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+{"text":"\\section*{Introduction}\n\nLet us recall that we have introduced in paper~\\cite{I} some\n``one-particle'' eigenstates for the model based upon solutions\nof the tetrahedron equation. In the same paper, we have also\nconstructed some ``two-particle'' states. However, some\nspecial condition arised in this construction, and the\nsuperposition of two {\\em arbitrary\\\/} one-particle states\nwas not achieved. Even the ``creation operators'' of paper~\\cite{II}\ndid not give a clear answer concerning multi-particle states.\n\nOn the other hand, in paper~\\cite{IV} we have brought in correspondence\nto a one-particle state some new state that could be\ncalled ``one open string''. It was done using some special ``kagome\ntransfer matrix''. Here we will show that the superposition\nof such one-string states is easier to construct, because of\ndegeneracy of kagome transfer matrix: it turns into zero the\n``obstacles'' that hampered constructing of multi-particle states.\n\nThe scheme of string---particle ``marriage'' in~\\cite{IV} was\nas follows: take a one-particle state from \\cite{I,II}, and apply\nto it a kagome transfer matrix with boundary conditions corresponding\nto the presence of two string tails in the infinity, e.g.\\ like this:\n\\allowbreak\n\\unitlength=1mm\n\\linethickness{0.4pt}\n\\begin{picture}(25.00,4.50)\n\\put(25.00,2.00){\\oval(50.00,4.00)[l]}\n\\end{picture}\\,.\n\nIn this paper, we are going to complicate this scheme in the following\nway: the boundary conditions will correspond to the presence\nof an even number\nof string tails at the infinity, and instead of a one-particle\nstate, we will use some special multi-particle vector~$\\Psi$.\nIts peculiarity will be in the fact that $\\Psi$ is {\\em no longer\nan eigenstate\\\/} of the hedgehog transfer matrix~$T$ defined in~\\cite{I}.\nInstead, it will obey the condition\n\\begin{equation}\nT\\Psi = \\lambda\\Psi + \\Psi',\n\\label{V-int-1}\n\\end{equation}\nwhere $\\lambda={\\rm const}$, and $\\Psi'$ is annulated by the\nkagome transfer matrix of paper~\\cite{IV} which we will\ndenote~$K$.\n\nRecall that we have defined $T$ in such a way that its degrees\ncould be described geometrically as ``oblique slices''\nof the cubic lattice.\nThe transfer matrix~$T$ can be passed through the transfer matrix~$K$:\n\\begin{equation}\nTK=KT,\n\\label{V-int-2}\n\\end{equation}\nthe boundary conditions (such as the number and form of tails\nat the infinity) for $K$ being intact. Define vector~$\\Phi$ as\n$$\n\\Phi=K\\Psi.\n$$\nThis together with (\\ref{V-int-1}) and (\\ref{V-int-2}) gives\n\\begin{equation}\nT\\Phi=\\lambda\\Phi,\n\\label{V-int-4}\n\\end{equation}\nexactly as needed for an eigenvector.\n\nWe also present in this paper\nthe ``dispersion relations'' for our particles--strings\nin a workable form---something that was missing in\npapers~\\cite{I,II}.\n\n\n\\section{Eigenvectors of the ``several open strings'' type for the infinite\nlattice}\n\\label{secV-1}\n\nLet there be $n$ one-particle amplitudes $\\varphi_{\\ldots}^{(1)}, \\ldots,\n\\varphi_{\\ldots}^{(n)}$ of the same type as those described\nin the work~\\cite{I}. Let us compose an ``$n$-particle vector''\n$\\Psi$, i.e.\\ put in correspondence to each unordered\n$n$-tuple of vertices $A^{(1)},\\ldots,A^{(n)}$ of the kagome lattice\nthe symmetrized amplitude in the following way:\n\\begin{equation}\n\\psi_{A^{(1)},\\ldots,A^{(n)}} = \\sum_s \\varphi_{A^{s(1)}}^{(1)}\n\\ldots \\varphi_{A^{s(n)}}^{(n)},\n\\label{V-1-1}\n\\end{equation}\nwhere $s$ runs through the group of all permutations\nof the set $\\{1,\\ldots,n\\}$.\n\nAs for the boundary conditions for the transfer matrix~$K$ described\nin the Introduction, let us assume that there are exactly $2n$ string tails,\nand they all go in positive directions,\nthat is between the east and the north.\nThus, in each of the points $A^{(1)},\\ldots,A^{(n)}$ a string is\ncreated, and they are not annihilated.\n\nThe vector (\\ref{V-1-1}) is not an eigenvector of transfer matrix~$T$\ndue to problems arising when two or more points $A^{(k)}$ get close\nto one another. Nevertheless, the vector $\\Phi=K\\Psi$ {\\em is\\\/}\nan eigenvector, because for it those problems disappear due to the\nsimple fact:\n{\\em creation of two or more strings within one triangle of the kagome\nlattice is geometrically forbidden}.\n\n\n\\section{Eigenvectors of the ``closed string'' type for the infinite\nlattice}\n\\label{secV-2}\n\nIn this section, we will put in correspondence\nto each unordered pair of vertices\nof the infinite kagome lattice an ``amplitude'' $\\Psi_{AB}$\naccording to the following rules. If one of the vertices,\nsay $A$, {\\em precedes\\\/} the other one, say $B$, in the sense\nthat they can be linked by a path---a broken line---going along\nlattice edges in positive directions, namely northward, eastward,\nor to the north-east, then let us put\n\\begin{equation}\n\\Psi_{AB}= \\varphi_A \\psi_B - \\psi_A \\varphi_B,\n\\label{V-2-1}\n\\end{equation}\nwhere $\\varphi_{\\ldots}$ and $\\psi_{\\ldots}$ are two one-particle\namplitudes of the same type as in paper~\\cite{I}.\nIn the case if vertices $A$ and $B$ cannot be joined by a path\nof such kind, let us put\n$$\n\\Psi_{AB}=0.\n$$\n\nThe values $\\Psi_{AB}$ are components of the vector~$\\Psi$\nthat belong to the two-particle subspace of tensor product of\ntwo-dimensional spaces situated in all kagome lattice vertices.\nWhat prevents $\\Psi_{AB}$ from being an eigenvector of the hedgehog\ntransfer matrix is discrepancies arising near those pairs $A,B$\nthat lie at the ``border'' between such pairs where one of the\nvertices precedes the other (so to speak, ``the interval $AB$\nis timelike''),\nand such pairs where it does not (``the interval $AB$\nis spacelike'').\n\nThose discrepancies, however, disappear for the vector $\\Phi=K\\Psi$,\nwhere $K$ is the kagome transfer matrix described in the Introduction\nwith the boundary conditions reading {\\em no string tails at the\ninfinity}. This is because if a string cannot, geometrically,\nbe created at the point $A$ (or $B$) and then annihilated\nat the point $B$ (or $A$), then the amplitude $\\Psi_{AB}$\ndoesn't influence at all the vector~$\\Phi$.\nThe only thing that remains to be checked for (\\ref{V-int-4})\nto hold is a situation where $A$ and $B$ are in the same kagome lattice\ntriangle that will be turned inside out by one of the hedgehogs\nof transfer matrix~$T$, as in Figure~\\ref{figV-1}.\n\\begin{figure}[ht]\n\\begin{center}\n\\unitlength=1mm\n\\linethickness{0.4pt}\n\\begin{picture}(65.00,33.00)\n\\put(3.00,12.00){\\line(1,0){20.00}}\n\\put(23.00,12.00){\\line(0,1){20.00}}\n\\put(23.00,32.00){\\line(-1,-1){20.00}}\n\\put(30.00,17.00){\\vector(1,0){6.00}}\n\\put(43.00,2.00){\\line(0,1){20.00}}\n\\put(43.00,22.00){\\line(1,0){20.00}}\n\\put(63.00,22.00){\\line(-1,-1){20.00}}\n\\put(3.00,12.00){\\circle*{1.00}}\n\\put(23.00,12.00){\\circle*{1.00}}\n\\put(23.00,32.00){\\circle*{1.00}}\n\\put(43.00,22.00){\\circle*{1.00}}\n\\put(63.00,22.00){\\circle*{1.00}}\n\\put(43.00,2.00){\\circle*{1.00}}\n\\put(13.00,22.00){\\vector(1,1){1.00}}\n\\put(23.00,22.00){\\vector(0,1){1.00}}\n\\put(13.00,12.00){\\vector(1,0){1.00}}\n\\put(43.00,12.00){\\vector(0,1){1.00}}\n\\put(53.00,12.00){\\vector(1,1){1.00}}\n\\put(53.00,22.00){\\vector(1,0){1.00}}\n\\put(1.00,12.00){\\makebox(0,0)[rc]{$A$}}\n\\put(25.00,32.00){\\makebox(0,0)[lc]{$B$}}\n\\put(41.00,2.00){\\makebox(0,0)[rc]{$B'$}}\n\\put(65.00,22.00){\\makebox(0,0)[lc]{$A'$}}\n\\end{picture}\n\\end{center}\n\\caption{}\n\\label{figV-1}\n\\end{figure}\nActing in the same manner as in Section~\\ref{sec-twop}\nof work~\\cite{I}, write\n\\begin{equation}\n\\pmatrix{ \\varphi_{A'} \\cr \\varphi_{B'} }=\n\\pmatrix{\\alpha & \\beta \\cr \\gamma & \\delta}\n\\pmatrix{ \\varphi_A \\cr \\varphi_B }, \\qquad\n\\pmatrix{ \\psi_{A'} \\cr \\psi_{B'} }=\n\\pmatrix{\\alpha & \\beta \\cr \\gamma & \\delta}\n\\pmatrix{ \\psi_A \\cr \\psi_B },\n\\label{V-2-3}\n\\end{equation}\nwhere\n\\begin{equation}\n\\alpha=-\\delta, \\qquad \\alpha\\delta-\\beta\\gamma=-1.\n\\label{V-2-4}\n\\end{equation}\nIt follows from the formulas (\\ref{V-2-3}) and (\\ref{V-2-4}) that\n$$\n\\varphi_A \\psi_B-\\varphi_B \\psi_A=\n\\varphi_{B'} \\psi_{A'}-\\varphi_{A'} \\psi_{B'},\n$$\ni.e.\\\n$$\n\\Psi_{AB}=\\Psi_{B'A'},\n$$\nexactly what was needed to comply with the fact that an\n$S$-operator-hedgehog acts as a unity operator in the\ntwo-particle subspace.\n\n\n\\section{Dispersion relations}\n\\label{V-sec-disp}\n\nThe constructed eigenvectors of transfer matrix~$T$ are of course\neigenvectors for translation operators through periods of kagome\nlattice as well. Let us consider here relations between\nthe corresponding eigenvalues, starting from the simplest\none-particle eigenstate.\n\nConsider once again some triangle $ABC$ of the kagome lattice,\nand its image $A'B'C'$ under the action of $S$-matrix-hedgehog,\nas in Figure~\\ref{figV-3-1}.\n\\begin{figure}[ht]\n\\begin{center}\n\\unitlength=1.00mm\n\\special{em:linewidth 0.4pt}\n\\linethickness{0.4pt}\n\\begin{picture}(65.00,32.50)\n\\put(3.00,12.00){\\line(1,0){20.00}}\n\\put(23.00,12.00){\\line(0,1){20.00}}\n\\put(23.00,32.00){\\line(-1,-1){20.00}}\n\\put(30.00,17.00){\\vector(1,0){6.00}}\n\\put(43.00,2.00){\\line(0,1){20.00}}\n\\put(43.00,22.00){\\line(1,0){20.00}}\n\\put(63.00,22.00){\\line(-1,-1){20.00}}\n\\put(3.00,12.00){\\circle*{1.00}}\n\\put(23.00,12.00){\\circle*{1.00}}\n\\put(23.00,32.00){\\circle*{1.00}}\n\\put(43.00,22.00){\\circle*{1.00}}\n\\put(63.00,22.00){\\circle*{1.00}}\n\\put(43.00,2.00){\\circle*{1.00}}\n\\put(1.00,12.00){\\makebox(0,0)[rc]{$A$}}\n\\put(25.00,32.00){\\makebox(0,0)[lc]{$C$}}\n\\put(41.00,2.00){\\makebox(0,0)[rc]{$C'$}}\n\\put(65.00,22.00){\\makebox(0,0)[lc]{$A'$}}\n\\put(41.00,22.00){\\makebox(0,0)[rc]{$B'$}}\n\\put(25.00,12.00){\\makebox(0,0)[lc]{$B$}}\n\\end{picture}\n\\end{center}\n\\caption{}\n\\label{figV-3-1}\n\\end{figure}\nLet us write out some relations of the type (\\ref{V-2-3}), namely\n\\begin{equation}\n\\pmatrix{\\varphi_{A'} \\cr \\varphi_{B'}}=\n\\pmatrix{a&b\\cr c&d} \\pmatrix{\\varphi_A \\cr \\varphi_B},\n\\label{V-3-1}\n\\end{equation}\n\\begin{equation}\n\\pmatrix{\\varphi_{B'} \\cr \\varphi_{C'}}=\n\\pmatrix{\\tilde a&\\tilde b\\cr \\tilde c&\\tilde d}\n\\pmatrix{\\varphi_B \\cr \\varphi_C},\n\\label{V-3-2}\n\\end{equation}\nwhere $\\varphi_{\\ldots}$ is any one-particle vector, and the numbers\n$a, \\ldots ,\\tilde d$ satisfy conditions of type~(\\ref{V-2-4}), i.e.\\\n$$\n\\matrix{ a=-d,& \\qquad ad-bc=-1, \\cr\n\\tilde a=-\\tilde d,& \\qquad \\tilde a\\tilde d-\\tilde b\\tilde c=-1.}\n$$\n From (\\ref{V-3-1}) follows\n\\begin{equation}\n{\\varphi_B\\over \\varphi_{B'}}=\n{-a(\\varphi_A\/\\varphi_{A'})+1\\over (\\varphi_A\/\\varphi_{A'})-a},\n\\label{V-3-3}\n\\end{equation}\nand from (\\ref{V-3-2}) follows\n$$\n{\\varphi_C\\over \\varphi_{C'}}=\n{-\\tilde a(\\varphi_B\/\\varphi_{B'})+1\n\\over (\\varphi_B\/\\varphi_{B'})-\\tilde a}.\n$$\nSurely, the numbers $a$ and $\\tilde a$ depend on an\n$S$-operator-hedgehog. On the other hand, this latter is parameterized\nby exactly two parameters. So, it seems that it makes sense to take\n$a$ and $\\tilde a$ as those parameters.\n\nWe can take for eigenvalue of the hedgehog transfer matrix~$T$\neither $\\varphi_{A'}\/\\varphi_A$, or $\\varphi_{B'}\/\\varphi_B$,\nor $\\varphi_{C'}\/\\varphi_C$.\nThese variants correspond, strictly speaking, to different definitions\nof~$T$, but each of them is consistent with the requirement that\nthe degrees of~$T$ must be represented graphically as ``oblique\nlayers'' of cubic lattice (the difference being that, with the three\ndifferent definitions, the action of transfer matrix~$T$ corresponds\nto the shifts through cubic lattice periods along three different axes).\nOur goal is to express the eigenvalues of translation operators\nacting within the kagome lattice for a given one-particle state\nthrough, say, $\\varphi_{A'}\/\\varphi_A$.\n\nIf we speak about translation through one lattice\nperiod {\\em to the right\\\/}\nin the sense of Figures \\ref{figV-3-1} and~\\ref{figV-3-2},\n\\begin{figure}[ht]\n\\begin{center}\n\\unitlength=1mm\n\\linethickness{0.4pt}\n\\begin{picture}(50.00,50.00)\n\\put(25.00,0.00){\\line(0,1){50.00}}\n\\put(0.00,25.00){\\line(1,0){50.00}}\n\\put(0.00,20.00){\\line(1,1){30.00}}\n\\put(20.00,0.00){\\line(1,1){30.00}}\n\\put(4.00,27.00){\\makebox(0,0)[rb]{$A$}}\n\\put(23.00,46.00){\\makebox(0,0)[rb]{$C$}}\n\\put(46.00,23.00){\\makebox(0,0)[lt]{$D$}}\n\\put(26.00,3.00){\\makebox(0,0)[lt]{$E$}}\n\\put(27.00,27.00){\\makebox(0,0)[lb]{$B$}}\n\\put(5.00,25.00){\\circle*{1.00}}\n\\put(25.00,5.00){\\circle*{1.00}}\n\\put(25.00,25.00){\\circle*{1.00}}\n\\put(25.00,45.00){\\circle*{1.00}}\n\\put(45.00,25.00){\\circle*{1.00}}\n\\end{picture}\n\\end{center}\n\\caption{}\n\\label{figV-3-2}\n\\end{figure}\nthen this eigenvalue is $\\varphi_D\/\\varphi_A$. It is clear that\n$$\n{\\varphi_D\\over \\varphi_B}={\\varphi_{A'}\\over \\varphi_{B'}}\n$$\n---the ratios of values $\\varphi_{\\ldots}$ in the triangle $DBE$ are\nthe same as in $A'B'C'$. Thus,\n\\begin{equation}\n{\\varphi_D\\over \\varphi_A}={\\varphi_{A'}\\over \\varphi_{B'}}\n{\\varphi_B\\over \\varphi_A}={\\varphi_{A'}\\over \\varphi_A} \\cdot\n{-a(\\varphi_A\/\\varphi_{A'})+1 \\over (\\varphi_A\/\\varphi_{A'})-a}\n\\label{V-3-6}\n\\end{equation}\n(we have used (\\ref{V-3-3}). A similar relation can be written out\nfor the translation through one lattice period in {\\em upward\\\/}\ndirection in the sense of Figures \\ref{figV-3-1} and~\\ref{figV-3-2},\nnamely\n\\begin{equation}\n{\\varphi_C\\over \\varphi_E}=\n{\\varphi_{B'}\\over \\varphi_B} \\cdot\n{-\\tilde a(\\varphi_B\/\\varphi_{B'})+1 \\over\n(\\varphi_B\/\\varphi_{B'})-\\tilde a},\n\\label{V-3-7}\n\\end{equation}\nwhere one has to substitute the expression (\\ref{V-3-3})\nfor $\\varphi_B\/\\varphi_{B'}$.\n\nIt is clear that the ``dispersion relations'' of type\n(\\ref{V-3-6}--\\ref{V-3-7}) survive also for a string ``created\nby a particle'', if we substitute the eigenvalue of transfer matrix~$T$\ninstead of $\\varphi_{A'}\/ \\varphi_A$, and the eigenvalues\nof translation operators instead of $\\varphi_D\/ \\varphi_A$\nand $\\varphi_C\/ \\varphi_E$. As for the multi-string states,\nall of the eigenvalues are obtained for them as products\nof corresponding eigenvalues for each string.\n\n\n\n\\section{Discussion}\n\\label{sec-V-discussion}\n\nWe have shown in this paper that the string---particle ``marriage''\n from paper~\\cite{IV} makes possible a simple and clear construction\nof at least some multi-string states. Recall that, from all\nthe corresponding multi-particle states, we only could\nexplicitely construct some two-particle states~\\cite{I},\nwith an additional restriction that could be formulated\nas ``the total momentum of two particles is zero''.