\ref{table:state-of-the-art}. All results come from the related literatures. We have the following observations.\n\nDeep learning based methods on graphs (including DCNN, PSCN, NgramCNN, SAEN and ours) are superior to those conventional methods in most cases. The conventional kernel methods usually require the calculation on graph kernels with high-computational complexity. In contrast, these graph neural networks attempt to learn more abstract high-level features by performing inference-forward, which need relatively low computation cost.\n\nCompared with recent graph convolution methods, ours can achieve better performance on most datasets, such as PTC, NCI1, NCI109, ENZYMES and PROTEINS. The main reason should be that local variations of subgraphs are accurately described with Gaussian component analysis.\n\nThe proposed GIC achieves state-of-the-art results on most datasets. The best performance is gained in some bioinformatics datasets and some social network datasets including PTC, NCI1, ENZYMES, PROTEINS, COLLAB, IMDB-BINARY and IMDB-MULTI. Although NgramCNN, DyF, WL and SEAN approaches have obtained the best performance on MUTAG, REDDIT-BINARY, NCI109, REDDIT-MULTI-5K and REDDIT-MULTI-12K respectively, our method is fully comparable to them.\n\n\n\\subsection{Node Classification}\n\nFor node classification, one node is assigned one\/multiple labels. It is challenging if the label set is large. During training, we only use a fraction of nodes and their labels. The task is to predict the labels for the remaining nodes. Following the setting in~\\cite{hamilton2017inductive}, we conduct the experiments on Reddit data and PPI data. For a fair comparison to graphSAGE~\\cite{hamilton2017inductive}, we use the same initial graph data, mini-batch iterators, supervised loss function and neighborhood sample. The other network parameters are similar to graph classification except removing the coarsening layer.\n\nTabel~\\ref{table:multi-label} summarizes the comparison results. Our GIC can obtain the best performance 0.661 on PPI data and a comparable result 0.952 on Reddit data. The raw features provide an important initial information for node multi-label classification. Based on the raw features, deep walk~\\cite{perozzi2014deepwalk} improves about 0.36 (micro-F1 scores) on Reddit data. Meanwhile, we conduct an experiment of node2vec and use regression model to classification. Our method gains better performance than node2vec~\\cite{grover2016node2vec}. Comparing different aggregation methods like GCN~\\cite{kipf2016semi}, mean and LSTM, our GIC has a significant improvement about 0.16 on PPI data and gains a competitive performance on Reddit data. The results demonstrate our approach is robust to infer unknown labels of partial graphs.\n\n\n\\subsection{Model Analysis}\n\n\\begin{table}[!t]\n\t\\centering\n\t\\caption{The verification of our convolution and coarsening.}\n\t\\label{table:convandpooling}\n\t\\begin{sc}\n\t\t\\scalebox{0.68}{\n\t\t\t\\begin{tabular}{l c c c c}\n\t\t\t\t\\toprule\n\t\t\t\t\\multirow{2}{*}{Dataset}        & ChebNet   & GCN   & GIC  &\\multirow{2}{*}{GIC} \\\\\n\t\t\t\t& w\/ VI-GMM   & w\/ VI-GMM & w\/o VI-GMM & \\\\\n\t\t\t\t\\midrule\n\t\t\t\tMUTAG          & 89.44 $\\pm$ 6.30      & 92.22 $\\pm$ 5.66   & 93.33 $\\pm$ 4.84  & \\textbf{94.44 $\\pm$ 4.30} \\\\\n\t\t\t\tPTC            & 68.23 $\\pm$ 6.28      & 71.47 $\\pm$ 4.75   & 68.23 $\\pm$ 4.11  & \\textbf{77.64 $\\pm$ 6.98} \\\\\n\t\t\t\tNCI1           & 73.96 $\\pm$ 1.87      & 76.39 $\\pm$ 1.08   & 79.17 $\\pm$ 1.63  & \\textbf{84.08 $\\pm$ 1.77}\t\\\\\t\t\n\t\t\t\tNCI109         & 72.88 $\\pm$ 1.85      & 74.92 $\\pm$ 1.70   & 77.81 $\\pm$ 1.88  & \\textbf{82.86 $\\pm$ 2.37} \\\\\n\t\t\t\tENZYMES        & 52.83 $\\pm$ 7.34      & 51.50 $\\pm$ 5.50   & 52.00 $\\pm$ 4.76  & \\textbf{62.50 $\\pm$ 5.12} \\\\\n\t\t\t\tPROTEINS       & 78.10 $\\pm$ 3.37      & \\textbf{80.09 $\\pm$ 3.20}    & 78.19 $\\pm$ 2.04 & 77.65 $\\pm$ 3.21 \\\\\n\t\t\t\t\\bottomrule\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{sc}\n\\end{table}\n\\begin{table}[!t]\n\t\\centering\n\t\\caption{Comparisons on $K$ and $C_1$.}\n\t\\label{table:GMM}\n\t\\begin{sc}\n\t\t\\scalebox{0.68}{\n\t\t\t\\begin{tabular}{l c c c c}\n\t\t\t\t\\toprule\n\t\t\t\tDataset        & $K,C_1=1$                & $K,C_1=3$                  & $K,C_1=5$                  & $K,C_1=7$       \\\\\n\t\t\t\t\\midrule\n\t\t\t\tMUTAG         & 67.77 $\\pm$ 11.05        & 83.88 $\\pm$ 5.80           & 90.55 $\\pm$ 6.11           & \\textbf{94.44 $\\pm$ 4.30}    \\\\\n\t\t\t\tPTC           & 72.05 $\\pm$ 8.02         & 77.05 $\\pm$ 4.11           & 76.47 $\\pm$ 5.58           & \\textbf{77.64 $\\pm$ 6.98}    \\\\\n\t\t\t\tNCI1         & 71.21 $\\pm$ 1.94         & 83.26 $\\pm$ 1.17        & \\textbf{84.47 $\\pm$ 1.64}  & 84.08 $\\pm$1.77\\\\\t\t\t\t\n\t\t\t\tNCI109         & 70.02 $\\pm$ 1.57         & 81.74 $\\pm$ 1.56           & \\textbf{83.39 $\\pm$ 1.65}  & 82.86 $\\pm$ 2.37            \\\\\n\t\t\t\tENZYMES        & 33.83 $\\pm$ 4.21         & \\textbf{64.00 $\\pm$ 4.42}  & 63.66 $\\pm$ 3.85           & 62.50 $\\pm$ 5.12            \\\\\n\t\t\t\tPROTEINS       & 75.49 $\\pm$ 4.00         & 77.47 $\\pm$ 3.37           & \\textbf{78.10 $\\pm$ 2.96}  & 77.65 $\\pm$ 3.21            \\\\\n\t\t\t\t\\bottomrule\t\t\t\n\t\t\t\t\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{sc}\n\\end{table}\n\\begin{table}[!t]\n\t\\centering\n\t\\caption{Comparisons on the layer number.}\n\t\\label{table:layer}\n\t\\begin{sc}\t\n\t\t\\scalebox{0.68}{\n\t\t\t\\begin{tabular}{l c c c c}\n\t\t\t\t\\toprule\n\t\t\t\tDataset       & $N=2$                  & $N=4$                        & $N=6$                      & $N=8$          \\\\\n\t\t\t\t\\midrule\n\t\t\t\tMUTAG         & 86.66 $\\pm$ 8.31       & 91.11 $\\pm$ 5.09             & 93.88 $\\pm$ 5.80           & \\textbf{94.44 $\\pm$ 4.30}   \\\\\n\t\t\t\tPTC           & 64.11 $\\pm$ 6.55       & 74.41 $\\pm$ 6.45             & 75.29 $\\pm$ 6.05           & \\textbf{77.64 $\\pm$ 6.98}   \\\\\n\t\t\t\tNCI1          & 71.82 $\\pm$ 1.85       & 81.36 $\\pm$ 1.07             & 83.01 $\\pm$ 1.54           & \\textbf{84.08 $\\pm$ 1.77}   \\\\\n\t\t\t\tNCI109        & 71.09 $\\pm$ 2.41       & 80.02 $\\pm$ 1.67             & 81.60 $\\pm$ 1.83           & \\textbf{82.86 $\\pm$ 2.37}    \\\\\n\t\t\t\tENZYMES       & 42.33 $\\pm$ 4.22       & 61.83 $\\pm$ 5.55             & \\textbf{64.83 $\\pm$ 6.43}  & 62.50 $\\pm$ 5.12          \\\\\n\t\t\t\tPROTEINS      & 77.38 $\\pm$ 2.97       & \\textbf{79.81 $\\pm$ 3.84}    & 78.37 $\\pm$ 4.00           & 77.65 $\\pm$ 3.21          \\\\\n\t\t\t\t\\bottomrule\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{sc}\n\\end{table}\n\n\\textbf{EI-GMM and VI-GMM}: To directly analyze convolution filtering with EI-GMM, we compare our method with ChebNet~\\cite{defferrard2016convolutional} and GCN~\\cite{kipf2016semi} approaches by using the same coarsening mechanism VI-GMM.  As shown in Table~\\ref{table:convandpooling}, under the same coarsening operation, our GIC is superior to ChebNet+VI-GMM and GCN+VI-GMM. It indicates EI-GMM can indeed encode the variations of subgraphs more effectively. On the other hand, we remove the coarsening layer from our GIC. For different size graphs, we pad new zero vertices into a fixed size and then concatenate attributes of all vertices for classification. As shown in this table, the performance of GIC still outperforms GIC without VI-GMM coarsening, which verifies the effectiveness of the coarsening layer VI-GMM.