\nAs for the present paper, the momenta of ``particles'' generating\nthe multi-string states of Sections \\ref{secV-1} and~\\ref{secV-2}\ncan change independently.\n\nThese states have been constructed for the infinite kagome lattice.\nWe have to recognize that constructing such states on a finite\nlattice remains an open problem.\n\nIt is also unclear how to combine the results\nof Sections \\ref{secV-1} and~\\ref{secV-2}, i.e.\\ construct\nsuch states with string ``creation'' and ``annihilation'' where\nthe total number of ``creating'' and ``annihilating'' particles\nwould be more than two. Note that in Section~\\ref{secV-1} we use\nthe symmetrized product of one-particle amplitudes, while\nin Section~\\ref{secV-2}---the antisymmetrized one.\n\nConcerning the dispersion relations of Section~\\ref{V-sec-disp},\nlet us remark that perhaps there are too many of them.\nIt is probably caused by the fact that, for now, we managed\nto construct not all one-particle and\/\\allowbreak or one-string states.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n \n\nWe consider a system of Schr\\\"odinger equations \n\\begin{equation}\\label{eq:schro}\n \\left\\{\\begin{array}{l}\ni\\varepsilon\\partial_t\\psi^\\varepsilon_t = -\\frac{\\varepsilon^2}{2} \\Delta_q\\psi^\\varepsilon_t +V(q)\\psi^\\varepsilon_t,\\;\\;(t,q)\\in{\\mathbb R}\\times{\\mathbb R}^d\\\\\n\\psi^\\varepsilon_{t=0}=\\psi^\\varepsilon_0,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $V$ is a smooth function on ${\\mathbb R}^d$, whose values are $2\\times2$ real symmetric matrices, \n$$\nV(q) = \\alpha(q){\\rm Id} + \\begin{pmatrix}\\beta(q)&\\gamma(q)\\\\ \\gamma(q) & -\\beta(q)\\end{pmatrix},\\qquad q\\in{\\mathbb R}^d.\n$$\nThe smooth functions $\\alpha,\\beta,\\gamma\\in{\\mathcal C}^\\infty({\\mathbb R}^d,{\\mathbb R})$ are of subquadratric growth such that the Schr\\\"odinger operator in equation (\\ref{eq:schro}) is essentially self-adjoint, and there exists a unique solution for all times $t\\in{\\mathbb R}$.  We aim at the semi-classical limit $\\varepsilon\\to0$ in situations, where the two eigenvalues $\\lambda^+(q)\\ge \\lambda^-(q)$ of the potential matrix~$V(q)$ get close to each other and  non-adiabatic transitions occur to leading order in $\\varepsilon$. Our method of proof allows for rather general initial data $(\\psi^\\varepsilon_0)_{\\varepsilon>0}$ that are uniformly bounded in $L^2({\\mathbb R}^d,{\\mathbb C}^2)$.\n\n\\medskip\nSchr\\\"odinger systems with matrix-valued potentials can be rigorously derived in the context of the Born--Oppenheimer approximation of molecular quantum dynamics, and we refer to~\\cite{ST} and~\\cite{MS} where this theory is carefully carried out.  Born--Oppenheimer theory also applies for the present two-level system, provided that the eigenvalues $\\lambda^+$ and $\\lambda^-$ are uniformly separated, \nthat is, if there exists a small gap parameter $\\delta_0>0$, independent of the semi-classical parameter $\\varepsilon>0$, such that the gap function\n\\begin{equation}\\label{def:g}\ng(q)= \\lambda^+(q)-\\lambda^-(q)\n\\end{equation}\nsatisfies $g(q)\\ge \\delta_0$ for all $q\\in{\\mathbb R}^d$. In this situation, the eigenspaces are adiabatically decoupled in the following sense: If $\\Pi^\\pm(q)$ denote the eigenprojectors onto the eigenspaces of $V(q)$, then for initial data with  $\\psi^\\varepsilon_0=\\Pi^+\\psi^\\varepsilon_0$, one obtains only a small non-adiabatic contribution $\\|\\Pi^-\\psi^\\varepsilon(t)\\|_{L^2} = O(\\varepsilon)$ at time $t$, and the analogous statement holds true if $\\psi^\\varepsilon_0=\\Pi^-\\psi^\\varepsilon_0$. If the gap-condition is violated because the gap becomes small (with respect to $\\varepsilon$) or vanishes, then adiabatic decoupling no longer holds. \nFor concrete molecular systems, the semi-classical parameter $\\varepsilon$ and the gap parameter $\\delta_0$ are given numbers, and an asymptotic analysis just taking into account the smallness of the semi-classical parameter $\\varepsilon$ does not provide the necessary information.\n\n\n\\medskip\nEspecially for Schr\\\"odinger systems, non-adiabatic transitions have been of interest over decades, \nsince they occur in many applications ranging from atmospheric chemistry to photochemistry, see \nthe recent perspective article \\cite{Tu2}. Typically, non-adiabatic phenomena are attributed to avoided or conical crossings of eigenvalues. Conical crossings occur, when the eigenvalue gap vanishes and the eigenprojectors have a conical singularity at these points. Generic conical crossings have been classified by symmetry, see \\cite{Hag1}, and have been analysed by semiclassical wavepackets \\cite{Hag1} as well as by pseudodifferential operators \\cite{CdV1,FG02,LT}, \nsee also the kinetic model for graphene \\cite{FM} that encorporates a conical crossing. Here we aim at the analysis of avoided crossings. For them,  the following definition has been proposed \\cite{Hag2}:\n\n\\begin{definition}\\label{def:avoided_hag} \nSuppose $V(q,\\delta)$ is a family of real symmetric $2\\times 2$ matrices depending smoothly on $q\\in\\Omega$ and \n$\\delta\\in I$, $V\\in\\mathcal C^\\infty(\\Omega\\times I,{\\mathbb R}^{2\\times 2})$, where $\\Omega$ is an open subset of ${\\mathbb R}^d$ and $I\\subset{\\mathbb R}$ some interval containing $0$. Suppose that $V(q,\\delta)$ has two eigenvalues $\\lambda^+(q,\\delta)$ and $\\lambda^-(q,\\delta)$ that depend continuously on $q$ and $\\delta$. Assume that \n$$\n\\{q\\in\\Omega,\\;\\; \\lambda^+(q,0)=\\lambda^-(q,0)\\}\n$$ \nis a non-empty submanifold of $\\Omega$, such  that $\\lambda^+(q,\\delta)\\neq\\lambda^-(q,\\delta)$ for all $q\\in\\Omega$ and $\\delta\\neq0$. Then we say that $V(q,\\delta)$ has an avoided crossing of eigenvalues. \n\\end{definition}  \n\nAvoided crossings have a similar symmetry classification \\cite{Hag2} as the conical intersections. \nIn \\cite{HJ1,HJ2,Rou}, the perturbation parameter $\\delta$ of Definition~\\ref{def:avoided_hag} has been linked with the semiclassical parameter $\\varepsilon$, and a leading order analysis of \n$$\ni\\varepsilon\\partial_t\\psi^\\varepsilon_t = -\\tfrac{\\varepsilon^2}{2}\\Delta_q \\psi^\\varepsilon_t + V(q,\\sqrt\\varepsilon)\\psi^\\varepsilon_t,\\qquad \n\\psi^\\varepsilon_{t=0} = \\psi^\\varepsilon_0\n$$\nhas been carried out for families of initial data $(\\psi^\\varepsilon_0)_{\\varepsilon>0}$ which are semiclassical wavepackets. \nFor Schr\\\"odinger systems in one space dimension, avoided crossings have also been considered without assuming that the perturbation $\\delta$ and the semiclassical parameter $\\varepsilon$ are coupled. In this situation, the non-adiabatic contributions are exponentially small with respect to $\\varepsilon$. In \\cite{HJ3}, it is assumed that the potential $V(q)$ of the Schr\\\"odinger system~(\\ref{eq:schro}) belongs to a family $V(q,\\delta)$ with an avoided crossing such that $V(q)=V(q,\\delta_0)$ for all $q\\in{\\mathbb R}$ and some fixed $\\delta_0>0$. Then, the scattering wave function of semiclassical wavepackets is determined together with its non-adiabatic contributions. In \\cite{BGT,BG}, superadiabatic representations of one-dimensional avoided crossings have been developed together with an explicit heuristic formula for the outgoing nonadiabatic component. Our results interpolate between the existing ones in the following sense. On the one hand, \nwe allow for general families $(\\psi^\\varepsilon_0)_{\\varepsilon>0}$ of initial data in $L^2({\\mathbb R}^d,{\\mathbb C}^2)$ without restricting to coherent states or a single space dimension. Also, we will not explicitly link the semiclassical parameter $\\varepsilon$ and the gap parameter $\\delta$. On the other hand, we will only reach for the leading order behaviour with respect to $\\varepsilon$.\n\n \n\n\n\\subsection{Wigner transforms}  It is impossible to directly study the densities \n$$\nn_\\pm^\\varepsilon(q,t)= | \\Pi^\\pm(q)\\psi^\\varepsilon(q,t) |_{{\\mathbb C}^2}^2\n$$ \nor the  dynamics of the so-called level populations\n$$\nt\\mapsto \\int_{{\\mathbb R}^d} n_\\pm^\\varepsilon(t,q) dq\n$$ \nfor general initial data. Thus, we focus on providing an asymptotic description for the time evolution of the\nWigner transform of~$\\psi^\\varepsilon(q,t)$ in a suitable\n$\\varepsilon$-dependent scaling,\n$$W^\\varepsilon\\!\\left(\\psi^\\varepsilon_t\\right)\\!(q,p)=\n(2\\pi)^{-d}\\int_{{\\mathbb R}^d} \\psi^\\varepsilon\\!\\left(q-\\tfrac{\\varepsilon}{2}v,t\\right)\\otimes \\overline \\psi^\\varepsilon\\!\\left(q+\\tfrac{\\varepsilon}{2}v,t\\right)\\,{\\rm e}^{i\\,v\\cdot p}\\,{\\rm d} v$$\nwith $(q,p)\\in{\\mathbb R}^{2d}$. The Wigner transform plays\nthe role of a generalized probability density on phase space. For\nsquare integrable wave functions $\\psi\\in L^2({\\mathbb R}^d,{\\mathbb C}^2)$, the Wigner function\n$W^\\varepsilon(\\psi)$ is a square integrable function on phase space with\nvalues in the space of Hermitian $2\\times 2$ matrices. One recovers for\nexample the position density by\n$$n_\\pm^\\varepsilon(q,t)=\n{\\rm tr}\\int_{{\\mathbb R}^d}\n\\Pi^\\pm(q)W^\\varepsilon(\\psi^\\varepsilon_t)(q,p)\\,{\\rm d} p.\n$$\nBesides, the action of the Wigner function against compactly\nsupported smooth test functions\n$a\\in{\\mathcal C}^\\infty_c({\\mathbb R}^{2d},{\\mathbb C}^{2\\times 2})$ is simply expressed in\nterms of the semi-classical pseudodifferential operator of symbol\n$a$, which is defined by\n$$\n{\\rm op}_\\varepsilon(a)\\psi(q)=(2\\pi\\varepsilon)^{-d} \\int_{{\\mathbb R}^{2d}}\na\\left(\\tfrac12(q+q'),p\\right){\\rm\ne}^{\\tfrac{i}{\\varepsilon}p\\cdot(q-q')}\\psi(q')\\,{\\rm d} q'\\,{\\rm d} q\n$$\nfor $\\psi\\in L^2({\\mathbb R}^d,{\\mathbb C}^2)$. Indeed, we have\n$$\\int_{{\\mathbb R}^{2d}}\\,{\\rm tr}\\left(W^\\varepsilon(\\psi)(q,p)a(q,p)\\right)\\,{\\rm d}\nq\\,{\\rm d}\np=\\left({\\rm op}_\\varepsilon(a)\\psi\\,,\\,\\psi\\right)_{L^2({\\mathbb R}^d,{\\mathbb C}^{2})}.\n$$\nThe Wigner transform is perfectly suited for the analysis of quadratic functions of the wave function, which do not require all the phase information.\n\nOur aim is the study of the diagonal parts of the Wigner transform\n\\begin{eqnarray*}\n\\Pi^\\pm(q) W^\\varepsilon(\\psi^\\varepsilon_t)\\Pi^\\pm(q)=w^\\pm_\\varepsilon(t)\\Pi^\\pm(q),\\\\\nw_\\pm^\\varepsilon(t)={\\rm tr} \\left(\\Pi^\\pm(q) W^\\varepsilon(\\psi^\\varepsilon_t)\\Pi^\\pm(q)\\right),\n\\end{eqnarray*}\nand to describe the evolution of the coefficients $w_\\pm^\\varepsilon(t)$ in terms of $w_+^\\varepsilon(0)$  and $w_-^\\varepsilon(0)$ as $\\varepsilon\\to0$. The oscillatory dynamics of the off-diagonal part of the Wigner function implies that it can be neglected far from the crossing set (see Remark~\\ref{rem:appendix} in the Appendix). However, these effects could  restrict our results, see the comments after our main Theorem~\\ref{theorem} and the corresponding numerical experiment in \\S\\ref{sec:dual}.\n\n\n\\subsection{Egorov's theorem}\n\nWe consider the classical flow\n$$\n\\Phi^t_\\pm: {\\mathbb R}^{2d}\\to{\\mathbb R}^{2d}\\,,\\quad\n\\Phi^t_\\pm(q_0,p_0)=\\left(q^\\pm(t),p^\\pm(t)\\right)\n$$\nassociated with the Hamiltonian curves of $\\Lambda^\\pm(q,p)=\\tfrac{|p|^2}{2}+\\lambda^\\pm(q)$. These curves   are solutions to the\nHamiltonian systems\n\\begin{equation}\\label{eq:clastraj}\n\\left\\{\\begin{array}{l} \\dot q^\\pm(t)=p^\\pm(t),\\;\\;\\dot\np^\\pm(t)=-\\nabla \\lambda^\\pm\\left(q^\\pm(t)\\right),\\\\\nq^\\pm(0)=q_0,\\;\\;p^\\pm(0)=p_0 \\end{array}\\right.\n\\end{equation}\nwhich can be solved for all $t\\in{\\mathbb R}$, since the maps $q\\mapsto \\lambda^\\pm(q)$ are smooth for eigenvalues, which do not intersect each other.\n\n\\medskip\nIf the eigenvalues are uniformly separated from each other, \nthen the classical flows~$\\Phi^t_\\pm$ are enough for an\napproximate description of the dynamics up to an error of order\n$\\varepsilon$. Indeed, \nthe action of the diagonal part of the Wigner transform on scalar test functions~$a\\in{\\mathcal C}^\\infty_c({\\mathbb R}^{2d+1},{\\mathbb C})$ obeys\n\\begin{equation}\n\\label{eq:ct} \\int_{{\\mathbb R}^{2d+1}}\n\\left(w^\\varepsilon_\\pm(t)- w^\\varepsilon_\\pm(0)\n\\circ\\Phi_\\pm^{-t}\\right)\\!(q,p)\\,a(t,q,p)\\,{\\rm d}(t,q,p) = O(\\varepsilon).\n\\end{equation}\nSuch dynamical descriptions in the spirit of Egorov's theorem are\nwell established, see for example \\cite{GMMP}. \n\n\\medskip\nIf the gap $g(q)=\\lambda^+(q)-\\lambda^-(q)$ is not uniformly bounded from below and small, but not too small, this description is still valid. More precisely, one proves in \\cite{FL08} (see also the proof in Appendix~A) that as long as the trajectories of $\\Phi^t_\\pm$ which reach the support of the observable $a$ remain in a zone where  $g(q)>R\\sqrt\\varepsilon = \\varepsilon^{3\/8}$ for $R=R(\\varepsilon) = \\varepsilon^{-1\/8}$, then\n\\begin{equation*}\n \\int_{{\\mathbb R}^{2d+1}}\n\\left(w_\\pm^\\varepsilon(t)-\nw_\\pm^\\varepsilon(0)\\circ\\Phi_\\pm^{-t}\\right)\\!(q,p)\\,a(t,q,p)\\,{\\rm d}(t,q,p)=O(\\varepsilon^{1\/8}) ,\n\\end{equation*}\nwhere the error estimate just depends on derivatives of the potential matrix $V$ and the symbol $a$, while it is independent of the gap parameter $\\delta_0$.\n\n\\medskip\nHowever, on regions with smaller eigenvalue gap the approximation of the diagonal Wigner\ncomponents $w_\\pm^\\varepsilon(t)$ by mere classical transport is no longer valid, and there are\nnon-adiabatic transitions between the levels.  \nThe components propagated until the  crossing region on one level may pass (partially or\nutterly) the other level.\n\n\\subsection{Surface hopping}\nFor a particular isotropic conical crossing \\cite{LT} and later for general conical crossings \\cite{FL08}, it has been proved\n that the diagonal parts of the Wigner transform can effectively be described by the following random walk construction: We consider a classical trajectory $\\Phi_+^t(q,p)$ with associated weight $w_+^\\varepsilon(q,p,t)$. If the gap function \n$$\nt\\mapsto g(q^+(t))\n$$ \nattains a local minimum at time $t^*$ for the phase space point $(q^*,p^*)$, such that $g(q^*)\\le R\\sqrt\\varepsilon$, then one opens a new trajectory $\\Phi_-^{t-t^*}(q^*, p^*_{out})$ with\n$$\n p^*_{out}=p ^*+\\omega^*,\\;\\;\\omega^*=g(q^*)\\tfrac{p^*}{|p^*|^2}.\n$$\nThe two trajectories $\\Phi_+^{t-t^*}(q^*, p^*)$ and   $\\Phi_-^{t-t^*}(q^*, p^*_{out})$ are equipped  with the weights\n\\begin{eqnarray*}\nw^+_\\varepsilon(q^*,p^*, (t^*)^+) & = & \\left(1-T_{\\varepsilon}(q^*,p^*)\\right)w^+_\\varepsilon(q^*,p^*,(t^*)^-),\\\\\nw^-_\\varepsilon(q^*,p^*_{out}, (t^*)^+)& = & T_{\\varepsilon}(q^*,p^*)w^+_\\varepsilon(q^*,p^*,(t^*)^-),\n\\end{eqnarray*}\nrespectively. The transition probability is given by the Landau--Zener formula\n\\begin{equation}\\label{eq:transfert}\nT_{\\varepsilon}(q^*,p^*)=\\exp\\!\\left(-\\frac{\\pi}{4\\, \\varepsilon} \\frac{g(q^*)^2}{|{\\rm det}(p^*\\cdot\\nabla_qV_0(q^*))|^{1\/2}}\\right),\n\\end{equation}\nwhere $V_0(q^*)$ denotes the trace-free part of the potential matrix $V(q^*)$. The analogous construction applies to the classical trajectories entering the region of small gap on the other eigenvalue surface. \n\n\n\\medskip\nThis combination of classical transport and Landau--Zener transitions yields an easy algorithm for the numerical simulation of non-adiabatic quantum dynamics, see \\cite{FL12} and its applications to a three-dimensional model of the pyrazine molecule~\\cite{LS} and the twelve-dimensional ammonia cation \\cite{BDLT}. Its striking properties are, that only classical trajectories, local gap minima along classical trajectories and the Landau--Zener formula~\\aref{eq:transfert} are required. \nMany other surface hopping\nalgorithms exist in the chemical literature starting with the pioneering work of Tully and Preston~\\cite{TP}, and it is worth mentioning that they are equally applied for systems with avoided or conical eigenvalue crossings. In high dimensions, surface hopping algorithms are computationally much less demanding than the discretization of the full wave function and thus often a popular choice. Despite the intense research activity in chemical physics on these algorithms, \nthere are very few mathematical results on their justification.\n\n\\subsection{Aim and organisation of the paper.}\nWe are interested in extending the Landau--Zener random walk  through conical crossings~\\cite{LT,FL08} to the case of avoided crossings, thus obtaining a unified treatment for conical intersections and avoided crossings, regardless of the respective sizes of the gap and the semi-classical parameter. As far as we know, \nthis unified treatment of both crossing types and its rigorous mathematical analysis is new, see also \\cite{FL12} for comments on the subject. Following~\\cite{Hag2},  we assume that the potential matrix $V$ presents an \navoided crossing in the sense of the definition below.\n \n\\begin{definition}\\label{def:avoided} \n Let the potential\n \\[\n V(q) = \\alpha(q){\\rm Id} + \\begin{pmatrix}\\beta(q) & \\gamma(q)\\\\ \\gamma(q) & -\\beta(q)\\end{pmatrix},\\qquad q\\in{\\mathbb R}^d,\n \\]\n be a smooth function on ${\\mathbb R}^d$, whose values are $2\\times 2$ real symmetric matrices. Denote by $\\lambda^\\pm(q)$ and $g(q) = \\lambda^+(q)-\\lambda^-(q)$ the eigenvalues and the gap function of $V(q)$ and by $V_0(q)$ its trace-free part, that is, \n\\begin{equation}\\label{def:V0}\nV_0(q) = \\begin{pmatrix}\\beta(q) & \\gamma(q)\\\\ \\gamma(q) & -\\beta(q)\\end{pmatrix},\\qquad q\\in{\\mathbb R}^d.\n\\end{equation} \nWe say that $V$ has an {\\em avoided gap} in an open subset $\\Omega\\subset{\\mathbb R}^d$ if it satisfies the following conditions :\n\\begin{enumerate}\n\\item\nThere exists $\\delta_0>0$, which is the minimum of $g$ in $\\Omega$, \n\\item Let \n$$\nS_0=\\{q\\in{\\mathbb R}^d,\\;\\; g(q)=\\delta_0\\}. \n$$\nThe set $S_0\\cap\\Omega$ is a hypersurface.\n\\item \nThere exists a system of local coordinates $(y_1,y')$ with $S_0\\cap\\Omega=\\{y_1=0\\}\\cap\\Omega$ and\n$$V_0(y)=y_1V_1(y)+V_2(y')$$\nwhere $V_1(y)$ is an invertible matrix, while $V_1(y)$ and $V_2(y)$ are linearly independent for all $y\\in\\Omega$.\n\\end{enumerate}\n \\end{definition}\n \nAn illustrative easy example for an avoided crossing in the sense of Definition~\\ref{def:avoided} is provided by\n\\begin{equation}\\label{eq:simple_example}\nV(q) = \\begin{pmatrix}q_1 & \\delta_0\\\\ \\delta_0 & -q_1\\end{pmatrix},\\qquad q\\in{\\mathbb R}^d,\\quad \\delta_0>0.\n\\end{equation}\nHere, the eigenvalues \n\\[\n\\lambda^\\pm(q) = \\pm\\sqrt{q_1^2+\\delta_0^2}\n\\] \nhave a global minimal gap of size $\\delta_0$ at the hyperplane $S_0 = \\{q\\in{\\mathbb R}^d, q_1= 0\\}$, and the potential matrix can be written as $V(q) = q_1 V_1 + V_2$ with\n\\[\nV_1 = \\begin{pmatrix}1 & 0 \\\\ 0 & -1\\end{pmatrix},\\qquad V_2 = \\begin{pmatrix}0 & \\delta_0\\\\ \\delta_0 & 0\\end{pmatrix}.\n\\] \nOn the contrary, the matrix \n$$V(q) = \\sqrt {q_1^2+\\delta_0^2}\\,  \\begin{pmatrix}1 & 0 \\\\ 0 & -1\\end{pmatrix}$$\nnot satisfy the assumptions of Definition~\\ref{def:avoided}.\n \n\n\\begin{remark}\nWe note that if one can write the minimal gap set in local coordinates as $S_0\\cap\\Omega = \\{y_1=0\\}$ and $V_0(y) = y_1 V_1(y) + V_2(y')$ with $V_1(y)$ an invertible matrix, \nthen $V_1(y)$ and $V_2(y')$ are necessarily linearly independent, in the sense, that there exists no smooth function $f:\\Omega\\to{\\mathbb R}$ with $V_1(y) = f(y) V_2(y')$ for all $y\\in\\Omega$.\n\\end{remark} \n \n \n\\begin{remark}\nThe avoided crossing of Definition~\\ref{def:avoided} is associated with a minimal gap manifold $S_0\\subset{\\mathbb R}^d$ of codimension one. \nIn the symmetry classification of avoided crossings given in \\cite{Hag2}, higher codimensions also occur. \nWe expect that our analysis of the codimension one case can be generalized, see also Remark~\\ref{rem:generalize}.\n\\end{remark}\n\nIf potential $V$ has an \navoided gap with minimal gap size~$\\delta_0>0$ (in the sense of Definition~\\ref{def:avoided}), then one can construct  a family $V(q,\\delta)$ with an avoided crossing of eigenvalues in the sense of Definition~\\ref{def:avoided_hag} with two crucial properties. First, we have\n$$\nV(q) = V(q,\\delta_0),\\qquad q\\in\\Omega,\n$$ \nand second the mapping $(q,\\delta)\\mapsto V(q,\\delta)$ has a conical intersection of its eigenvalues for $(q,\\delta)\\in S_0\\times\\{\\delta=0\\}$, \nsee Theorem~\\ref{prop:parametrization} below. Considering momenta $p\\in{\\mathbb R}^d$ which are transverse to the hypersurface~$S_0$  ensures that the crossing is generic in $T^*({\\mathbb R}^d_q\\times{\\mathbb R}_\\delta)$ in the sense of \\cite{CdV1} and \\cite{FG03}. This link between avoided and conical crossings -- which is already indicated in Colin de Verdi\\`ere's paper \\cite[\\S2.8]{CdV1} -- allows us to redevelop the proof strategy of \\cite{FL08}.\nA crucial step in our new proof is an elementary normal form construction inspired by \\cite{FG02} that is explicit enough to keep $\\delta$ as a controlled parameter. \n\n\n\\medskip\nWe start in Section~\\ref{sec:result} by providing the precise mathematical statement of our result. In \nSection~\\ref{sec:strategy} we prove the relation between avoided and conical crossings. In Section~\\ref{sec:reduction} \nwe perform an elementary reduction to a Landau--Zener model parametrized by the gap parameter $\\delta$. Both these new results are \ncrucial for the proof of the surface hopping approximation in Section~\\ref{sec:mainproof}. We then describe the \nassociated surface hopping algorithm in Section~\\ref{sec:numerics} and present numerical experiments for Tully's well-known avoided crossing models~\\cite{Tu1}. \nThe Appendix~\\ref{proof:prop} presents the proof of classical transport in the zone of large gap.\n\n\n\n\n\\section{An Theorem}\\label{sec:result}\n\nWe now give a precise statement for the Egorov type description of the propagation of the diagonal part of the Wigner transform for systems with an avoided eigenvalue crossing in the sense of Definition~\\ref{def:avoided}. We consider the classical trajectories $(q^\\pm(t),p^\\pm(t))$ of the Hamiltonian systems \\aref{eq:clastraj} and monitor \nthe phase space points, where the classical trajectories attain a {\\em local\nminimal gap between the two eigenvalues.} At such points we have\n\\begin{equation}\\label{jumpcond}\n\\frac{{\\rm d}}{{\\rm d} t}\\left(g(q^\\pm(t))\n\\right)=p^\\pm(t)\\cdot \\nabla_q g(q^\\pm(t))=0.\n\\end{equation}\nTherefore, one performs an effective\nnon-adiabatic transfer of weight, whenever a trajectory passes the\nset\n$$\n\\Sigma_{\\varepsilon} = \\left\\{(q,p)\\in{\\mathbb R}^{2d}\\mid \n\\;g(q)\\le R\\sqrt\\varepsilon\n ,\\;\\;p\\cdot\\nabla g(q)=0\\right\\}\n$$\nwhere $R=R(\\varepsilon) = \\varepsilon^{-1\/8}\\gg1$. This choice of $R$ is motivated from the analysis in~\\cite{FL08} and ensures that the largest occuring  error terms \n$R^3\\sqrt\\varepsilon$ and $R^{-5}\\varepsilon^{-1\/2}$ are of the same order $\\eta_\\varepsilon=\\varepsilon^{1\/8}$.  \n\n\n\\subsection{The random trajectories}\nWe attach the labels~$j=-1$ and~$j=+1$ to the phase space ${\\mathbb R}^{2d}$\nand consider the random  trajectories\n$$\n{\\mathcal T}_{\\varepsilon}^{(q,p,j)}:[0,+\\infty)\\rightarrow\n{\\mathbb R}^{2d}\\times\\{-1,+1\\},\n$$\nwith\n$$\n{\\mathcal T}_{\\varepsilon}^{(q,p,j)}(t)=\\left(\\Phi^t_j(q,p),j\\right)\\quad\\mbox{as long as} \\quad\\Phi^t_{j}(q,p)\\not\\in \\Sigma_{\\varepsilon}.$$ \nWhenever the deterministic flow $\\Phi^t_j(q,p)$ hits the set\n$\\Sigma_{\\varepsilon}$ at a point $(q^*,p^*)$, a jump \n$$\n(q^*,p^*,j) \\to (q^*,p^*+j\\,\\omega^*,-j)\n$$ \noccurs with the transition probability $T_{\\varepsilon}(q^*,p^*)$  defined in~\\aref{eq:transfert}. The drift\n \\begin{equation}\\label{def:drift}\n \\omega^* = \\omega^*(q^*,p^*) = \\frac{g(q^*)}{|p^*|^2}\\, p^*,\n \\end{equation}\nis applied to preserve the energy of the trajectories\n$$\n\\Lambda^\\pm(q,p) = \\tfrac12|p|^2 +  \\lambda^\\pm(q) = \\tfrac12|p|^2 + \\alpha(q) \\pm\\tfrac12 g(q)\n$$\nup to order $R^2\\varepsilon$.\nIndeed, let us suppose that the incoming trajectory is on the plus mode. Then, one chooses the momentum $p^*_{out}=p^*+ \\omega^*$ of the trajectory generated on the minus mode such that its energy $\\Lambda^-(q^*,p^*_{out})$ satisfies\n$$\\Lambda^-(q^*,p^*_{out})=\\Lambda^+(q^*,p^*)+O(R^2\\varepsilon).$$ Since \n$\\Lambda^-(q^*,p^*_{out})=\n \\tfrac{1}{2} |p^*+ \\omega^*|^2 +\\alpha(q^*)-\\tfrac{1}{2}g(q^*)$, \nit is enough to choose $\\omega^*$ such that \n$ \\omega^*\\cdot p^* = g(q^*),\n$\nwhence~\\aref{def:drift}. \n\n\\begin{remark}\\label{rem:energy}\nLet us comment about various aspects of the drift. The drift $p^*\\pm\\omega^*$ will be crucial later on for localizing the solution at a distance of size~$R\\sqrt\\varepsilon$ to the energy surfaces $\\{\\tau+\\Lambda^\\pm(q,p)=0\\}$. \nThe drift is performed in the momentum coordinates, since the difference of the two Hamiltonian vector fields \n\\[\nH_{\\lambda^\\pm}(q,p) = (p, -\\nabla\\lambda^\\pm(q))\n\\] \nvanishes identically in the position coordinates. Moreover, after a change of space-time coordinates, see Section~\\ref{subsubsec:drift}, the drift is exact when performed in this direction. A similar drift has been used in \\cite{HJ1} and~\\cite{HJ2} for analysing wave packet propagation through avoided crossings.\n\nThe transversality condition that will be stated in assumption~(A0) of our main Theorem~\\ref{theorem} excludes trajectories \nwith small momenta at the jump manifold. Therefore, \n$\\omega^*$ is of the order of the gap size and thus bounded by $R\\sqrt\\varepsilon$.\nOutside the jump manifold $\\Sigma_\\varepsilon$ the gap and thus the drift are large. However, the Landau--Zener transition rates are exponentially small there. \nConsequently, in this regime the drift would be harmless (if performed).\n\\end{remark}\n\n\n\n\\subsection{The semigroup} Within a\nbounded time-interval $[0,T]$, each path\n$$\n(q,p,j)\\to {\\mathcal T}_{\\varepsilon}^{(q,p,j)}(t)\n$$\nonly has a finite number of jumps, remains in bounded regions of\n${\\mathbb R}^{2d}\\times\\{-1,+1\\}$, and is smooth away from the jump\nmanifold $\\Sigma_{\\varepsilon}\\times\\{-1,+1\\}$. Hence, the random\ntrajectories define a Markov process\n$$\n\\left\\{ X^{(q,p,j)} \\mid (q,p,j)\\in{\\mathbb R}^{2d}\\times\\{-1,+1\\}\\right\\}.\n$$\nThe associated transition function $P(p,q,j;t,\\Gamma)$ describes\nthe probability of being at time $t$ in the set $\\Gamma\\subset\n{\\mathbb R}^{2d}\\times \\{-1,+1\\}$ having started in $(q,p,j)$. Its action\non the set \n$$\n{\\mathcal B} = \\left\\{ f:{\\mathbb R}^{2d}\\times\\{-1,+1\\}\\to{\\mathbb C} \\mid f \\;\\mbox{is measurable, bounded}\\right\\}\n$$ \ndefines a semigroup\n$({\\mathcal L}_{\\varepsilon}^t)_{t\\ge0}$ by\n$$\n{\\mathcal L}_{\\varepsilon}^t\\,f(q,p,j) =\n\\int_{{\\mathbb R}^{2d}\\times\\{-1,+1\\}} f(x,\\xi,k)\\,P(q,p,j;t,{\\rm d}(x,\\xi,k)).\n$$\n\n\\begin{remark}\nWe associate with $f\\in{\\mathcal B}$ two functions $f_\\pm:{\\mathbb R}^{2d}\\to{\\mathbb C}$ via\n\\begin{equation}\\label{fpm}\nf_\\pm(q,p)=f(q,p,\\pm 1).\n\\end{equation}\nReversely,  relation~\\aref{fpm} implies that two bounded measurable functions $f_\\pm$ on ${\\mathbb R}^{2d}$ define a function $f\\in{\\mathcal B}$. We shall use this identification all over the paper. \n\\end{remark}\n\nWe now define the action of the semigroup on Wigner functions\nby duality. More precisely, let $\\psi\\in L^2({\\mathbb R}^d,{\\mathbb C}^2)$ and $W^\\varepsilon(\\psi)$ be its Wigner transform.  Denote by\n$$ \nw^\\varepsilon_\\pm(\\psi)(q,p) =  {\\rm tr }\\left(\\Pi^\\pm(q)W^\\varepsilon(\\psi)(q,p)\\right)\n$$ \nthe diagonal components of $W^\\varepsilon(\\psi)$ and define $w^\\varepsilon(\\psi)\\in{\\mathcal B}$ according to relation \\aref{fpm}.\nFor $a\\in{\\mathcal B}$ such that $a_+$ and $a_-$ have compact support, we set\n$$\n\\left( w^\\varepsilon(\\psi),a\\right) =\\int_{{\\mathbb R}^{2d}}w^\\varepsilon_+(\\psi)(q,p) \\,a_+(q,p)\\, {\\rm d}(q,p) + \\int_{{\\mathbb R}^{2d}} w^\\varepsilon_-(\\psi)(q,p)\\, a_-(q,p)\\, {\\rm d}(q,p)\n$$\nand define ${\\mathcal L}_{\\varepsilon}^t w^\\varepsilon(\\psi)\\in{\\mathcal B}$ by\n$$\n\\left({\\mathcal L}_{\\varepsilon}^t w^\\varepsilon(\\psi), a\\right) =\\left(w^\\varepsilon(\\psi),{\\mathcal L}_{\\varepsilon}^t a\\right).\n$$\n\n\n\\subsection{The result}\\label{subsec:result}\nLet $V$ be a potential matrix presenting an avoided crossing of eigenvalues in the sense of Definition~\\ref{def:avoided}, the notations of which we shall use in the following. \nThe semi-group $({\\mathcal L^t_\\varepsilon})_{t>0}$ approximates the non-adiabatic dynamics generated by this avoided crossing in $\\Omega\\subset{\\mathbb R}^d$, if we assume the following: \n\n\\subsubsection{Initial data (A0)}\nThe initial data $(\\psi^\\varepsilon_0)_{\\varepsilon>0}$ is a bounded family in\n$L^2\\!\\left({\\mathbb R}^d,{\\mathbb C}^2\\right)$ associated either with ${\\rm Ran}\\Pi^+$\nor ${\\rm Ran}\\Pi^-$, meaning that either\n$$\n\\left|\\left|\\Pi^-\\psi^\\varepsilon_0\\right|\\right|_{L^2({\\mathbb R}^d,{\\mathbb C}^2)}=\nO(\\varepsilon^{\\beta_1}),\\qquad \\beta_1\\ge 1\/32,\n$$\nor the analogous condition on $\\Pi^+\\psi^\\varepsilon_0$ holds. We suppose\nthat the initial data are localized away from $S_0\\cap\\Omega$, \n that is, there is some $C_0>0$ such that\n$$\n\\int_{\\{{\\rm d}(q,S_0)<C_0\\}} | W^\\varepsilon(\\psi^\\varepsilon_0)(q,p)|\\, {\\rm d} q\\, {\\rm d} p =\nO(\\varepsilon^{\\beta_2}),\\qquad \\beta_2\\ge1\/32.\n$$\nThe classical trajectories issued from the support of $W^\\varepsilon(\\psi^\\varepsilon_0)$ reach their minimal gap points in $S_0\\cap\\Omega$. Moreover, \nthe initial data is localized away from the set \n$$\n\\left\\{(q_0,p_0)\\in{\\mathbb R}^{2d}\\mid \\exists t>0: q^\\pm(t)\\in S_0,\n\\;  p^\\pm(t) \\in T_{q^\\pm(t)}S_0\\right\\},\n$$\nwhich contains those points, which are transported to the minimal gap manifold $S_0$ and have gained momenta which are not transverse to $S_0$.