\n\n\\textbf{$K$ and $C_1$}: The kernel size $K$ and the number of Gaussian components $C_1$ are the most crucial parameters. Generally, the $C_1$ is proportional to the $K$. The reason is that the larger receptive field usually contains more vertices ({i.e.}} \\def\\etal{{et.al}, a relative large subgraph). Thus we simply take the equal values for them, $K=C_1 = \\{1,3,5,7\\}$. The experimental results are shown in Table~\\ref{table:GMM}. With the increase of $K, C_1$, the performance improves at most cases. The reasons are two folds: i) with increasing receptive field size, the convolution will cover the farther hopping neighbors; ii) with the increase of $C_1$, the variations of subgraphs are encoded more accurately. But for the larger values of $K$ and $C_1$ will increase the computational burden. Moreover, the overfitting phenomenon might occur with the increase of model complexity. Take the example of NCI109, in the first convolution layer, the encoded attributes (in Eqn.~(\\ref{eqn:gau_fea})) will be $2\\times39\\times7=546$ for each scale of receptive field, where $39$ is the dimension of attributes (w.r.t the number of node labels) and $7$ is the number of Gaussian components. Thus, for 7 scales of receptive field, the final encoded attributes will be $546\\times7=3822$ dimensions, which will be mapping to 64 dimensions by the function $f=[f_1,\\cdots,f_{C_1}]$ in Eqn.~(\\ref{eqn:gau_fea}). Thus the model parameter is $3822\\times64=244608$ in the first layer. Similarly, if the number of node label is 2, the model parameter will sharply decrease into $18816$. Besides, the parameter complexity is related to the number of classes and nodes. The comparison results in Table~\\ref{table:GMM} demonstrate the trend of the parameters $K$ and $C_1$ in our GIC framework.\n\n\\textbf{Number of stacked layers}: Here we test on the number of stacked network layers with $N=2,4,6,8$. When $N=2$, only one fully connected layer and one softmax layer are preserved. When $N=4$, we add two layers: the convolution layer and the coarsening layer. When continuing to stack both, the depth of network will be 6 and 8. The results are shown in Table~\\ref{table:layer}. Deeper networks can gain better performance in most cases, because the larger receptive field is observed and more abstract structures will be extracted in the topper layer. Of course, there is an extra risk of overfitting due to the increase of model complexity.\n\n\\textbf{An analysis of computation complexity}: In the convolution layer, the computational costs of receptive fields and Gaussian encoding are about $O(Km^2)$ and $O(C_1d^2)$ respectively, where $m, d$ are number of nodes and the feature dimensionality. Generally, $K=C_1\\ll d<m$. In the coarsening layer, the time complexity is about $O(pm^2+md)$, where $p$ is iteration number of the EM algorithm. In all, suppose the whole GIC alternatively stacks $n$ convolution and coarsening layers, the entire time complexity is $O(n(K+p)m^2+nC_1d^2+nmd)$.\n\n\n\\section{Conclusion}\nIn this paper, we proposed a novel Gaussian-induced convolution network to handle with general irregular graph data. Considering the previous spectral and spatial methods do not well characterize local variations of graph, we derived edge-induced GMM to adaptively encode subgraph structures by projecting them into several Gaussian components and then performing different filtering operations on each Gaussian direction like the standard CNN filters on images. Meanwhile, we formulated graph coarsening into vertex-induced GMM to dynamically partition a graph, which was also proven to be equal to graph cut. Extensive experiments in two graphic tasks (i.e. graph and node classification) demonstrated the effectiveness and superiority of our GIC compared with those baselines and state-of-the-art methods. In the future, we would like to extend our method into more applications to irregular data.\n\n\n\\section{Acknowledgments}\nThe authors would like to thank the Chairs and the anonymous reviewers for their critical and constructive comments and suggestions. This work was supported by the National Science Fund of China under Grant Nos. 61602244, 61772276, U1713208 and 61472187 and Program for Changjiang Scholars.\n \n\n\\small\n\\bibliographystyle{aaai}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Supplementary Material}\n\\hrulefill\n\n\\vspace{4 cm}\n\n\\twocolumngrid\n\n\\section{Minimization of the perpendicular field component}\nAs evidenced in Fig.~\\ref{fig:fr_Qi_perp} (a) and (b), $B_{\\perp}$ shifts strongly the \nresonance frequency. We use this susceptibility to minimize the spurious perpendicular field \ncomponent during in-plane sweeps. For each value of $B_{\\parallel}$, we trace the phase \nresponse of the resonator at a fixed frequency (close to $f_{\\mathrm{r}}$) in a narrow field range of $B_{\\perp}$ (cf. Fig.~\\ref{fig:field_alignment}). The maximum of the phase response, corresponding to the optimal compensating $B_{\\perp}$, is determined from quadratic fit. We observe a compensation field linearly-dependent on the in-plane magnetic field, hence, a minor chip tilt with respect to Helmholtz coils is the most probable origin for the unwanted perpendicular field component. A typical compensation field of $\\approx~\\SI{1.94}{\\milli\\tesla}$ required for $B_{\\parallel}~=~\\SI{200}{\\milli\\tesla}$ (cf. Fig.~\\ref{fig:field_alignment}) corresponds to a misalignment angle below \\SI{1}{\\degree}.\n\n\\begin{figure}[htb]\n  \\center{\\includegraphics[width=\\columnwidth]\n        {comp_field}}\n  \\caption{Typical measurement used to calibrate the compensation field $B_{\\perp}$. The phase response of the resonator at a fixed frequency close to $f_{\\mathrm{r}}$ is shown in red and the quadratic fit in blue. We choose the value of the compensation field which maximizes the phase response. For the example shown here, res. A and $B_{\\parallel}~=~\\SI{0.2}{\\tesla}$, the compensation field is $B_{\\perp}~\\approx~\\SI{1.94}{\\milli\\tesla}$.}\n  \\label{fig:field_alignment}\n\\end{figure} \n\n\\section{In-plane field sweeps along orthogonal axes}\nA comparison between the resonance frequency shift for two different in-plane magnetic field directions is shown for res. B in Fig.~\\ref{fig:B_para_comp}. Magnetic field sweeps along the resonator's axis (red) and perpendicular to the resonator's axis (gray) overlap closely, which indicates that in our case the orientation of the in-plane field does not significantly influence the superconducting properties of the film. Since the effective areas of the resonator parallel and perpendicular to the its axis are 60 $\\times$ different, the suppression of the superconductor's gap appears to be the main mechanism responsible for the measured change of kinetic inductance under in-plane magnetic field (cf. Eq.~\\ref{eq:delta_fr} in the main text).  \n\n\\begin{figure}[htb]\n  \\center{\\includegraphics[width=\\columnwidth]\n        {B_para_comp}}\n  \\caption{Resonance frequency shift versus in-plane magnetic field for res. B (measured in the cylindrical waveguide), applied along two orthogonal directions: parallel (red) and perpendicular (gray) to the resonator's axis. The fact that the measurements overlap, for different effective side areas of the resonator exposed to in-plane field, indicates that screening currents do not play a significant role. Active compensation of the spurious perpendicular field component is performed for both sweeps, according to the procedure shown in Fig.~\\ref{fig:field_alignment}.