\n\n\\subsubsection{Observables (A1)}\nThe observable $a\\in{\\mathcal B}$ satisfies $a_\\pm\\in{\\mathcal C}_c^\\infty\\left({\\mathbb R}^{2d},{\\mathbb C}\\right)$ and has its support at a distance larger than $\\varepsilon^{\\beta_3}$\nfrom $S_0$, i.\\ e.\\\n$$\n{\\rm d}({\\rm supp}_{(q,p)}(a_\\pm),S_0)\\gg \\varepsilon^{\\beta_3},\\qquad\\beta_3\\ge 1\/32.\n$$\n\n\\subsubsection{Time-interval (A2)}\nLet $T>0$. Within the time-interval $[0,T]$, the classical trajectories issued from the support of $W^\\varepsilon(\\psi^\\varepsilon_0)$ reach their minimal gap points only once.\n\n\n\\medskip\nThese assumptions on the initial data, the observables, and the time interval allow us to effectively describe the dynamics through an avoided crossing by surface hopping.\n\n\n\n\\begin{theorem}\\label{theorem}\nLet $\\varepsilon>0$ and $\\psi^\\varepsilon$ be the solution of the Schr\\\"odinger equation\n$$\ni\\varepsilon\\partial_t\\psi^\\varepsilon_t = -\\tfrac{\\varepsilon^2}{2} \\Delta_q\\psi^\\varepsilon_t +V(q)\\psi^\\varepsilon_t,\\qquad \\psi^\\varepsilon_{t=0}=\\psi^\\varepsilon_0,\n$$\nwhere the potential $V$ has an \navoided crossing in the sense of Definition~\\ref{def:avoided} with a \ngap parameter $\\delta_0\\in ]0,1]$.\nAssuming (A0), (A1) and (A2), we have for all test functions $\\chi\\in{\\mathcal C}_c^\\infty([0,T])$ a constant $C>0$\n\\begin{equation}\n\\label{eq:approx}\n\\left| \\int_0^T \\chi(t) \\left(w^\\varepsilon(\\psi^\\varepsilon_t)-{\\mathcal L^t_\\varepsilon w^\\varepsilon(\\psi^\\varepsilon_0)},a\\right) {\\rm d} t \\right| \\le C\\,\\varepsilon^{1\/32},\n\\end{equation}\nwhere the constant $C$ depends on a finite number of upper bounds of derivatives of the smooth functions $\\alpha,\\beta,\\gamma$ defining the potential $V$ and $a,\\chi$ and of lower bounds of the determinant of the matrix $V_1$.\n\\end{theorem}\n\n\n\\medskip\nThe semigroup $(\\mathcal{L}^t_\\varepsilon)_{t\\ge0}$ crucially depends on the jump manifold $\\Sigma_\\varepsilon$, that comprises those points in phase space with $g(q)\\le R\\sqrt\\varepsilon$, \n$R=R(\\varepsilon) = \\varepsilon^{-1\/8}$, where the classical trajectories $(q^\\pm(t),p^\\pm(t))$ attain locally minimal surface gaps.  If the minimal gap size $\\delta_0>0$ of the avoided crossing is larger than $R\\sqrt\\varepsilon$, then the jump manifold is the empty set, so that Theorem~\\ref{theorem} reduces to a leading order description of expectation values for block-diagonal observables by mere classical transport, see Appendix~\\ref{proof:prop}. As a consequence, $\\delta_0\\le R\\sqrt\\varepsilon$ is the only  interesting regime, and the key issue is to prove the hopping formula locally, close to any point of  $S_0\\cap\\Omega$.\nThis is done by\nusing the possibility to parametrically link the avoided crossing with a conical intersection. This construction is carried out in Section~\\ref{sec:strategy},  where we prove that close to any point of $\\Omega$, the potential $V$ is embeddable in a parametrized family of potentials. We then prove a result  analogous to Theorem~\\ref{theorem} for the family of solutions to the Schr\\\"odinger equation associated with the parametrized potentials (see Theorem~\\ref{theorem:delta-result} below). The different steps of the proof of the surface hopping approximation are then developed in Section~\\ref{sec:mainproof}, via a reduction to a Landau--Zener model performed in Section~\\ref{sec:reduction}. \n\n\\medskip\nAn interesting feature of Theorem~\\ref{theorem} is that it justifies using a surface algorithm without assessing the size of the gap with respect to~$\\varepsilon$. This is of major interest for applications, since for ``real'' molecular quantum systems the explicit comparison of $\\delta_0$ and $\\varepsilon$ might be difficult. The algorithm takes into account the three main regimes:\n\\begin{enumerate}\n\\item $\\delta_0\\gg\\sqrt\\varepsilon$ propagation along the eigenvalue surfaces\n\\item $\\delta_0\\sim \\sqrt\\varepsilon$ partial transition between eigenspaces\n\\item $\\delta_0\\ll\\sqrt\\varepsilon$ total transition between the eigenspaces\n\\end{enumerate}\nIn the context of semi-classical wave packet propagation, \\cite{HJ1} and \\cite{HJ2} have analysed the second regime, while \\cite{Rou} has considered the first and third one.\nIn particular, if there are several sizes of minimal gaps in different open subsets, Theorem~\\ref{theorem} proves that the algorithm can be used and the transition process will be automatically adapted to the gap size. The Born--Oppenheimer result \\eqref{eq:ct} uses plain classical transport and has an error constant \nthat tends to infinity for shrinking minimal gap size $\\delta_0$. In constrast, the error bound of Theorem~\\ref{theorem} only depends on bounds of the potential $V$ that can be controlled with respect to $\\delta_0$. (The lower bound on the determinant of $V_1$ is not related to the gap size.)\n\n\\medskip\nWe note that the transversality condition of assumption (A0) is crucial for the microlocal normal form we use for effectively describing the \nnonadiabatic transitions. For the simple example \\eqref{eq:simple_example}, it means that the set $\\{(q,p)\\in{\\mathbb R}^{2d}, q_1=p_1=0\\}$ is negligible for the \ntrajectories of\n\\[\n\\dot q = p,\\qquad \\dot p = \\mp \\frac{q_1}{\\sqrt{q_1^2+\\delta_0^2}} (1,0,\\ldots,0) \n\\]\nthat are issued from the support of the initial Wigner function $W^\\varepsilon(\\psi^\\varepsilon_0)$.  The first condition of assumption (A0) can be relaxed to initial data associated with \nboth ${\\rm Ran}\\, \\Pi^+$ and ${\\rm Ran}\\,\\Pi^-$, \nprovided that the trajectories for both modes do not arrive simultaneously at the same phase space point of the  jump manifold $\\Sigma_\\varepsilon$. However, as illustrated by the numerical experiments for the dual avoided crossing in Section~\\ref{sec:dual}, simultaneous arrival at the jump manifold is the situation where the off-diagonal components of the Wigner transform become relevant such that the present surface hopping approximation breaks down.\n\n\\begin{remark}\\label{rem:generalize}\nThe avoided crossing of Definition~\\ref{def:avoided} has a minimal gap manifold $S_0\\subset{\\mathbb R}^d$ of codimension one.\nThe results of \\cite{FG03,Fe06,FL08} on eigenvalue crossings of codimension three and five allow to extend Theorem~\\ref{theorem} to avoided crossings with minimal gap manifolds of higher codimension as well. \n\\end{remark}\n\n\n\n\\section{Reduction to a conical intersection}\\label{sec:strategy}\n\nWe now introduce a family of potentials $(V_0(q,\\delta))_{\\delta\\in I}$ locally extending the trace-free part $V_0(q)$ of our original potential. We verify \nthat $V_0(q,\\delta)$ viewed as a function on ${\\mathbb R}^d\\times I$ has a generic codimension~two crossing for $q\\in S_0$ and $\\delta=0$ in the sense of ~\\cite{CdV1} and~\\cite{FG03}, respectively. Then, we describe the parametrized Schr\\\"odinger system that we shall consider afterwards.\n\n\n\\subsection{Parametrization of the gap}\n\nWe start by constructing the family of trace-free potentials $(V_0(q,\\delta))_{\\delta\\in I}$ that locally extends the original potential $V_0(q)$ by adding the gap size as an additional coordinate.\n\n\\begin{theorem}\\label{prop:parametrization} \nLet $V$ have an\navoided crossing in $\\Omega\\subset{\\mathbb R}^d$, in the sense of Definition~\\ref{def:avoided}, with minimal gap hypersurface $S_0$.  \nThen there  exists an open interval $I\\subseteq{\\mathbb R}$ with $0,\\delta_0\\in I$, an open subset $\\widetilde\\Omega\\subseteq\\Omega$ and two functions $\\beta(q,\\delta)$ and  $\\gamma(q,\\delta)$ smooth on $\\widetilde\\Omega\\times I$ and affine in $\\delta$,  such that the matrix\n$$V_0(q,\\delta)=\\begin{pmatrix} \\beta(q,\\delta) & \\gamma(q,\\delta) \\\\ \\gamma(q,\\delta) & -\\beta(q,\\delta)\\end{pmatrix}$$\nsatisfies the following properties:\n\\begin{enumerate}\n\\item We have $V_0(q,\\delta_0)=V_0(q)$ for all $q\\in\\widetilde\\Omega$.\n\\item The eigenvalue gap\n$$\ng(q,\\delta)=2\\sqrt{\\beta(q,\\delta)^2+\\gamma(q,\\delta)^2}\n$$ \nof $V_0(q,\\delta)$ is of minimal size $|\\delta|$ on $S_0$, that is, \n$$\ng(q,\\delta)\\ge|\\delta|\\;\\mbox{for all}\\;q\\in\\widetilde\\Omega,\\quad\ng(q,\\delta)=|\\delta|\\;\\mbox{if and only if}\\;q\\in S_0.\n$$\n\\item There exists a smooth orthogonal matrix $R(q)$ and two smooth functions $\\widetilde \\beta(q,\\delta)$  and $\\widetilde \\gamma(q)$, where $\\widetilde \\beta$ is affine in $\\delta$, such that \n$$R(q)V_0(q,\\delta)R(q)^*=\\begin{pmatrix}\n\\widetilde\\beta(q,\\delta) & \\delta \\widetilde\\gamma(q) \\\\\n\\delta \\widetilde\\gamma(q)  & - \\widetilde\\beta(q,\\delta)\n\\end{pmatrix}$$\nand $\\widetilde\\gamma(q)\\neq 0$ for all $q\\in\\widetilde\\Omega$. All derivatives of $\\widetilde\\beta$ are bounded.\n\\item\nIf $y=(y_1,y')$ are local coordinates such that $S_0\\cap\\widetilde\\Omega = \\{y_1=0\\}\\cap\\widetilde\\Omega$, then \n\\[\n\\forall k\\in{\\mathbb N} \\,\\exists c_k>0 : \\sup_{y\\in\\widetilde\\Omega} |\\partial^k_{y_1}\\widetilde\\gamma(y)| < c_k\n\\]\nand\n\\[\n\\forall \\alpha\\in{\\mathbb N}^{d-1} \\,\\exists c_\\alpha>0 : \\sup_{y\\in\\widetilde\\Omega} |\\partial^\\alpha_{y'}\\widetilde\\gamma(y)| < c_\\alpha {|y_1|\\over \\delta_0},\n\\] \nwhile all other derivatives of the function $\\widetilde\\gamma$ are of the order $1\/\\delta_0$. The derivative bounds  involve a lower bound on the determinant of the matrix~$V_1(y)$ \nof the decomposition $V_0(y) = y_1 V_1(y) + V_2(y')$.\n\\end{enumerate} \n\\end{theorem}\n\n\\begin{proof}\nWe work close to some $q_0\\in S_0 $ in  local coordinates $y=(y_1,y')$ such that $S_0\\cap\\Omega=\\{y_1=0\\}\\cap\\Omega$ and \n$$\nV_0(y)=y_1V_1(y)+V_2(y')\\;\\;\n\\text{with}\\;\\;V_1(y)\\;\\text{invertible}\n$$ \nfor all $y\\in\\Omega$.\nSetting $y_1=0$, one obtains that $V_2$ and consequently $V_1$ are trace-free on $\\Omega$. We denote \n$$\nV_1(y)=\\begin{pmatrix} a_1(y)& b_1(y)\\\\ b_1(y) & -a_1(y)\\end{pmatrix},\\quad\nV_2(y')=\\begin{pmatrix} a_2(y')& b_2(y')\\\\ b_2(y') & -a_2(y')\\end{pmatrix}\n$$\nand write the gap as\n\\begin{align*}\ng(y)^2 &= 4y_1^2 \\left(a_1(y)^2 + b_1(y)^2\\right) + 8y_1\\left(a_1(y)a_2(y') + b_1(y)b_2(y')\\right)\\\\\n& \\quad + 4\\left(a_2(y')^2 + b_2(y')^2\\right) \n\\end{align*}\nfor all $y\\in\\Omega$. From the relation $g(y)^2=\\delta_0^2$ for all $y\\in S_0$ we then deduce \n\\[\n\\delta_0^2 = 4(a_2(y')^2+b_2(y')^2)\\;\\;\\mbox{for all}\\;\\; y\\in\\Omega.\n\\]\nWe define for $\\delta\\in{\\mathbb R}$\n$$\nV_0(y,\\delta) = y_1 V_1(y)+{\\frac{\\delta}{\\delta_0}} V_2(y')\n$$\nsuch that for all $y\\in\\Omega$\n$$\nV_0(y,\\delta_0)=V_0(y),\\qquad g(y,\\delta_0)=g(y)\n$$\nand\n\\begin{eqnarray*}\ng(y,\\delta)^2 = 4y_1^2\\left(a_1(y)^2 + b_1(y)^2\\right)+ 8y_1 \\frac{\\delta}{\\delta_0}\n\\left(a_1(y)a_2(y')+b_1(y)b_2(y')\\right) + \\delta^2.\n\\end{eqnarray*} \nWith respect to the original coordinates, this construction means\n$$\nV_0(q,\\delta)= \\begin{pmatrix}\\beta(q,\\delta) & \\gamma(q,\\delta)\\\\ \\gamma(q,\\delta) & -\\beta(q,\\delta)\\end{pmatrix}\n$$\nwith\n$$\n\\beta(q,\\delta)=y_1a_1(y)+\\frac{\\delta}{\\delta_0} a_2(y'),\\quad\n\\gamma(q,\\delta)=y_1b_1(y)+\\frac{\\delta}{\\delta_0} b_2(y').\n$$\n\nLet us prove now that the gap $g(y,\\delta)$ is minimal on $S_0$ for $\\delta$ in some open interval~$I$ which contains $[0,\\delta_0]$.\nSince the gap $g(y)$ is minimal on $S_0\\cap\\Omega$, we have $\\nabla_y (g(y)^2)=0$ for all $y\\in S_0\\cap\\Omega$. Consequently, \n\\[\na_1(0,y')a_2(y')+b_1(0,y')b_2(y') = 0,\\qquad y=(0,y')\\in S_0\\cap\\Omega.\n\\]\nTherefore, there exist an open subset $\\widetilde\\Omega\\subseteq\\Omega$ and a continuous function $\\Gamma:\\widetilde\\Omega\\to{\\mathbb R}$ such that \n$$a_1(y)a_2(y')+b_1(y)b_2(y')=y_1\\Gamma(y)$$\n and \n\\[\ng(y,\\delta)^2=4y_1^2\\left(a_1(y)^2+b_1(y)^2+2\\frac{\\delta}{\\delta_0}\\Gamma(y)\\right) + \\delta^2.\n\\]\nNow it remains to find an open interval $I$ such that\n\\begin{equation}\\label{relation}\na_1(y)^2+b_1(y)^2+2\\frac{\\delta}{\\delta_0}\\Gamma(y) >0\n\\end{equation}\nfor all $(y,\\delta)\\in\\widetilde\\Omega\\times I$ with $y_1\\neq0$. We observe that \n$$\ng(y,\\delta_0)^2=g(y)^2>\\delta_0^2\\;\\;{\\rm for}\\;\\; y_1\\not=0\n$$\nimplies\n$$\n4y_1^2\\left(a_1(y)^2+b_1(y)^2+2\\Gamma(y)\\right)>0\\;\\;{\\rm for}\\;\\; y_1\\not=0,\n$$\nwhile the invertibility of $V_1(y)$, $y\\in\\widetilde\\Omega$, implies\n$$\na_1(y)^2+b_1(y)^2 > 0,\\qquad y\\in\\widetilde\\Omega.\n$$\nTherefore, the affine function\n$$\n\\delta\\mapsto a_1(y)^2 + b_1(y)^2 + 2\\frac{\\delta}{\\delta_0} \\Gamma(y) \n$$\ntakes nonnegative values in $\\delta=0$ and $\\delta=\\delta_0$ and thus for any $\\delta\\in]0,\\delta_0[$, which yields \\eqref{relation}. \nWe note that the we can choose the interval $I$ small enough such that the quotient $\\delta\/\\delta_0$ remains bounded for all $\\delta\\in I$. \nConsequently, the functions $\\beta(\\cdot,\\delta)$ and $\\gamma(\\cdot,\\delta)$ have smooth bounded derivatives.\n\n\\medskip\nFor the rotation of $V_0(q,\\delta)$ we set\n$$\nA(q)= - {b_1(y(q))\\over\\sqrt{b_1(y(q))^2+a_1(y(q))^2}},\\;\\; B(q)={a_1(y(q))\\over\\sqrt{b_1(y(q))^2+a_1(y(q))^2}}\n$$\nand note that $A$ and $B$ are smooth functions with bounded derivatives, where the bound involves a lower bound on the determinant of $V_1$.\nWe also define the smooth rotation matrix $R(q) $ of angle $\\theta(q)$ such that \n$$\n{\\rm cos} \\left(2\\theta(q)\\right)=B(q),\\qquad{\\rm sin} \\left(2\\theta(q)\\right)=A(q).\n$$\nThen, we have \n\\begin{eqnarray*}\nR(q) V_0(q,\\delta) R(q)^* & = & \\begin{pmatrix}\n{\\rm cos} \\,\\theta & -{\\rm sin} \\,\\theta\\\\\n{\\rm sin} \\,\\theta & {\\rm cos} \\,\\theta \n\\end{pmatrix}\n \\begin{pmatrix}\\beta & \\gamma\\\\ \\gamma & -\\beta\\end{pmatrix}\n \\begin{pmatrix}\n{\\rm cos} \\,\\theta & {\\rm sin} \\,\\theta\\\\\n-{\\rm sin} \\,\\theta & {\\rm cos} \\,\\theta \n\\end{pmatrix}\\\\\n& = &  \\begin{pmatrix}\n\\beta\\,{\\rm cos} (2\\theta)-\\gamma\\,{\\rm sin}(2\\theta) & \\beta\\,{\\rm sin} (2\\theta)+\\gamma\\,{\\rm cos}(2\\theta)\\\\\n\\beta\\,{\\rm sin} (2\\theta)+\\gamma\\,{\\rm cos}(2\\theta) & -\\beta\\,{\\rm cos} (2\\theta)+\\gamma\\,{\\rm sin}(2\\theta) \n\\end{pmatrix}\\\\\n& = & \n\\begin{pmatrix}\n\\widetilde\\beta(q,\\delta) & \\delta \\widetilde\\gamma(q) \\\\\n\\delta \\widetilde\\gamma(q)  & - \\widetilde\\beta(q,\\delta)\n\\end{pmatrix},\n\\end{eqnarray*}\nwhere we define the $\\delta$-affine function \n\\begin{eqnarray*}\n&&\\widetilde \\beta(q,\\delta) = B(q)\\beta(q,\\delta) -A(q)\\gamma(q,\\delta)\\\\\n&&= y_1(q)\\sqrt{a_1(y(q))^2+b_1(y(q))^2} + \\frac{\\delta}{\\delta_0} {a_2(y'(q))a_1(y(q))+b_1(y(q))b_2(y'(q))\\over \\sqrt {b_1(y(q))^2+a_1(y(q))^2}},\n\\end{eqnarray*}\nwhose derivatives are bounded functions, since $\\delta\/\\delta_0$ is uniformly bounded. The smooth function $\\widetilde\\gamma(q)$ is defined by\n\\begin{eqnarray*}\n\\widetilde \\gamma(q) & = & {1\\over \\delta} \\left(A(q)\\beta(q,\\delta)+B(q)\\gamma(q,\\delta)\\right)\\\\\n& = & \\frac{-b_1(y(q)) a_2(y'(q))+a_1(y(q)) b_2(y'(q))}{\\delta_0 \\sqrt{b_1(y(q))^2+a_1(y(q))^2}}.