}\n  \\label{fig:B_para_comp}\n\\end{figure} \n\n\\section{Self-Kerr effect vs. in-plane field}\n\n\\begin{figure}[hb!]\n  \\center{\\includegraphics[width=\\columnwidth]\n        {self_Kerr}}\n  \\caption{Self-Kerr coefficient of res. A under in-plane field. (a) Resonance frequency shift vs. the average photon number, $\\overline{n}$, for $B_{||}~=~\\SI{0}{\\tesla}, \\SI{0.3}{\\tesla}, \\SI{0.5}{\\tesla}, \\SI{0.65}{\\tesla}$. The self-Kerr coefficient is extracted from the data above $\\overline{n}~\\approx~1~\\cdot~10^{5}$ (indicated by vertical black dashed line) using a linear fit, $\\Delta f_{\\mathrm{r}}~=~-K_{11}\\overline{n}$. This resonator bifurcates at $\\approx~4~\\cdot~10^{5}$ photons. (b) Fitted self-Kerr coefficient vs. $B_{||}$. }\n  \\label{fig:self_Kerr}\n\\end{figure} \n\nThe resonance frequency shift as a function of the number of circulating photons in the resonator, known as the self-Kerr effect, is measured under in-plane magnetic field for res. A (cf. Fig.~\\ref{fig:self_Kerr}~(a)).\nNote that for $B_{||}~>~\\SI{0.5}{\\tesla}$ the resonance frequency shift is no longer linear in the low photon number region. Furthermore at \\SI{0.65}{\\tesla}, $\\Delta~f_{\\mathrm{r}}$ first increases and then decreases. The exact cause of this behaviour is unclear. It can be attributed to induced superconducting fluxons, due to imperfect compensation of the perpendicular field in the end regions of the resonator, at the current nodes. These fluxons can act as quasiparticle traps\\cite{Nsanzineza2014}, which become more efficient at higher circulating power in the resonator when the mobility of the quasiparticles is higher. Trapping quasiparticles into these areas could slightly reduce the effective quasiparticle density, $n_{\\mathrm{qp}}$, and as $f_{\\mathrm{r}}~\\propto~1\/\\sqrt{L_{\\mathrm{k}}}~\\propto~1\/\\sqrt{n_{\\mathrm{qp}}}$, result in an increased $f_{\\mathrm{r}}$.\n\nThe extracted self-Kerr coefficient under in-plane magnetic field up to \\SI{650}{\\milli\\tesla} is shown in Fig.~\\ref{fig:self_Kerr}~(b). As discussed in Ref.~\\cite{Maleeva2018}, the self-Kerr coefficient of grAl is $K_{11}~\\propto~f_{\\mathrm{r}}^2\/j_{\\mathrm{c}}$, where $f_{\\mathrm{r}}$ and $j_{\\mathrm{c}}$ are the resonant frequency and the critical current density. Because both $f_{\\mathrm{r}}^{2}$ and $j_{\\mathrm{c}}$ are $\\propto~1\/L_{\\mathrm{k}}$, $K_{11}$ is expected to be field-independent\\cite{Maleeva2018}.\n\n\\section{Onset of the plastic regime under perpendicular magnetic field}\n\nThe transition of resonator A to the plastic regime (cf. Fig.~\\ref{fig:fr_Qi_perp}) is visible during three consecutive sweeps with $B_{\\perp}^{\\mathrm{max}} = \\SI{0.7}{\\milli\\tesla}, \\SI{0.8}{\\milli\\tesla}~\\mathrm{and}~\\SI{0.9}{\\milli\\tesla}$ (cf. Fig.~\\ref{fig:onset_plastic}). When $B_{\\perp}$ is ramped up to \n\\SI{0.8}{\\milli\\tesla}, the onset of the plastic regime is evidenced by a sharp drop in \nthe $Q_{\\mathrm{i}}$ for $B_{\\perp}~>~\\SI{0.7}{\\milli\\tesla}$, which is explained by fluxons permeating the film (cf. Fig.~\\ref{fig:onset_plastic} (c)).\n\n\\begin{figure}[hb!]\n  \\center{\\includegraphics[width=\\columnwidth]\n        {onset_plastic}}\n  \\caption{Measurements of $Q_{\\mathrm{i}}$ versus perpendicular magnetic field for res. A, evidencing the onset of the plastic regime during the sweeps to $B_{\\perp}^{\\mathrm{max}} = \\SI{0.7}{\\milli\\tesla}$ (a - b), $B_{\\perp}^{\\mathrm{max}} = \\SI{0.8}{\\milli\\tesla}$ (c - d) and $B_{\\perp}^{\\mathrm{max}} = \\SI{0.9}{\\milli\\tesla}$ (e - f). Horizontal blue and red arrows indicate the directions of the magnetic field sweep. \n}\n  \\label{fig:onset_plastic}\n\\end{figure}\n\n\\end{document}\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}