\n\\end{eqnarray*}\nObserving that $a_2$ and $b_2$ only depend on $y'$ and that \n\\[\na_2(y')^2+b_2(y')^2 = \\tfrac{1}{4} \\delta_0^2 \\le\\delta_0^2,\\qquad y=(y_1,y')\\in\\widetilde\\Omega,\n\\] \nwe deduce that \n$\\partial_{y_1}^k \\widetilde\\gamma$ is a smooth bounded function for any $k\\in{\\mathbb N}$. Using that $a_1a_2+b_1b_2=0$ on $S_0$, we observe \n\\[\n\\widetilde\\beta(q,\\delta) = 0,\\qquad q\\in S_0\\cap\\widetilde\\Omega,\n\\]\nand $\\delta^2 = \\beta(\\cdot,\\delta)^2 + \\gamma(\\cdot,\\delta)^2 = \n\\widetilde\\beta(\\cdot,\\delta)^2 + \\delta^2\\,\\widetilde\\gamma^2$,  \nand we deduce that \n$\\widetilde \\gamma^2 = 1$ on $S_0$. Since $\\widetilde\\gamma$ is non-vanishing, \ndue to the linear independence of $V_1(y)$ and $V_2(y')$, we then conclude that \n$\\partial_{y'}^\\alpha\\widetilde\\gamma(0,y')=0$ for all $\\alpha\\in{\\mathbb N}^{d-1}$.\nA Taylor expansion together with the rough estimate, \nthat derivatives of $\\widetilde\\gamma$ are of the order $1\/\\delta_0$, yields the claimed bound on \n$\\partial_{y'}^\\alpha \\widetilde\\gamma(y)$ for $y\\in\\widetilde\\Omega$.\n\\end{proof}\n\n\n\\subsection{The geometry of the crossing}\nWe add half the trace to the $\\delta$-parametrized trace-free family of Theorem~\\ref{prop:parametrization} and consider\n\\begin{equation}\\label{def:V}\nV(q,\\delta) = \\alpha(q){\\rm Id} + V_0(q,\\delta),\\qquad (q,\\delta)\\in\\widetilde\\Omega\\times I,\n\\end{equation}\nwith $\\alpha(q) = \\frac12{\\rm tr}V(q)$, such that the original potential can be written as\n\\[\nV(q) = V(q,\\delta_0),\\qquad q\\in\\widetilde\\Omega.\n\\]\nWe now verify that the symbol of the corresponding time-dependent Schr\\\"odinger operator, the matrix-valued function \n$$\nP(q,p,\\tau,\\delta):=\\left(\\tau+\\tfrac{|p|^2}{2}\\right){\\rm Id} +V(q,\\delta),\n$$\nhas a generic codimension two crossing on $S_0\\times\\{\\delta=0\\}$ in the sense of~\\cite[\\S1]{CdV1} and \\cite[\\S1]{FG03}. It has to satisfy the following two properties: \n\\begin{enumerate}\n\\item The gap function satisfies\n$$\ng(q,\\delta)=0\\quad\\text{if and only if}\\quad (q,\\delta)\\in (S_0\\cap\\widetilde\\Omega)\\times\\{\\delta=0\\},\n$$\naccording to point (2) of Theorem~\\ref{prop:parametrization}. \n\\item The Poisson bracket \n$$\n\\left\\{\\tau+\\tfrac{|p|^2}{2}+\\alpha(q)\\;,\\; V_0(q,\\delta)\\right\\}=p\\cdot\\nabla_q V_0(q,\\delta)\n$$\nis invertible for $(q,\\delta)\\in (S_0\\cap\\widetilde\\Omega)\\times\\{\\delta=0\\}$  and those momenta $p\\in{\\mathbb R}^d$ which are transverse to~$S_0$ at $q$.  This is implied by the following\nLemma~\\ref{lem:transverse}.\n\\end{enumerate}\n\n\\begin{lemma}\\label{lem:transverse} \nLet $V$ have an \n avoided crossing in the sense of Definition~\\ref{def:avoided} with minimal gap hypersurface $S_0$.  \nLet $q_0\\in S_0$ and $p_0\\in{\\mathbb R}^d$ transverse to $S_0$ at $q$. Then,  there exists a neighborhood $\\Omega_1\\subset{\\mathbb R}^{2d}$ of $(q_0,p_0)$, independent of $\\delta$ such that for all $\\delta\\in I$ and $(q,p)\\in\\Omega_1$,\n$$\np\\cdot \\nabla_q V_0(q,\\delta)\\;\\text{is invertible and}\\;\\; p\\cdot\\nabla_q\\widetilde\\beta (q,\\delta)<0,\n$$\nwhere $(V_0(q,\\delta))_{\\delta\\in I}$ and $\\widetilde\\beta(q,\\delta)$ are defined as in Theorem~\\ref{prop:parametrization}.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\widetilde\\Omega$ be the open set of Theorem~\\ref{prop:parametrization}.\nWe again work close to some point $q_0\\in S_0\\cap\\widetilde\\Omega$ in  local coordinates $y=(y_1,y')$ such that $S_0\\cap\\widetilde\\Omega=\\{y_1=0\\}\\cap\\widetilde\\Omega$ and \n$$\n\\widetilde \\beta(q,\\delta) = \\frac{y_1}{\\sqrt{a_1(y)^2 + b_1(y)^2}} \\left( a_1(y)^2 + b_1(y)^2 + \\frac{\\delta}{\\delta_0} \\Gamma(y)\\right),\n$$\nwhere by~(\\ref{relation}),\n$$\na_1(y)^2 + b_1(y)^2 + 2\\frac{\\delta}{\\delta_0} \\Gamma(y)\\ge 0,\\qquad y\\in\\widetilde\\Omega,\\quad \\delta\\in I.\n$$\nThen, for all $(q,p)$,\n$$\np\\cdot \\nabla_q \\widetilde\\beta(q,\\delta) = \\frac{p\\cdot\\nabla_q  y_1}{\\sqrt{a_1(y)^2 + b_1(y)^2}} \\left( a_1(y)^2 + b_1(y)^2 + \\frac{\\delta}{\\delta_0} \\Gamma(y)\\right)+ y_1p\\cdot \\gamma(y,\\delta)\n$$\nwhere the function $p\\cdot \\gamma(y,\\delta)$ is bounded for $\\delta \\in I$ and $(p,q)$ in any bounded set. \nSince $y_1=0$ is an equation of the hypersurface $S_0$ in $\\widetilde\\Omega$ and $p_0$ is transverse to $S_0$ at $q_0$, we have $p_0\\cdot\\nabla_q y_1(q_0)\\not=0$. Therefore, if necessary, we turn $y_1$ into $-y_1$, so that \n$$\np_0\\cdot\\nabla_q \\widetilde \\beta(q_0,\\delta)<0,\n$$\nfor all $\\delta\\in I$. Besides, by setting a bound on $y_1$ and $p$, we can find a $\\delta$-independent neighborhood $\\Omega_1\\subset\\widetilde\\Omega$ of $(q_0,p_0)$ such that \n$$\n\\forall \\delta\\in I\\,\\forall (q,p)\\in\\Omega: \\;p\\cdot\\nabla_q \\widetilde \\beta(q,\\delta)<0.\n$$\nThe invertibility of $p\\cdot\\nabla_q V_0(q,\\delta)$ in $I\\times\\Omega_1$ is then implied by\n\\[\n-\\det(p\\cdot\\nabla_q V_0(q,\\delta)) = (p\\cdot\\nabla_q\\widetilde\\beta(q,\\delta))^2 + \\delta^2 (p\\cdot\\nabla_q\\widetilde\\gamma(q))^2.\n\\]\n\\end{proof}\n\nWe now work in the set $\\Omega_1$ and denote by \n$$\n\\lambda^\\pm(q,\\delta) = \\alpha(q) \\pm\\sqrt{\\widetilde\\beta(q,\\delta)^2+\\delta^2\\widetilde\\gamma(q)^2}\n$$\nthe eigenvalues of the matrix $V(q,\\delta)$ and still denote by \n$$\n\\Phi_\\pm^t: {\\mathbb R}^{2d}\\to{\\mathbb R}^{2d},\\qquad \\Phi_\\pm^t(q_0,p_0)=\\left(q_\\delta^\\pm(t),p_\\delta^\\pm(t)\\right)\n$$ \nthe flow associated with the $\\delta$-dependent Hamiltonian system \n$$\n\\dot q_\\delta^\\pm(t)=p_\\delta^\\pm(t),\\;\\; \\dot p^\\pm_\\delta(t)=-\\nabla_q \\lambda^\\pm(q_\\delta^\\pm(t),\\delta),\n$$\nthat becomes singular on the hypersurface $S_0$ if $\\delta=0$. However, by the analysis of~\\cite[\\S3]{FG02} and \\cite[\\S2]{FG03}, one can pass through the \nsingularity in the following sense.  We denote by $H_{\\Lambda^\\pm}$ the Hamiltonian vector fields of \n$$\n\\Lambda^\\pm(q,p,\\delta)= \\tfrac{|p|^2}{2} + \\lambda^\\pm(q,\\delta)\n$$\nand consider $(q,p)\\in S_0\\times{\\mathbb R}^d$ with $p\\cdot \\nabla V_0(q,0)$ invertible and $\\delta=0$. Then, \n\\begin{eqnarray}\\label{def:H}\n\\lim_{t},{0^-} H_{\\Lambda^+} (\\Phi^t_+(q,p))& =&  \\lim_{t},{0^+} H_{\\Lambda^-} (\\Phi^t_-(q,p))\\\\\n\\nonumber\n&=& p\\cdot\\nabla_q-\\nabla_q \\alpha(q)\\cdot\\nabla_p-\\nabla_q\\widetilde\\beta(q,0)\\cdot\\nabla_p =:H, \\\\\n\\label{def:H'}\n\\lim_{t},{0^+} H_{\\Lambda^+} (\\Phi^t_+(q,p)) &=& \\lim_{t},{0^-} H_{\\Lambda^-} (\\Phi^t_-(q,p))\\\\\n\\nonumber\n&=& p\\cdot\\nabla_q-\\nabla_q \\alpha(q)\\cdot\\nabla_p+\\nabla_q\\widetilde\\beta(q,0)\\cdot\\nabla_p =:H'.\n\\end{eqnarray}\nMoreover,  the standard symplectic product of $H$ and $H'$ has a sign, since \n$$\n\\omega(H,H') = 2 p\\cdot \\nabla_q \\widetilde\\beta(q,0) <0\n$$\naccording to Lemma~\\ref{lem:transverse}. This continuation of the classical trajectories through the crossing at $S_0\\times\\{\\delta=0\\}$ will be a crucial element of our analysis. \n  \n\\begin{remark}\\label{rem:beta<0}\nFor $\\delta=0$, on ingoing trajectories, that is, on trajectories entering the conical crossing, we have \n$$\n{{\\rm d}\\over {\\rm d} t} g(q^\\pm_{\\delta=0}(t),0) = p_{\\delta=0}^\\pm(t)\\cdot \\nabla_q g(q_{\\delta=0}^\\pm(t),0)\\leq 0,\n$$\nwhich implies that they are included in the set\n$$\n\\left\\{\\left(p\\cdot\\nabla_q\\widetilde \\beta(q,0)\\right)\\widetilde \\beta(q,0)\\le 0\\right\\}\\subset\\left\\{\\widetilde\\beta(q,0)\\ge0\\right\\}.\n$$ \nSimilarly,  outgoing trajectories are included in $\\left\\{\\widetilde\\beta(q,0)\\le 0\\right\\}$. \n\\end{remark}\n  \n     \n\n\n\\subsection{The parametrized Schr\\\"odinger system}\\label{sec:delta}\n\nWe analyse the time-dependent Schr\\\"odinger systems\n$$\n \\left\\{\\begin{array}{l}\ni\\varepsilon\\partial_t\\psi^\\varepsilon_t = -\\frac{\\varepsilon^2}{2} \\Delta_q\\psi^\\varepsilon_t +V(q,\\delta)\\psi^\\varepsilon_t,\\;\\;(t,q)\\in{\\mathbb R}\\times{\\mathbb R}^d,\\\\\n\\psi^\\varepsilon_{t=0}=\\psi^\\varepsilon_0,\n\\end{array}\n\\right.\n$$\ndefined by the family of potential matrices $V(q,\\delta) = \\tfrac12 {\\rm tr}\\, V(q) + V_0(q,\\delta)$ with $\\delta\\in I$ of Theorem~\\ref{prop:parametrization} and equation~\\eqref{def:V}.\n\n\n\\medskip\nLiterally as in Section~\\ref{sec:result}, we construct a surface hopping semigroup $({\\mathcal L}^t_\\varepsilon)_{t\\ge0}$ for all $\\delta\\in I$, and thus obtain an effective dynamical description comprising both the original avoided crossing at $\\delta=\\delta_0$ and the  conical intersection at $\\delta=0$. On the one hand we use classical transport along the flows $\\Phi^t_\\pm:{\\mathbb R}^{2d}\\to{\\mathbb R}^{2d}$ of\n\\[\n\\dot q_\\delta^\\pm(t) = p_\\delta^\\pm(t),\\qquad \\dot p_\\delta^\\pm(t) = -\\nabla\\lambda^\\pm(q_\\delta^\\pm(t),\\delta). \n\\]\nOn the other hand we monitor the gap function along the classical trajectories and detect local minima by checking whether\n$$\n\\frac{{\\rm d}}{{\\rm d} t} g(q^\\pm_\\delta(t),\\delta) = p^\\pm_\\delta(t) \\cdot \\nabla_q g(q^\\pm_\\delta(t),\\delta) = 0.\n$$\nThe corresponding jump manifold reads\n$$\n\\Sigma_{\\varepsilon} = \\left\\{(q,p)\\in{\\mathbb R}^{2d}\\mid \n\\;g(q,\\delta)\\le R\\sqrt\\varepsilon\n ,\\;\\;p\\cdot\\nabla_q g(q,\\delta)=0\\right\\}.\n$$\nThe non-adiabatic transition probability for $(q^*,p^*)\\in\\Sigma_\\varepsilon$ is given by \n\\begin{equation}\\label{Tepsdelta}\nT_{\\varepsilon}(q^*,p^*,\\delta)=\\exp\\!\\left(-\\frac{\\pi}{4\\, \\varepsilon} \\frac{g(q^*,\\delta)^2}{|{\\rm det}(p^*\\cdot\\nabla_qV_0(q^*,\\delta))|^{1\/2}}\\right).\n\\end{equation}\nFinally, the diagonal parts of the Wigner transform $W^\\varepsilon(\\psi)$ of a wave function $\\psi\\in L^2({\\mathbb R}^d,{\\mathbb C}^2)$ are defined with respect to the eigenprojectors $\\Pi^\\pm(q,\\delta)$, that is, by\n$$\nw^\\varepsilon_\\pm(\\psi)(q,p,\\delta)={\\rm tr} \\left(\\Pi^\\pm(q,\\delta) W^\\varepsilon(\\psi)(q,p)\\right).\n$$ \nThe semigroup $({\\mathcal L}^t_\\varepsilon)_{t\\ge0}$ then acts on the function $w^\\varepsilon(\\psi)\\in{\\mathcal B}$ constructed from the diagonal components \n$w^\\varepsilon_\\pm(\\psi)$ according to relation~\\eqref{fpm}.\n\n\\medskip\nThe assumptions $(A0)$, $(A1)$, $(A2)$ of Theorem~\\ref{theorem} refer to the potential $V(q)=V(q,\\delta_0)$, and we denote by $(A0)_\\delta$, $(A1)_\\delta$, and $(A2)_\\delta$ the corresponding assumptions with respect to $V(q,\\delta)$. \nOur aim is to prove the following result:\n\n\\begin{theorem}\\label{theorem:delta-result}\nLet $V$ be a potential matrix with an avoided crossing in the sense of Definition~\\ref{def:avoided} and $V(\\cdot,\\delta)$, $\\delta\\in I$, the corresponding \nparametrized family of Theorem~\\ref{prop:parametrization}. We consider the time-dependent Schr\\\"odinger equation\n$$\ni\\varepsilon\\partial_t\\psi^\\varepsilon_t = -\\tfrac{\\varepsilon^2}{2} \\Delta_q\\psi^\\varepsilon_t +V(q,\\delta)\\psi^\\varepsilon_t,\\qquad \\psi^\\varepsilon_{t=0}=\\psi^\\varepsilon_0,\n$$\nand assume that $(A0)_\\delta$, $(A1)_\\delta$ and $(A2)_\\delta$ hold for all $\\delta\\in I$. Then, for all cut-off functions $\\chi\\in{\\mathcal C}^\\infty_c([0,T])$, \nthere exists a constant $C>0$ such that\n\\[\n\\left| \\int_0^T \\chi(t) \\left(w^\\varepsilon(\\psi^\\varepsilon_t)-{\\mathcal L^t_\\varepsilon w^\\varepsilon(\\psi^\\varepsilon_0)},a\\right) {\\rm d} t \\right| \\le C\\,\\varepsilon^{1\/32}.\n\\]\nThe constant $C$ depends on a finite number of upper bounds of derivatives of the smooth functions $\\alpha,\\beta,\\gamma$ defining the potential $V$ and $a,\\chi$ and of lower bounds of the determinant of the matrix $V_1$.\n\\end{theorem}\n\nThe particular choice $\\delta=\\delta_0$ then implies Theorem~\\ref{theorem}.\n\n\\begin{remark}\\label{rem:delta}\nBy the construction of Theorem~\\ref{prop:parametrization}, the gap function $g(q,\\delta)$ has $|\\delta|$ as its minimal value. Hence, if  $|\\delta|>R\\sqrt\\varepsilon$, then the jump manifold $\\Sigma_\\varepsilon$ is empty. In this situation the semigroup $(\\mathcal L^t_\\varepsilon)_{\\varepsilon>0}$ reduces to mere classical transport, that is proven in Appendix~\\ref{proof:prop}. \n\\end{remark}\n\n\\begin{remark}\nIf the off-diagonal function $\\widetilde\\gamma$ had derivatives uniformly bounded with respect to the gap parameter $\\delta_0$, then Theorem~\\ref{theorem:delta-result} would hold with an error of the order $\\varepsilon^{1\/8}$. We will indicate in Remark~\\ref{rem:1\/8},~\\ref{rem:1\/8bis} and~\\ref{rem:1\/8ter} how the analysis would simplify, if uniform estimates were available.\n\\end{remark}\n\n\n\n\n\n\n\n\\section{Reduction to a Landau--Zener model}\\label{sec:reduction}\n\nWe now focus on points in the minimal gap hypersurface $S_0$ and construct a  symplectic change of space-time phase space coordinates that allows an elementary microlocal normal form reduction to a Landau--Zener model in \\S\\ref{sec:normal}. Contrary to the approach in \\cite[\\S2.8]{CdV1}, the minimal gap size $\\delta$ is not treated as another coordinate but as a controlled parameter. \n\n\n\n\\subsection{The new symplectic coordinates}\\label{sec:coord}\nFollowing the ideas of~\\cite[\\S6.1]{FG02}, we now construct a symplectic coordinate transformation locally around points in the critical set\n\\[\nS = \\left\\{(q,t,p,\\tau)\\in{\\mathbb R}^{2d+2}, \\; q\\in S_0,\\; \\tau + \\tfrac12|p|^2 + \\alpha(q) = 0\\right\\},\n\\]\nthat restricts both the energy shells $E^+$ and $E^-$ to the conical crossing situation for $(q,\\delta)\\in S_0\\times\\{\\delta=0\\}$.\n\n\\begin{proposition}\\label{normalform}\nConsider $\\rho_0=(q_0,t_0,p_0,\\tau_0)\\in{\\mathbb R}^{2d+2}$ with $q_0\\in S_0$ and $p_0\\in{\\mathbb R}^d$ transverse to $S_0$ at $q_0$. There exists a \nneighborhood $\\Omega_2\\subset{\\mathbb R}^{2d+2}$ of the point $\\rho_0$ and for all $\\delta\\in I$ a positive function $\\lambda(\\cdot,\\delta):\\Omega_2\\to{\\mathbb R}$ such that\n$$\n\\sigma(\\rho,\\delta)=- \\lambda(\\rho,\\delta)\\left(\\tau+\\tfrac12|p|^2+\\alpha(q)\\right)\\;\\;{\\rm and} \\;\\; s(\\rho,\\delta)=\\lambda(\\rho,\\delta)\\widetilde\\beta(q,\\delta)\n$$\nsatisfy  \n$$\n\\{\\sigma(\\cdot,\\delta),s(\\cdot,\\delta)\\}=1\\;\\;\\text{on}\\;\\;\\Omega_2,\n$$\nand \n\\begin{equation}\\label{eq:lambda}\n\\lambda(\\rho,\\delta)^2 =-(p\\cdot\\nabla_q\\widetilde\\beta(q,\\delta))^{-1},\\qquad \\rho\\in S\\cap\\Omega_2.\n\\end{equation}\nMoreover, all derivatives of $\\lambda(\\cdot,\\delta)$ are uniformly bounded with respect to $\\delta\\in I$.\n\\end{proposition}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:transverse}, we have for $q\\in S_0$ and $p\\in{\\mathbb R}^d$ transverse to $S_0$ at $q$\n$$\n\\left\\{\\tau+\\tfrac12|p|^2+\\alpha(q),\\widetilde\\beta(q,\\delta)\\right\\}=p\\cdot\\nabla  \\widetilde\\beta(q,\\delta)<0.\n$$\nBy \\cite[Lemma~21.3.4]{Ho}, we can find $\\Omega_2$ and a positive function $\\lambda(\\cdot,\\delta)$ such that the functions \n$\\sigma(\\cdot,\\delta)$ and $s(\\cdot,\\delta)$ satisfy for all $\\delta\\in I$\n$$\n\\{\\sigma(\\cdot,\\delta),s(\\cdot,\\delta)\\}=1\\quad\\text{on}\\;\\;\\Omega_2.\n$$ \nIndeed, the proof of \\cite[Lemma 21.3.4]{Ho} relies on solving differential equations in the variable $(q,p)$, which requires to restrict the set $\\Omega_2$. When this is done with coefficients depending smoothly on $\\delta$, for $\\delta $ in the bounded interval $I$, the restriction can be taken uniformly in $\\Omega_2$. Therefore, the set $\\Omega_2$ does not depend on~$\\delta$. It remains to compute for $\\rho\\in S\\cap\\Omega_2$,\n$$\n1 = \\{\\sigma(\\rho,\\delta),s(\\rho,\\delta)\\} = -\\lambda(\\rho,\\delta)^2 \\; p\\cdot\\nabla_q  \\widetilde\\beta(q,\\delta),\n$$\nand to observe that the derivatives of $\\lambda(\\cdot,\\delta)$ inherit the boundedness of the derivatives of $\\widetilde\\beta(\\cdot,\\delta)$, \nsee Theorem~\\ref{prop:parametrization}.\n\\end{proof}\n\nWe will use this germ of symplectic coordinates and the rotation matrix $R(q)$ introduced in Theorem~\\ref{prop:parametrization} to construct a normal form. \nBy the Darboux Theorem (see \\cite[Theorem 21.1.6]{Ho}) close to  a point $\\rho_0=(q_0,t_0,p_0,\\tau_0)\\in{\\mathbb R}^{2d+2}$ with $q_0\\in S_0$ and $p_0$ transverse to $S_0$ at $q_0$, there exists a locally defined canonical transform\n $$\\kappa_\\delta:\\;(s,z,\\sigma,\\zeta)\\mapsto (q,t,p,\\tau)$$\nwith $s,\\sigma\\in{\\mathbb R}$ and $(z,\\zeta)\\in{\\mathbb R}^{2d}$, such that \n\\begin{equation}\\label{def:B}\n(RP  R^*)\\circ \\kappa_\\delta = \\frac{1}{\\lambda\\circ\\kappa_\\delta} \\left(-\\sigma +\n\\begin{pmatrix}\ns & \\delta \\check \\gamma\\\\\n\\delta\\check\\gamma & -s \n\\end{pmatrix}\\right),\\qquad \\check \\gamma =( \\lambda \\widetilde \\gamma)\\circ \\kappa_\\delta,\n\\end{equation}\nand the function $\\check\\gamma$ is nonzero everywhere.\n\n\\medskip\nThis local change of coordinates preserves the symplectic structure of the phase space ${\\mathbb R}^{2d+1}_{q,t}\\times{\\mathbb R}^{2d+1}_{p,\\tau}$: The variables $\\sigma$ and $\\zeta$ are the dual variables of $s$ and~$z$, respectively. Besides, in the new variables $(s,z,\\sigma,\\zeta)$, \nthe geometry of the conical crossing for $\\delta=0$ is simple, since  we have \n\\[\nE^\\pm=\\{-\\sigma\\pm \\sqrt {s^2+\\delta^2\\check\\gamma^2}=0\\},\\qquad S=\\{ s=0,\\;\\sigma=0\\}.\n\\]\nIn particular, by Remark~\\ref{rem:beta<0}, the ingoing and outgoing trajectories are included in the sets $\\{s\\ge0\\}$ and $\\{s\\le0\\}$, respectively. The off-diagonal function $\\check\\gamma$ satisfies additional properties that will be useful later on:\n\n\\begin{lemma}\\label{lem:gamma}\nLet $\\delta\\in I$ and $\\rho\\in S\\cap\\Omega_2$. Then, \n$$\n\\delta^2 (\\check\\gamma^2 \\circ\\kappa^{-1}_\\delta)(\\rho) = \\tfrac14 \\,g(q,\\delta)^2 \\left( |\\det p\\cdot \\nabla_q V_0(q,\\delta)|^{-1\/2}  + O(\\delta^2)\\right).\n$$\n\\end{lemma}\n\n\\begin{proof}\nFor $q\\in S_0$ we have $\\widetilde\\beta(q,\\delta)=0$ and \n$g(q,\\delta)^2= 4 \\delta^2 \\, \\widetilde \\gamma(q)^2$. We also observe that\n$$\np\\cdot\\nabla_q\\widetilde \\gamma(q) =  \\{\\tfrac{|p|^2}{2},\\widetilde\\gamma\\} = \n-\\tfrac{1}{\\lambda}\\{\\sigma,\\widetilde\\gamma\\} - \\sigma \\{\\tfrac{1}{\\lambda},\\widetilde\\gamma\\} = \n-\\tfrac{1}{\\lambda}\\partial_s(\\widetilde\\gamma\\circ\\kappa_\\delta) \\circ\\kappa_\\delta^{-1}\n$$\nfor $\\rho=(q,t,p,\\tau)\\in S$, and that $\\partial_s(\\widetilde\\gamma\\circ\\kappa_\\delta)$ is a derivative normal to $S_0=\\{s=0\\}$. \nTherefore, by Theorem~\\ref{prop:parametrization}, the product $p\\cdot\\nabla_q\\widetilde\\gamma(q)$ is bounded. \nUsing equation \\eqref{eq:lambda}, we then obtain\n\\begin{align*}\n\\delta^2 (\\check\\gamma^2 \\circ\\kappa^{-1}_\\delta)(\\rho) &= \\delta^2 \\lambda(\\rho,\\delta)^2 \\widetilde\\gamma(q)^2 = \n-\\tfrac14 g(q,\\delta)^2 (p\\cdot\\nabla_q \\widetilde\\beta(q,\\delta))^{-1}\\\\\n&=\n\\tfrac14 g(q,\\delta)^2 \\left(\\left((p\\cdot\\nabla_q \\widetilde\\beta(q,\\delta))^2 + \\delta^2(p\\cdot\\nabla_q\\widetilde\\gamma(q))^2\\right)^{-1\/2} + O(\\delta^2)\\right)\\\\\n&=\n\\tfrac14 \\,g(q,\\delta)^2 \\left( |\\det p\\cdot \\nabla_q V_0(q,\\delta)|^{-1\/2}  + O(\\delta^2)\\right).\n\\end{align*}\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{The quantization of the normal form}\\label{sec:normal}\n\nWe now lift the classical normal form~(\\ref{def:B}) to the quantum level using  Fourier integral operator theory. \nWe define the matrix-valued symbol \n$$\nB(q,t,p,\\tau,\\delta) = \\sqrt{\\lambda(q,t,p,\\tau,\\delta)} R(q)\n$$ \nfor $\\delta\\in I$ and $(q,t,p,\\tau)\\in\\Omega_2\\subset{\\mathbb R}^{2d+2}$. The following quantization process retains the minimal gap size $\\delta$ as a controlled parameter.\n\n\\begin{proposition}\\label{prop:quant}\nConsider $\\rho_0=(q_0,t_0,p_0,\\tau_0)\\in{\\mathbb R}^{2d+2}$ with $q_0\\in S_0$ and $p_0\\in{\\mathbb R}^d$ transverse to $S_0$ at $q_0$. \nThen there exist neighborhoods $\\widetilde I\\subset I$ of $\\delta=0$ and $\\Omega_3\\subset\\Omega_2$ of the point $\\rho_0$,  a matrix-valued function $B_\\varepsilon=B+\\varepsilon B_1$ defined on $\\Omega_3$, a canonical transform~$\\kappa_\\varepsilon$ which is a perturbation of order $\\varepsilon$ of the canonical transform $\\kappa_{\\delta}$, and a unitary operator $K_\\varepsilon$ of ${\\mathcal L}(L^2({\\mathbb R}^d))$ such that the transformed solution \n$$\nv^\\varepsilon=K_\\varepsilon^* \\,{\\rm op}_\\varepsilon(B^*_\\varepsilon)^{-1}\\psi^\\varepsilon_t\n$$\nsatisfies\n\\begin{equation}\\label{eq:systreduit}\n{\\rm op}_\\varepsilon(\\varphi)\\;{\\rm op}_\\varepsilon\\!\n\\begin{pmatrix}\n-\\sigma + s & \\delta \\check \\gamma \\\\\n\\delta\\check\\gamma& -\\sigma -s \n\\end{pmatrix}v^\\varepsilon  = O(\\varepsilon^2),\n\\end{equation}\nfor any compactly supported function $\\varphi\\in{\\mathcal C}^\\infty_c({\\mathbb R}^{2d+2})$. \n\\end{proposition}\n\nProposition~\\ref{prop:quant} allows us to microlocally trade the original Schr\\\"odinger equation ${\\rm op}_\\varepsilon(P)\\psi^\\varepsilon_t = 0$ for the reduced system~(\\ref{eq:systreduit}).  When saying that the canonical transform $\\kappa_\\varepsilon$ is  a perturbation of order $\\varepsilon$ of $\\kappa_\\delta$, we mean that $\\kappa_\\varepsilon$ is defined on a subset $\\Omega_3$ of the open set where $\\kappa_\\delta $ is defined and that for all \n$a\\in{\\mathcal C}_c^\\infty(\\kappa_\\varepsilon^{-1}(\\Omega_3))$, \n$a\\circ \\kappa_\\varepsilon =a\\circ\\kappa_\\delta +O(\\varepsilon)$ with respect to the semi-norms of the derivatives of $a$.\n\n\\medskip \n\nWe provide the proof of Proposition~\\ref{prop:quant} for the sake of completeness. It relies on  the Fourier integral operator construction of~\\cite[\\S2.2]{FG02}, which is based on Egorov's Theorem, and follows part of the schedule of Colin de Verdi\\`ere's normal form approach (see \\cite[Theorem~3]{CdV1} and also~\\cite[Theorem~1]{Fe06}).\n\n \n\\begin{proof}\nThe first step uses~\\cite[\\S2.2]{FG02}: There exists a Fourier integral operator $K_0$ associated with $\\kappa_\\delta$ such that \nfor any $a\\in{\\mathcal C}_c^\\infty(\\Omega_2, {\\mathbb C}^{2\\times 2})$,\n$$\nK_0^* {\\rm op}_\\varepsilon(a) K_0- {\\rm op}_\\varepsilon(a\\circ\\kappa_\\delta)  = O(\\varepsilon^2)\n$$\nin ${\\mathcal L}(L^2({\\mathbb R}^{d+1}))$, \nwhere the $O(\\varepsilon^2)$ contains semi-norms of derivatives of $a$.  We note that the the Fourier integral operator is a diagonal operator with the same scalar operator on each position of the diagonal. \n\n\\medskip\nThe second steps turns to the classical normal form \\eqref{def:B} to obtain \n\\[\nK_0^*{\\rm op}_\\varepsilon(B P B^*) K_0={\\rm op}_\\varepsilon\\!\\begin{pmatrix}-\\sigma+s & \\delta\\check\\gamma\\\\ \\delta \\check\\gamma & -\\sigma-s\\end{pmatrix}+ O(\\varepsilon^2).\n\\]\nThis suggests the change of unknown $\\psi^\\varepsilon \\mapsto K_0^*{\\rm op}_\\varepsilon(B^*)^{-1}\\psi^\\varepsilon$. However, this change of unknown generates unsatisfactory terms of order $\\varepsilon$, since we have \nby symbolic calculus\n$${\\rm op}_\\varepsilon(BPB^*)={\\rm op}_\\varepsilon(B){\\rm op}_\\varepsilon(P){\\rm op}_\\varepsilon(B^*) + \\varepsilon\\, {\\rm op}_\\varepsilon(R) +O(\\varepsilon ^2),$$\nwhere $R$ is the self-adjoint matrix defined by\n$$R= {1\\over 2i}( B\\{P,B^*\\} + \\{B,P\\}B^*).$$\nMore precisely, we have \n\\[\nK_0^*{\\rm op}_\\varepsilon(B){\\rm op}_\\varepsilon( P){\\rm op}_\\varepsilon( B^*) K_0={\\rm op}_\\varepsilon\\!\\begin{pmatrix}-\\sigma+s & \\delta\\check\\gamma\\\\ \\delta \\check\\gamma & -\\sigma-s\\end{pmatrix} +\\varepsilon\\,{\\rm op}_\\varepsilon(R\\circ\\kappa_\\delta)+ O(\\varepsilon^2).\n\\]\nWe note that here the function $\\delta\\check\\gamma$ is treated as a whole. Since it has bounded derivatives, the above remainder is uniform with respect to $\\delta$.\nNext we will remove the term of order $\\varepsilon$ by modifying $B$ and $\\kappa_\\delta$ at order~$\\varepsilon$, which will give the stated change of unknown $v^\\varepsilon=K_\\varepsilon^*{\\rm op}_\\varepsilon(B^*_\\varepsilon)^{-1} \\psi^\\varepsilon$. \n\n\\medskip\nIn the rest of this proof, and only here, $\\tau$ will be a parameter belonging to $[0,1]$. We define the canonical transform $\\kappa_1(\\tau)$ which is a perturbation of identity and solves the Hamiltonian equation \n$$\\frac{{\\rm d}}{{\\rm d}\\tau} \\kappa_1(\\tau ) =H_{1+\\varepsilon\\varphi} \\kappa_1(\\tau),\\;\\; \\kappa_1(0)={\\rm Id},$$\nwhere $\\varphi\\in{\\mathcal C}_c^\\infty({\\mathbb R}^{2d+2})$ is a smooth function that we shall fix later on.  The canonical transform $\\kappa_1$ is a perturbation of order $\\varepsilon$ of the identity, so that \n$$\\kappa_\\varepsilon :=\\kappa_\\delta\\circ\\kappa_1(1)$$\n is the sought-after perturbation of order $\\varepsilon$ of $\\kappa_\\delta$.\n We associate with $\\kappa_\\varepsilon(\\tau)$ a Fourier integral operator $K_\\varepsilon(\\tau)$ by setting \n $$i\\varepsilon \\frac{{\\rm d}}{{\\rm d}\\tau} K_\\varepsilon(\\tau) = {\\rm op}_\\varepsilon (1+\\varepsilon \\varphi) K_\\varepsilon (\\tau),\\;\\; K_\\varepsilon (0)={\\rm Id}.$$\n The solution of our problem will be \n $$K_\\varepsilon:=K_0\\circ K_\\varepsilon(1).$$\n Note that this construction method \\cite[\\S2.2]{FG02} has been used for $K_0$: Given a canonical transform, one links it to the identity in a differentiable way,\nthereby defining a function $\\varphi$ \nand the operator $K_\\varepsilon(\\tau)$ as a solution of a differential system. \n\n\\medskip\n We define the matrix $B_\\varepsilon(\\tau)=B+\\varepsilon \\tau B_1$ where $B_1$ will be fixed later, such\n $$B_\\varepsilon:=B_\\varepsilon(1),$$\n will be the solution of the Proposition.\n Let us now investigate how $B_1$ and $\\varphi$ have to be chosen, which might require to restrict to smaller neighbourhood of $\\delta=0$ and $\\rho_0$. \n For $\\tau\\in (0,1)$, we set \n $$L_\\varepsilon(\\tau):= K_\\varepsilon(\\tau)^* \n K_0^* \\left[{\\rm op}_\\varepsilon(B_\\varepsilon(\\tau)){\\rm op}_\\varepsilon (P) {\\rm op}_\\varepsilon (B_\\varepsilon(\\tau)^*)-\\varepsilon\\,(1-\\tau) {\\rm op}_\\varepsilon(R)\n \\right]K_0 K_\\varepsilon(\\tau).$$\n We have \n $$\n L_\\varepsilon(0)={\\rm op}_\\varepsilon\\!\\begin{pmatrix}-\\sigma+s & \\delta\\check\\gamma\\\\ \\delta \\check\\gamma & -\\sigma-s\\end{pmatrix} + O(\\varepsilon^2)\n $$\n  and we are going to prove that we can find $B_1$ and $\\varphi$ such that $\\frac{{\\rm d}}{{\\rm d}\\tau} L_\\varepsilon(\\tau)=O(\\varepsilon^2)$, so that we shall get $L_\\varepsilon(1)=L_\\varepsilon(0)+ O(\\varepsilon^2)$ which will conclude our proof. \n\n\\medskip\nA simple computation shows that \n \\begin{eqnarray*}\n\\frac{{\\rm d}}{{\\rm d}\\tau} L_\\varepsilon(\\tau) & = & \\varepsilon\\,  K_\\varepsilon(\\tau)^*  K_0^* \\Bigl[\n {\\rm op}_\\varepsilon(B_1P B_0^*+B_0 P B_1^*)+ {\\rm op}_\\varepsilon(R) \\\\ \n &&+ {1\\over i} \\left[{\\rm op}_\\varepsilon (\\varphi), {\\rm op}_\\varepsilon(B_0 PB_0^*)\\right]\n\\Bigr] K_0 K_\\varepsilon(\\tau)  +O(\\varepsilon^2)\\\\\n & = &  \\varepsilon \\, K_\\varepsilon(\\tau)^*  K_0^* {\\rm op}_\\varepsilon\\left[ B_1 PB_0^* + B_0 P B_1^* + R + \\{\\varphi, B_0 P B_0^*\\}\n\\right] K_0 K_\\varepsilon(\\tau)\\\\ \n&&  +\\,O(\\varepsilon^2) .\n \\end{eqnarray*}\nThe choice of $B_1$ and $\\varphi$ such that $B_1 PB_0^* + B_0 P B_1^* + R + \\{\\varphi, B_0 P B_0^*\\}=0$ is possible by~\\cite[Lemma~5]{CdV1}. \n\\end{proof}\n \n \n\n\n\\subsection{The off-diagonal components}\nOur final step towards the Landau--Zener model is to remove the dependence of  the off-diagonal function $\\check\\gamma(s,z,\\sigma,\\zeta,\\delta)$ on the coordinates $s$ and $\\sigma$, following the method proposed in~\\cite[Lemma~5 and 6]{FG03}, see also~\\cite[Proposition~8]{FG02}.\nFrom now on, we restrict ourselves to \n\\[\n0<\\delta\\le R\\sqrt\\varepsilon,\\qquad R=R(\\varepsilon) = \\varepsilon^{-1\/8},\n\\] \nsee also Remark~\\ref{rem:delta}. Moreover, \nsince the scattering result for the Landau--Zener system, that we use in Section~\\ref{sec:LZ_form}, has an error estimate of the order $R^2\\sqrt\\varepsilon\/s$, \nwe also start focusing on regions, where \n\\[\n\\tfrac12 r R^2\\sqrt\\varepsilon \\le |s| \\le r R^2\\sqrt\\varepsilon \n\\] \nfor some $1\\ll r \\le R$, that will be chosen as $r = r(\\varepsilon) = \\varepsilon^{-1\/32}$ later on. These choices of $r$ and $R$ imply that \n$\\delta\\le |s|$ as soon as $\\varepsilon^{5\/32}\\le 1\/2$.\n\n\n\\begin{lemma}\\label{rid_of_s_and_sigma}\nOn $\\kappa_\\delta(\\Omega_3)\\times\\widetilde I$, there\nexist matrix-valued functions $M^\\varepsilon(s,z,\\sigma,\\zeta,\\delta)$ and $\\widetilde M^\\varepsilon(s,z,\\sigma,\\zeta,\\delta)$, such that \n$$\nM^\\varepsilon = M_0^\\varepsilon + \\delta M_1^\\varepsilon,\\qquad \\widetilde M^\\varepsilon = \\widetilde M_0^\\varepsilon + \\delta \\widetilde M_1^\\varepsilon, \n$$\nwith $M_0^\\varepsilon$ and $\\widetilde M_0^\\varepsilon$ unitary matrices, and for all $\\varphi\\in\\mathcal C^\\infty_c({\\mathbb R}^{2d+2})$ supported in a set \nwith $s=O(r R^2\\sqrt\\varepsilon)$, with $1\\ll r\\leq R$,  one has\n\\begin{align*}\n& {\\rm op}_\\varepsilon(\\varphi)\\,\n{\\rm op}_\\varepsilon(\\widetilde M^\\varepsilon)\\,  {\\rm op}_\\varepsilon\\!\\begin{pmatrix}-\\sigma+ s & \\delta \\check \\gamma \\\\\\delta\\check\\gamma& -\\sigma-s \n\\end{pmatrix} =\\\\\n&\\qquad {\\rm op}_\\varepsilon(\\varphi)\\, {\\rm op}_\\varepsilon\\!\\begin{pmatrix}\n-\\sigma + s & \\delta \\check \\gamma_0 \\\\\n\\delta\\check\\gamma_0& -\\sigma -s \n\\end{pmatrix} {\\rm op}_\\varepsilon(M^\\varepsilon) + O(r^2\\varepsilon^{7\/8})\n\\end{align*}\nin ${\\mathcal L}(L^2({\\mathbb R}^{d+1}))$, where \n$$\n\\check\\gamma_0(z,\\zeta,\\delta) = \\check\\gamma(0,z,0,\\zeta,\\delta).\n$$ \nMoreover, the families $({\\rm op}_\\varepsilon(\\varphi)\\, {\\rm op}_\\varepsilon(M^\\varepsilon))_{\\varepsilon,\\delta>0}$ and $({\\rm op}_\\varepsilon(\\varphi)\\, {\\rm op}_\\varepsilon(\\widetilde M^\\varepsilon))_{\\varepsilon,\\delta>0}$ are uniformly bounded in ${\\mathcal L}(L^2({\\mathbb R}^{d+1}))$.\n\\end{lemma}\n\n\\begin{remark}\\label{rem:1\/8}\nThe proof below shows that if $\\check\\gamma(\\cdot,\\delta)$ had bounded derivatives uniformly with respect to $\\delta$, then Lemma~\\ref{rid_of_s_and_sigma} would hold with a remainder estimate of the order $\\varepsilon\\delta$.\n\\end{remark}\n\n\\begin{remark}\\label{rem:CVeps}\nTo estimate the norm of operators such as ${\\rm op}_\\varepsilon(\\check\\gamma)$, \n we shall use  the scaling operator~$T_\\varepsilon$ defined by \n$$\\forall f\\in L^2({\\mathbb R}^{d+1}),\\;\\; T_\\varepsilon f(u)=\\varepsilon^{d+1\\over 4} f(\\sqrt\\varepsilon u).$$\nThis unitary operator is such that \n$$\\forall a\\in{\\mathcal C}_0^\\infty({\\mathbb R}^{2d+2}),\\;\\; T_\\varepsilon{\\rm op}_\\varepsilon(a) T_\\varepsilon^* ={\\rm op}_1(a(\\sqrt\\varepsilon\\cdot,\\sqrt\\varepsilon\\cdot)).$$\nThe Calder\\'on--Vaillancourt theorem yields the existence of $N\\in{\\mathbb N}$ and $C_N>0$ such that\n$$\\forall a\\in{\\mathcal C}_c^\\infty({\\mathbb R}^{2d+2}),\\;\\; \\| {\\rm op}_\\varepsilon(a)\\| _{{\\mathcal L}(L^2({\\mathbb R}^{d+1}))}\\leq C_N \\, \\sup_{\\beta\\in{\\mathbb N}^{2d+2},\\;\\;|\\beta|\\leq N}\\,\\sup_{\\rho\\in{\\mathbb R}^{2d+2}} \\left(\\varepsilon^{|\\beta|\\over 2} \\left| \\partial_\\rho ^\\beta a(\\rho) \\right| \\right)$$\nholds. As a consequence,  Theorem~\\ref{prop:parametrization} implies for all $\\varphi\\in\\mathcal C^\\infty_c({\\mathbb R}^{2d+2})$\n$$\\|{\\rm op}_\\varepsilon (\\varphi) {\\rm op}_\\varepsilon( \\check\\gamma)\\|_{{\\mathcal L}(L^2({\\mathbb R}^{d+1}))}\\leq C (1 +\\sqrt\\varepsilon\\,  \\delta^{-1}\\sup_{s\\in{\\rm supp}(\\varphi)} |s|).$$\n\\end{remark}\n\n\n\n\\begin{proof}\nWe consider the three matrices \n$$J=\\begin{pmatrix} 1 & 0 \\\\ 0 & -1\\end{pmatrix},\\;\\;\nK=\\begin{pmatrix} 0 & 1 \\\\ -1 & 0\\end{pmatrix},\\;\\;\nL=\\begin{pmatrix} 0 & 1 \\\\ 1 & 0\\end{pmatrix},$$\nthat satisfy\n$$\nJL = K = -LJ,\\qquad JK = L = -KJ,\\qquad J^2 = {\\rm Id}.\n$$\nWe write\n$$\n\\begin{pmatrix}\n-\\sigma + s & \\delta\\check\\gamma\\\\ \\delta\\check\\gamma & -\\sigma -s \n\\end{pmatrix} = -\\sigma {\\rm Id} + s J + \\delta\\check\\gamma L\n$$\nand proceed in two steps.\n\n\\medskip \n\nWe first remove the $s$-dependence of $\\check\\gamma$ and construct matrix-valued functions $D_j^\\varepsilon=D_j^\\varepsilon(s,z,\\sigma,\\zeta,\\delta)$, $j=0,1$, with\n\\begin{align*}\n& {\\rm op}_\\varepsilon(D_0^\\varepsilon+\\delta D_1^\\varepsilon)\\,{\\rm op}_\\varepsilon(-\\sigma{\\rm Id} + s J + \\delta \\check\\gamma L)\\\\\n&\\qquad = {\\rm op}_\\varepsilon(-\\sigma{\\rm Id} + s J + \\delta \\check\\gamma_* L) \\,{\\rm op}_\\varepsilon(D_0^\\varepsilon + \\delta D_1^\\varepsilon) + o(\\varepsilon),\n\\end{align*}\nwhere \n$$\n\\check\\gamma_*(z,\\sigma,z,\\delta) = \\check\\gamma(0,z,\\sigma,\\zeta,\\delta).\n$$\nSymbolic calculus provides that the above equation is equivalent to\n\\begin{align*}\n& {\\rm op}_\\varepsilon\\!\\left( (D_0^\\varepsilon+\\delta D_1^\\varepsilon)(-\\sigma{\\rm Id} + s J + \\delta \\check\\gamma L)\\right)\n+ \\frac{\\varepsilon}{2i}{\\rm op}_\\varepsilon\\!\\left( \\{ D_0^\\varepsilon+\\delta D_1^\\varepsilon,-\\sigma{\\rm Id} + s J + \\delta \\check\\gamma L\\} \\right)\\\\\n&= {\\rm op}_\\varepsilon\\!\\left( (-\\sigma{\\rm Id} + s J + \\delta \\check\\gamma_* L) (D_0^\\varepsilon + \\delta D_1^\\varepsilon)\\right)\n+ \\frac{\\varepsilon}{2i} {\\rm op}_\\varepsilon\\!\\left( \\{-\\sigma{\\rm Id} + s J + \\delta \\check\\gamma_* L,D_0^\\varepsilon + \\delta D_1^\\varepsilon\\} \\right)\\\\\n&\\qquad + {\\rm op}_\\varepsilon(\\rho^\\varepsilon).\n\\end{align*}\nwhere the remainder symbol $\\rho^\\varepsilon$ consists of second order derivatives terms times a factor $\\varepsilon^2$. \nSince any derivatives of $\\delta\\check\\gamma$ and $\\delta\\check\\gamma_*$ are of the order $|s|$, \nRemark~\\ref{rem:CVeps} yields that\n\\[\n{\\rm op}_\\varepsilon(\\varphi) {\\rm op}_\\varepsilon(\\rho^\\varepsilon) = O(|s|\\cdot |s|^2 \\delta^2) = O(r^3 \\,\\varepsilon^{3\/2}),\n\\]\nif the a priori estimate \n\\begin{equation}\\label{apriori}\n\\partial^{\\alpha}(D_0^\\varepsilon + \\delta D_1^\\varepsilon) = O((|s|\\delta\/\\varepsilon)^{|\\alpha|}),\\qquad \\alpha\\in{\\mathbb N}^{2d+2},\n\\end{equation}\nholds, that we will justify later during the proof. We neglect the term\n\\begin{align*}\n&\\frac{\\varepsilon}{2i} \\{D^\\varepsilon_0,\\delta\\check\\gamma L\\} + \\frac{\\varepsilon}{2i} \\{\\delta D^\\varepsilon_1,-\\sigma {\\rm Id} + sJ + \\delta \\check\\gamma L\\} \\\\\n&-\\frac{\\varepsilon}{2i} \\{\\delta\\check\\gamma_* L,D^\\varepsilon_0\\} - \\frac{\\varepsilon}{2i} \\{-\\sigma {\\rm Id} + sJ + \\delta \\check\\gamma_* L,\\delta D^\\varepsilon_1\\}, \n\\end{align*}\nwhich by the same argument produces an error of the order\n\\[\nO(|s|\\cdot|s|\\delta) = O(r^2 \\varepsilon^{7\/8})\n\\]\nin the region of observation. Then, we obtain the three relations\n\\begin{align*}\n& D_0^\\varepsilon(-\\sigma{\\rm Id} + sJ) = (-\\sigma{\\rm Id} + sJ) D_0^\\varepsilon,\\\\\n& D_1^\\varepsilon(-\\sigma{\\rm Id} + sJ) +\\check\\gamma D_0^\\varepsilon L = (-\\sigma{\\rm Id} + sJ) D_1^\\varepsilon + \\check\\gamma_* L D_0^\\varepsilon,\\\\\n& \\check\\gamma D_1^\\varepsilon L + \\frac{\\varepsilon}{2i\\delta^2}\\{D_0^\\varepsilon,-\\sigma {\\rm Id} + sJ \\} = \n\\check\\gamma_* L D_1^\\varepsilon + \\frac{\\varepsilon}{2i\\delta^2}\\{-\\sigma {\\rm Id} + sJ, D_0^\\varepsilon\\}.\n\\end{align*}\nWe make the ansatz\n$$\nD_0^\\varepsilon = \\begin{pmatrix}\\widetilde a_0^\\varepsilon & 0\\\\ 0& \\widetilde d_0^\\varepsilon\\end{pmatrix},\\qquad\nD_1^\\varepsilon = \\begin{pmatrix}0 & \\widetilde b_1^\\varepsilon\\\\ \\widetilde c_1^\\varepsilon & 0\\end{pmatrix},\n$$\nand rewrite the second of the three relations as\n\\begin{align*}\ns\\begin{pmatrix}0 & -2\\widetilde b_1^\\varepsilon\\\\ 2\\widetilde c_1^\\varepsilon & 0\\end{pmatrix} = \n\\begin{pmatrix}0 & -\\check\\gamma \\widetilde a_0^\\varepsilon+ \\check\\gamma_*\\widetilde d_0^\\varepsilon\\\\ \n-\\check\\gamma \\widetilde d_0^\\varepsilon+ \\check\\gamma_* \\widetilde a_0^\\varepsilon & 0\\end{pmatrix}.\n\\end{align*}\nThis requires \n$$\n\\widetilde a_0^\\varepsilon(0,z,\\zeta,\\delta) = \\widetilde d_0^\\varepsilon(0,z,\\zeta,\\delta)\n$$ \nand \n$$\n\\widetilde b_1^\\varepsilon = \\frac{1}{2s}(\\check\\gamma\\widetilde a_0^\\varepsilon - \\check\\gamma_*\\widetilde d_0^\\varepsilon),\\qquad\n\\widetilde c_1^\\varepsilon = \\frac{1}{2s}(\\check\\gamma_*\\widetilde a_0^\\varepsilon - \\check\\gamma\\widetilde d_0^\\varepsilon).\n$$\nThe third relation can be rewritten as\n$$\n\\begin{pmatrix}\\check\\gamma \\widetilde b_1^\\varepsilon -\\check\\gamma_* \\widetilde c_1^\\varepsilon & 0 \\\\ 0 & \\check\\gamma \\widetilde c_1^\\varepsilon -  \\check\\gamma_* \\widetilde b_1^\\varepsilon\\end{pmatrix} = \\frac{\\varepsilon}{i\\delta^2}\\begin{pmatrix}  -\\partial_s\\widetilde a_0^\\varepsilon -\\partial_\\sigma \\widetilde a_0^\\varepsilon & 0 \\\\ 0 & \n-\\partial_s\\widetilde d_0^\\varepsilon + \\partial_\\sigma \\widetilde d_0^\\varepsilon\\end{pmatrix}.\n$$\nWe define \n\\begin{align*}\n\\widetilde\\vartheta^\\varepsilon(s,z,\\sigma,\\zeta,\\delta) &= \\frac{i\\delta^2}{2\\varepsilon s}\n\\left(\\check\\gamma_*^2(z,\\sigma,\\zeta,\\delta) - \\check\\gamma^2(s,z,\\sigma,\\zeta,\\delta)\\right)\\\\\n& =- \\frac{i\\delta^2}{2\\varepsilon }\\int_0^1 \\partial_s (\\check \\gamma^2) (sr,z,\\sigma,\\zeta,\\delta) dr\n\\end{align*}\nand observe that all $s$-derivatives of $\\widetilde\\vartheta^\\varepsilon$ are of the order $\\delta^2\/\\varepsilon$, while any derivative with respect to $(z,\\sigma,\\zeta)$ is of the order $\\delta\/\\varepsilon$ in view of Theorem~\\ref{prop:parametrization}. We\nobtain the equations\n$$\n(\\partial_s +\\partial_\\sigma) \\widetilde a_0^\\varepsilon = \\widetilde\\vartheta^\\varepsilon \\, \\widetilde a_0^\\varepsilon,\\qquad\n(\\partial_s-\\partial_\\sigma) \\widetilde d_0^\\varepsilon = -\\widetilde\\vartheta^\\varepsilon \\, \\widetilde d_0^\\varepsilon,\n$$\nthat can be solved by\n\\begin{align*}\n\\widetilde a_0^\\varepsilon(s,z,\\sigma,\\zeta,\\delta) &= \\exp\\!\\left(\\int_0^s \\widetilde\\vartheta^\\varepsilon(\\tau,z,\\sigma-s+\\tau,\\zeta,\\delta) {\\rm d}\\tau\\right),\\\\\n\\widetilde d_0^\\varepsilon(s,z,\\sigma,\\zeta,\\delta) &= \\exp\\!\\left(-\\int_0^s \\widetilde\\vartheta^\\varepsilon(\\tau,z,\\sigma+s-\\tau,\\zeta,\\delta) {\\rm d}\\tau\\right)\n\\end{align*}\nsuch that $\\widetilde a_0^\\varepsilon(0,z,\\sigma,\\zeta,\\delta) = \\widetilde d_0^\\varepsilon(0,z,\\sigma,\\zeta,\\delta) = 1$.\nWe observe that \n\\[\n\\partial^\\alpha_{z,\\sigma,\\zeta} \\,\\widetilde a^\\varepsilon_0,\\, \\partial^\\alpha_{z,\\sigma,\\zeta}\\, \\widetilde d^\\varepsilon_0 = O( (|s|\\delta\/\\varepsilon)^{|\\alpha|})\n\\]\nfor any $\\alpha\\in{\\mathbb N}^{2d+1}$, while the $s$-derivatives satisfy\n\\[\n\\partial_s^k \\,\\widetilde a^\\varepsilon_0, \\, \\partial^k_s \\,\\widetilde d^\\varepsilon_0 = O(\\delta^2\/\\varepsilon) + O(|s|\\delta\/\\varepsilon)\n\\]\nfor any $k\\ge1$. We now write\n\\[\n\\delta \\widetilde b^\\varepsilon_1 = \\frac{\\delta}{2} \\,\\widetilde a^\\varepsilon_0 \\int_0^1 \\partial_s (\\check \\gamma) (sr,z,\\sigma,\\zeta,\\delta) dr  \n+ \\frac{\\delta}{2s}\\gamma_*(\\widetilde a^\\varepsilon_0 - d^\\varepsilon_0)\n\\]\nand derive a similar expression for $\\delta\\widetilde c^\\varepsilon_1$, such that \n\\[\n\\partial^\\beta( \\delta D^\\varepsilon_1) = O((|s|\\delta\/\\varepsilon)^{|\\beta|}) + O((\\delta^2\/\\varepsilon)^{|\\beta|})\n\\]\nfor all $\\beta\\in{\\mathbb N}^{2d+2}$.This implies the claimed a priori estimate \\eqref{apriori}.\n\n\n\\medskip\nWe now remove the $\\sigma$-dependence of the scalar function $\\check\\gamma_*$, taking advantage of the boundedness of any derivatives of the function $\\check\\gamma_* = \\check\\gamma \\mid_{\\{s=0\\}}$, see Theorem~\\ref{prop:parametrization}. We look for two matrix-valued functions \n$C_j ^\\varepsilon= C_j^\\varepsilon(\\sigma,z,\\zeta,\\delta)$, $j=0,1$, with the following properties. First, they satisfy the intertwining relation\n\\begin{align*}\n&{\\rm op}_\\varepsilon(J(C_0^\\varepsilon + \\delta C_1^\\varepsilon) J)\\;{\\rm op}_\\varepsilon(-\\sigma{\\rm Id} + s J + \\delta \\check\\gamma L)\\\\ \n&\\qquad = {\\rm op}_\\varepsilon(-\\sigma{\\rm Id} + sJ + \\delta \\check\\gamma_* L) \\;{\\rm op}_\\varepsilon(C_0^\\varepsilon + \\delta C_1^\\varepsilon) + o(\\varepsilon)\n\\end{align*}\nin $\\mathcal{L}(L^2({\\mathbb R}^{d+1}))$, \nwhere \n$$\n\\check\\gamma_0(s,z,\\zeta,\\delta) = \\check\\gamma_*(z,0,\\zeta,\\delta) = \\check\\gamma(0,z,0,\\zeta,\\delta).\n$$\nThis relation is equivalent to \n\\begin{align*}\n&{\\rm op}_\\varepsilon(C_0^\\varepsilon + \\delta C_1^\\varepsilon)\\;{\\rm op}_\\varepsilon(-\\sigma J + s {\\rm Id} + \\delta \\check\\gamma_* K)\\\\\n&\\qquad = {\\rm op}_\\varepsilon(-\\sigma J + s{\\rm Id} + \\delta \\check\\gamma_0 K) \\;{\\rm op}_\\varepsilon(C_0^\\varepsilon + \\delta C_1^\\varepsilon) + O(\\varepsilon^2\\delta),\n\\end{align*}\nusing the growth bound\n$$\n\\forall\\alpha\\in{\\mathbb N}^{2d+2}\\,  \\exists c_\\alpha>0\\, \\forall \\varepsilon,\\delta>0:  \\|\\partial^\\alpha C_j^\\varepsilon(\\cdot,\\delta)\\|_\\infty < c_\\alpha\\, (\\delta^2\/\\varepsilon)^{|\\alpha|}.\n$$\nSymbolic calculus yields\n\\begin{align*}\n&{\\rm op}_\\varepsilon\\!\\left( (C_0^\\varepsilon + \\delta C_1^\\varepsilon)(-\\sigma J + s {\\rm Id} + \\delta \\check\\gamma_* K)\\right)\n+ \\frac{\\varepsilon}{2i}\\,{\\rm op}_\\varepsilon\\!\\left(\\{ C_0^\\varepsilon + \\delta C_1^\\varepsilon, -\\sigma J + s {\\rm Id} + \\delta \\check\\gamma_* K\\}  \\right)\\\\\n& = {\\rm op}_\\varepsilon\\!\\left((-\\sigma J + s{\\rm Id} + \\delta \\check\\gamma_0 K)(C_0^\\varepsilon + \\delta C_1^\\varepsilon)\\right)\n+ \\frac{\\varepsilon}{2i} \\,{\\rm op}_\\varepsilon\\!\\left(\\{-\\sigma J + s{\\rm Id} + \\delta \\check\\gamma_0 K,C_0^\\varepsilon + \\delta C_1^\\varepsilon\\} \\right)\\\\\n& \\qquad + O(\\varepsilon^2\\delta),\n\\end{align*}\nwhere the neglected terms of the form $\\varepsilon^2$ times derivatives of the order $\\ge 2$ define the $O(\\varepsilon^2\\delta)$ remainder.\nWe now sort in powers of $\\delta$ and obtain the following three relations, \n\\begin{align}\\label{eq:delta0}\n& C_0^\\varepsilon(-\\sigma J + s{\\rm Id}) = (-\\sigma J+s{\\rm Id}) C_0^\\varepsilon,\\\\\\label{eq:delta1}\n& C_1^\\varepsilon(-\\sigma J + s{\\rm Id}) +\\check\\gamma_* C_0^\\varepsilon K = (-\\sigma J + s{\\rm Id}) C_1^\\varepsilon + \\check\\gamma_0KC_0^\\varepsilon,\\\\\\label{eq:delta2}\n& \\check\\gamma_* C_1^\\varepsilon K + \\frac{\\varepsilon}{2i \\delta^2}  \\{C_0^\\varepsilon,-\\sigma J + s{\\rm Id}\\} = \n\\check\\gamma_0 KC_1^\\varepsilon + \\frac{\\varepsilon}{2i\\delta^2} \\{-\\sigma J + s{\\rm Id}, C_0^\\varepsilon\\}.\n\\end{align}\nWe denote the components of the two matrices  by\n$$\nC_0^\\varepsilon = \\begin{pmatrix}a_0^\\varepsilon & b_0^\\varepsilon\\\\ c_0^\\varepsilon & d_0^\\varepsilon\\end{pmatrix},\\qquad\nC_1^\\varepsilon = \\begin{pmatrix}a_1^\\varepsilon & b_1^\\varepsilon\\\\ c_1^\\varepsilon & d_1^\\varepsilon\\end{pmatrix}.\n$$\nThe first relation \\eqref{eq:delta0} is equivalent to\n$[J,C_0^\\varepsilon] = 0$, that is, \n$$\nb_0^\\varepsilon = c_0^\\varepsilon = 0.\n$$\nThe second relation \\eqref{eq:delta1} is equivalent to $-\\sigma [C_1^\\varepsilon,J]  = -\\check\\gamma_* C_0^\\varepsilon K + \\check\\gamma_0 K C_0^\\varepsilon$, that is, \n$$\n\\sigma \\begin{pmatrix} 0 & 2b_1^\\varepsilon\\\\ -2c_1^\\varepsilon & 0\\end{pmatrix} = \n\\begin{pmatrix} 0 & -\\check\\gamma_* a_0^\\varepsilon +\\check\\gamma_0 d_0^\\varepsilon\\\\ \\check\\gamma_* d_0^\\varepsilon -\\check\\gamma_0 a_0^\\varepsilon& 0\\end{pmatrix}.\n$$\nThis requires\n$$\na_0^\\varepsilon(z,0,\\zeta,\\delta) = d_0^\\varepsilon(z,0,\\zeta,\\delta)\n$$\nand\n$$\nb_1^\\varepsilon = \\frac{1}{2\\sigma}\\left(-\\check\\gamma_* a_0^\\varepsilon +\\check\\gamma_0 d_0^\\varepsilon\\right),\\qquad\nc_1^\\varepsilon = \\frac{1}{2\\sigma}\\left(- \\check\\gamma_* d_0^\\varepsilon + \\check\\gamma_0 a_0^\\varepsilon\\right).\n$$\nThe third relation \\eqref{eq:delta2} is equivalent to\n$$\n\\check\\gamma_* C_1^\\varepsilon K - \\check\\gamma_0KC_1^\\varepsilon = \n\\frac{\\varepsilon}{i\\delta^2}\\left(-\\partial_\\sigma C_0^\\varepsilon - \\tfrac12 (J\\partial_s C_0^\\varepsilon + \\partial_s C_0^\\varepsilon J)\\right), \n$$\nthat is,\n$$\n\\check\\gamma_* \\begin{pmatrix} -b_1^\\varepsilon & a_1^\\varepsilon\\\\ -d_1^\\varepsilon & c_1^\\varepsilon\\end{pmatrix} \n- \\check\\gamma_0 \\begin{pmatrix} c_1^\\varepsilon & d_1^\\varepsilon\\\\ -a_1^\\varepsilon & -b_1^\\varepsilon\\end{pmatrix}\n= \\frac{\\varepsilon}{i\\delta^2}\\begin{pmatrix}-\\partial_\\sigma a_0^\\varepsilon  & 0\\\\ 0 & -\\partial_\\sigma d_0^\\varepsilon\\end{pmatrix}.\n$$\nThis can be satisfied by \n$$\na_1^\\varepsilon=d_1^\\varepsilon=0,\n$$ \nand requires  \n\\begin{align*}\n\\frac{a_0^\\varepsilon}{2\\sigma} \\left(\\check\\gamma_*^2-\\check\\gamma_0^2\\right) &= -\\frac{\\varepsilon}{i\\delta^2} \\partial_\\sigma a_0^\\varepsilon ,\\\\\n\\frac{d_0^\\varepsilon}{2\\sigma} \\left(\\check\\gamma_0^2-\\check\\gamma_*^2\\right) &= -\\frac{\\varepsilon}{i\\delta^2} \\partial_\\sigma d_0^\\varepsilon.\n\\end{align*}\nWe set\n$$\n\\vartheta^\\varepsilon(z,\\sigma,\\zeta,\\delta) = \\frac{i\\delta^2}{2\\varepsilon \\sigma} \\left(\\check\\gamma_0^2(z,\\zeta,\\delta)-\\check\\gamma_*^2(z,\\sigma,\\zeta,\\delta)\\right)\n$$\nand rewrite the above equations as\n$$\n\\vartheta^\\varepsilon a_0^\\varepsilon = \\partial_\\sigma a_0^\\varepsilon,\\qquad \\vartheta^\\varepsilon d_0^\\varepsilon = -\\partial_\\sigma d_0^\\varepsilon.\n$$\nThe functions\n\\begin{align*}\na_0^\\varepsilon(z,\\sigma,\\zeta,\\delta) &= \\exp\\!\\left(\\int_0^{\\sigma} \\vartheta^\\varepsilon(z,\\tau,\\zeta,\\delta)\\, {\\rm d}\\tau\\right),\\\\\nd_0^\\varepsilon(z,\\sigma,\\zeta,\\delta) &= \\exp\\!\\left(\\int_0^{\\sigma} \\vartheta^\\varepsilon(z,\\tau,\\zeta,\\delta)\\, {\\rm d}\\tau\\right)\n\\end{align*}\nsolve these equations and satisfy \n$$\na_0^\\varepsilon(z,0,\\zeta,\\delta) = d_0^\\varepsilon(z,0,\\zeta,\\delta) = 1.\n$$ \nWe conclude that the constructed matrices $C_0^\\varepsilon$ and $C_1^\\varepsilon$ have the desired properties. \n\\end{proof}\n\n\n\\subsection{Arriving at the Landau--Zener model}\\label{sec:arrival}\nWe now use Proposition~\\ref{prop:quant} and Lemma~\\ref{rid_of_s_and_sigma} to introduce\n$$\n\\widetilde v^\\varepsilon = {\\rm op}_\\varepsilon(M^\\varepsilon) v^\\varepsilon\\quad\\text{with}\\quad M^\\varepsilon = M_0^\\varepsilon + \\delta M_1^\\varepsilon.\n$$\nSince\n\\begin{align*}\n{\\rm op}_\\varepsilon\\!\\begin{pmatrix}-\\sigma+s & \\delta\\check\\gamma_0\\\\ \\delta\\check\\gamma_0 & -\\sigma -s\\end{pmatrix}\\widetilde v^\\varepsilon = \n{\\rm op}_\\varepsilon(\\widetilde M^\\varepsilon) \\!\\begin{pmatrix}-\\sigma+s & \\delta\\check\\gamma\\\\ \\delta\\check\\gamma & -\\sigma -s\\end{pmatrix}v^\\varepsilon + O(r^2\\varepsilon ^{7\/8}),\n\\end{align*}\nwe obtain for all $\\varphi\\in{\\mathcal C}^\\infty_c({\\mathbb R}^{2d+2})$ the doubly reduced system\n\\begin{equation}\\label{eq:L2sz}\n{\\rm op}_\\varepsilon(\\varphi) \\;{\\rm op}_\\varepsilon\\!\\begin{pmatrix}-\\sigma+s & \\delta\\check\\gamma_0\\\\ \\delta\\check\\gamma_0 & -\\sigma -s\\end{pmatrix}\\widetilde v^\\varepsilon =\nO(r^2\\varepsilon ^{7\/8}) \\;\\;\\text{in}\\;\\;L^2({\\mathbb R}^{d+1}).\n\\end{equation}\nThe estimate of \\cite[Proposition~7]{FG02} also implies, that $({\\rm op}_\\varepsilon(\\varphi)\\widetilde v^\\varepsilon)_{\\varepsilon>0}$ is a bounded sequence in \n$L^\\infty({\\mathbb R}_s,L^2({\\mathbb R}^d_z))$, and we compare the new function $\\widetilde v^\\varepsilon$ with the solution~$\\check v^\\varepsilon$ of the Landau--Zener type system \n\\begin{equation}\\label{eq:systreduitbis}\n{\\varepsilon\\over i} \\partial_s \\check v^\\varepsilon = {\\rm op}_\\varepsilon\n\\begin{pmatrix}\ns & \\delta \\check \\gamma_0\\\\\n\\delta\\check\\gamma_0 & -s \n\\end{pmatrix}\\check v^\\varepsilon,\\qquad\\check v^\\varepsilon|_{s=0}\\,=\\,\\widetilde v^\\varepsilon|_{s=0}.\n\\end{equation}\nThe order $r^2\\varepsilon ^{7\/8}$ right hand side of the doubly reduced system \\eqref{eq:L2sz} is small enough to be treated as a perturbation \nand we obtain a positive constant $C>0$ such that for all $s\\in{\\mathbb R}$ and $\\varepsilon>0$,\n\\begin{equation}\\label{eq:comparaison}\n\\left\\|\\widetilde v^\\varepsilon(s) - \\check v^\\varepsilon(s)\\right\\|_{L^2({\\mathbb R}^d_z)} \\le C |s| r^2\\varepsilon ^{-1\/8}.\n\\end{equation}\nThis implies that for all $\\varphi\\in \\mathcal C_c^\\infty({\\mathbb R}^{2d+2})$ that are supported in a region, where $s\\sim r R^2\\sqrt\\varepsilon = r\\varepsilon^{1\/4}$, we obtain\n\\begin{equation}\\label{eq:error}\n{\\rm op}_\\varepsilon(\\varphi)\\left(\\widetilde v^\\varepsilon - \\check v^\\varepsilon\\right) = O(r^3\\varepsilon ^{1\/8})\\;\\;\\text{in}\\;\\;L^2({\\mathbb R}^{d+1}_{s,z}),\n\\end{equation}\nand we have established the microlocal link of the original Schr\\\"odinger equation\n$$\\\n{\\rm op}_\\varepsilon(P)\\psi^\\varepsilon_t = 0\n$$\nto the Landau--Zener system~\\eqref{eq:systreduitbis},  provided we choose $r \\le \\varepsilon^{ -\\kappa}$ with $0<\\kappa<1\/24$.\n\n\\begin{remark}\\label{rem:1\/8bis}\nIf $\\check\\gamma(\\cdot,\\delta)$ had uniformly bounded derivatives, then,  in view of Remark~\\ref{rem:1\/8}, the term $ |s| r^2\\varepsilon ^{-1\/8}$ in~(\\ref{eq:comparaison}) would be replaced by $\\delta|s|$.  For bounded times, this remainder would be of order $\\varepsilon^{3\/8}$.\n\\end{remark}\n\n\n\\section{The proof of the main result}\\label{sec:mainproof}\n\nFor proving our main result Theorem~\\ref{theorem}, we now analyse the dynamics of the block diagonal components of the Wigner transform $w^\\varepsilon_\\pm(\\psi^\\varepsilon_t)$, that is, \n\\[\n\\int_{{\\mathbb R}^{2d+1}} \\chi(t) a(q,p) \\,w^\\varepsilon_\\pm(\\psi^\\varepsilon_t)(q,p,\\delta)\\, {\\rm d}(q,p,t)\n= \\left\\langle{\\rm op}_\\varepsilon(\\chi a\\Pi^\\pm)\\psi^\\varepsilon_t,\\psi^\\varepsilon_t\\right\\rangle\n\\]\nfor scalar observables $a\\in{\\mathcal C}_c^\\infty({\\mathbb R}^{2d})$ with ${\\rm supp}(a)\\subset\\Omega_3$ and $\\chi\\in{\\mathcal C}^\\infty_c([0,T])$. By the assumption (A1)$_\\delta$,  the observables have support away from the set of small eigenvalue gap, providing the derivative bounds\n\\begin{equation}\\label{eq:Pi}\n\\forall \\alpha,\\beta\\in{\\mathbb N}^{d}\\,\\exists c_{\\alpha,\\beta}>0\\,\\forall \\varepsilon,\\delta>0: \n\\left\\| \\partial^\\beta_q\\partial^\\alpha_p(a\\Pi^\\pm(\\cdot,\\delta)) \\right\\|_\\infty < c_{\\alpha,\\beta} (R^3\\sqrt\\varepsilon)^{-|\\beta|}.\n\\end{equation}\nIn the following, the scale $R\\sqrt\\varepsilon$ will play an important role. We note that, of course, $(R^3\\sqrt\\varepsilon)^{-|\\beta|}\\le (R\\sqrt\\varepsilon)^{-|\\beta|}$ for $\\beta\\in{\\mathbb N}^d$.\nOur first step in the proof is now the replacement of the eigenprojectors $\\Pi^\\pm(q,\\delta)$ in ${\\rm op}_\\varepsilon(\\chi a \\Pi^\\pm)\\psi^\\varepsilon_t$ by the localisation on the corresponding energy shell\n$$\nE^\\pm=\\left\\{(q,t,p,\\tau)\\in{\\mathbb R}^{2d+2},\\;\\;\\tau  + \\Lambda^\\pm (q,p,\\delta)=0\\right\\},\n$$\nthat is a subset of the space-time phase space.\n\n\n\\subsection{Localization in energy}\\label{sec:energy}\nThe localization is implemented by a smooth cut-off function $\\theta\\in{\\mathcal C}_c^\\infty({\\mathbb R})$ satisfying\n\\begin{align*}\n0\\leq \\theta(u)\\leq 1,\\quad \n\\theta(u)=0\\;\\text{for}\\;|u|>1,\\quad\n\\theta(u)=1\\;\\text{for}\\;|u|<1\/2.\n\\end{align*}\nWe combine it with the energy function and set\n$$\n\\theta^\\pm_{\\varepsilon,R} (q,p,\\tau,\\delta)= \\theta\\!\\left({\\tau + \\Lambda^\\pm(q,p,\\delta )\\over\nR\\sqrt\\varepsilon}\\right),\\qquad (q,p,\\tau)\\in{\\mathbb R}^{2d+1}.\n$$\n\n\\begin{lemma}\\label{lem:loc}\nFor all symbols $\\chi\\in{\\mathcal C}_c^\\infty({\\mathbb R})$ and $a\\in{\\mathcal C}^\\infty_c({\\mathbb R}^{2d})$ satisfying the derivative bounds \\eqref{eq:Pi}, we have \n\\[\n{\\rm op}_\\varepsilon\\!\\left(\\chi a \\Pi^\\pm\\right)\\psi^\\varepsilon_t = {\\rm op}_\\varepsilon\\!\\left(\\chi a \\theta^\\pm_{\\varepsilon,R}\\right)\\psi^\\varepsilon_t + O(R^{-2}) + O(R^{-1}\\sqrt\\varepsilon) \n\\]\nin $L^2({\\mathbb R}^{d+1}_{t,q})$.\n\\end{lemma}\n\n\\begin{proof} Following the lines of the proof of \\cite[Lemma~5.1]{FL08}, we observe that, since $1-\\theta$ vanishes identically close to $0$, we can write \n$$1-\\theta^\\pm_{\\varepsilon,R}(q,p,\\tau,\\delta)= \\frac{\\tau +\\Lambda^\\pm (q,p,\\delta)}{R\\sqrt\\varepsilon} G\\!\\left({\\tau +\n\\Lambda^\\pm (q,p,\\delta)\\over\nR\\sqrt\\varepsilon}\\right)$$\nfor some smooth function $G$. Since \n$$\n( \\tau + \\Lambda^\\pm(q,p,\\delta)) \\Pi^\\pm(q,\\delta)=\\Pi^\\pm(q,\\delta )P(q,p,\\tau,\\delta),\n$$\nwe have\n\\[\n(1-\\theta^\\pm_{\\varepsilon,R})\\Pi^\\pm\n =  \\frac{1}{R\\sqrt\\varepsilon} \\,G\\!\\left(\\frac{\\tau+\\Lambda^\\pm}{R\\sqrt\\varepsilon}\\right)\\Pi^\\pm P.\n\\]\nWe can use symbolic calculus to bring into play that $\\psi^\\varepsilon_t$ solves the Schr\\\"odinger equation \n${\\rm op}_\\varepsilon(P)\\psi^\\varepsilon_t = 0$.  \nThe derivative bounds \\eqref{eq:Pi} imply that\n\\[\n{\\rm op}_\\varepsilon\\!\\left(\\chi a(1-\\theta^\\pm_{\\varepsilon,R})\\Pi^\\pm\\right) \\psi^\\varepsilon_t = O(R^{-2}) + O(R^{-1}\\sqrt\\varepsilon)\\quad\\text{in}\\;\\; L^2({\\mathbb R}^{d+1}_{t,q}).\n\\]\nNow it remains to remove the matrix $\\Pi^\\pm(q,\\delta)$ from the right hand side of the equation\n\\[\n{\\rm op}_\\varepsilon\\!\\left(\\chi a\\Pi^\\pm\\right) \\psi^\\varepsilon_t = {\\rm op}_\\varepsilon\\!\\left(\\chi a\\theta_{\\varepsilon,R}^\\pm\\Pi^\\pm\\right) \\psi^\\varepsilon_t+ O(R^{-2}) + O(R^{-1}\\sqrt\\varepsilon).\n\\]\nIn view of \n \\begin{equation*}\\label{eq2}\n \\chi a \\theta^\\pm_{\\varepsilon,R} = \\chi a \\theta^\\pm_{\\varepsilon,R} \\Pi^\\pm + \\chi a \\theta^\\pm_{\\varepsilon,R} \\Pi^\\mp,\n \\end{equation*} \nwe only need to prove that \n\\begin{equation}\\label{eq:small}\n{\\rm op}_\\varepsilon( \\chi a \\theta^\\pm_{\\varepsilon,R} \\Pi^\\mp)\\psi^\\varepsilon_t = O(R^{-2}) + O(R^{-1}\\sqrt\\varepsilon)\\quad\\text{in}\\;\\; L^2({\\mathbb R}^{d+1}_{t,q}).\n\\end{equation}\nWe observe that, for $\\varepsilon$ small enough, \n$$\n\\theta^\\pm_{\\varepsilon,R}=\\theta^\\pm_{\\varepsilon,R}(1-\\theta^\\mp_{\\varepsilon,R}),\n$$\nsince \n$$\n\\left| \\tau+\\tfrac12|p|^2 + \\alpha(q) \\pm \\tfrac12 g(q,\\delta)\\right| \\le R\\sqrt\\varepsilon\n$$\non the support of $\\theta^\\pm_{\\varepsilon,R}$. Using again the Schr\\\"odinger equation, symbolic calculus and the estimate~(\\ref{eq:Pi}), we obtain the desired relation \\eqref{eq:small}. \n\\end{proof}\n\nWe now reconsider the Landau--Zener transformation of Section~\\ref{sec:reduction} in terms of expectation values. By Lemma~\\ref{lem:loc},\n$$\n\\left\\langle{\\rm op}_\\varepsilon(\\chi a \\Pi^\\pm)\\psi^\\varepsilon_t,\\psi^\\varepsilon_t\\right\\rangle = \\big\\langle{\\rm op}_\\varepsilon(\\chi a \\theta^\\pm_{\\varepsilon,R})\\psi^\\varepsilon_t,\\psi^\\varepsilon_t\\big\\rangle + O(R^{-2}) + O(R^{-1}\\sqrt\\varepsilon).\n$$\nNext, we rewrite the expectation value using the function $v^\\varepsilon = K^*_\\varepsilon{\\rm op}_\\varepsilon(B^*_\\varepsilon)^{-1}\\psi^\\varepsilon_t$ that has been introduced in Proposition~\\ref{prop:quant}. Since $B_\\varepsilon = B + \\varepsilon B_1$ with $B^*B = \\lambda$,  symbolic calculus implies\n\\begin{align*}\n&\\left\\langle{\\rm op}_\\varepsilon(\\chi a \\Pi^\\pm)\\psi^\\varepsilon_t,\\psi^\\varepsilon_t\\right\\rangle\\\\\n& \\quad= \n\\left\\langle{\\rm op}_\\varepsilon(\\chi a \\lambda \\theta^\\pm_{\\varepsilon,R}){\\rm op}_\\varepsilon(B^*_\\varepsilon)^{-1}\\psi^\\varepsilon_t,{\\rm op}_\\varepsilon(B^*_\\varepsilon)^{-1}\\psi^\\varepsilon_t\\right\\rangle + O(R^{-2}) + O(R^{-1}\\sqrt\\varepsilon).\n\\end{align*}\nIn the presence of the symbol $\\theta^\\pm_{\\varepsilon,R}$ that loses a factor $R\\sqrt\\varepsilon$ per derivative, the application of the Fourier integral operator $K^\\varepsilon$ yields\n\\begin{align*}\n&\\left\\langle{\\rm op}_\\varepsilon(\\chi a \\Pi^\\pm)\\psi^\\varepsilon_t,\\psi^\\varepsilon_t\\right\\rangle\\\\ \n&\\quad= \n\\big\\langle  {\\rm op}_\\varepsilon\\big((\\chi a \\lambda \\theta^\\pm_{\\varepsilon,R})\\circ\\kappa_\\delta\\big)  v^\\varepsilon, v^\\varepsilon\\big\\rangle + O(R^{-2}) + O(R^{-1}\\sqrt\\varepsilon).\n\\end{align*}\nIn the next step, we move towards $\\widetilde v^\\varepsilon = {\\rm op}_\\varepsilon(M^\\varepsilon_0+\\delta M^\\varepsilon_1)v^\\varepsilon$. Since the matrix $M_0^\\varepsilon$ is unitary and \n$|\\delta|\\le R\\sqrt\\varepsilon$, we have\n\\begin{align*}\n&\\left\\langle{\\rm op}_\\varepsilon(\\chi a \\Pi^\\pm)\\psi^\\varepsilon_t,\\psi^\\varepsilon_t\\right\\rangle\\\\\n&\\quad = \\big\\langle  {\\rm op}_\\varepsilon\\big((\\chi a \\lambda \\theta^\\pm_{\\varepsilon,R})\\circ\\kappa_\\delta\\big)  \\widetilde v^\\varepsilon, \\widetilde v^\\varepsilon\\big\\rangle + O(R^{-2}) + O(R\\sqrt\\varepsilon).\n\\end{align*}\nWe then arrive at the solution $\\check v^\\varepsilon$ of the Landau--Zener system~\\eqref{eq:systreduitbis} by\n\\begin{align}\\label{eq:exp_LZ}\n&\\left\\langle{\\rm op}_\\varepsilon(\\chi a\\Pi^\\pm)\\psi^\\varepsilon_t,\\psi^\\varepsilon_t\\right\\rangle \\nonumber\\\\ \n&\\quad= \n\\big\\langle {\\rm op}_\\varepsilon((\\chi a \\lambda \\theta^\\pm_{\\varepsilon,R})\\circ\\kappa_\\delta) \\check v^\\varepsilon,\\check v^\\varepsilon\\big\\rangle + O(R^{-2}) + O(R\\sqrt\\varepsilon) + O(r^3\\varepsilon^{1\/8}).\n\\end{align}\nAt this stage of the proof of Theorem~\\ref{theorem}, we have rewritten the expectation values for the Schr\\\"odinger solution $\\psi^\\varepsilon_t$ in terms of the \nLandau--Zener solution $\\check v^\\varepsilon$. Next, we reformulate the Markov process in the new coordinates. \n\n\n\\subsection{The Markov process for the normal form}\\label{sec:Markov}\n \nLet us now introduce a new Markov process for effectively describing the dynamics of the reduced Landau--Zener problem \\eqref{eq:systreduitbis}. We use the analogous building blocks as for the original process that defines the semigroup~$({\\mathcal L}_{\\varepsilon}^t)_{t\\ge0}$. \nWe shall prove that this new process is close to the image of the original Markov process by the canonical transform $\\kappa_\\delta$.\n\n\\subsubsection{The image of the classical trajectories by the canonical transform}\nWe observe that by the transformation in~(\\ref{def:B}), the eigenvalues of the reduced system~(\\ref{eq:systreduit}) satisfy \n$$\n-\\sigma + j \\, \\sqrt{s^2+\\delta^2\\check\\gamma(s,z,\\sigma,\\zeta)^2} = \\left(\\lambda \\left(\\tau+\\Lambda^\\mp(q,p,\\delta)\\right)\\right)\\circ\\kappa_\\delta,\n$$\nwhere the sign $j=\\pm1$ depends on  the numbering discussed below in \\S\\ref{sec:numbering}. As a consequence, \nthe integral curves of \n$$\nH_{-\\sigma \\pm\\sqrt{s^2+\\delta^2\\check\\gamma(s,z,\\sigma,\\zeta)^2}}\n$$ \nare mapped by the canonical transform $\\kappa_\\delta$ to those of \n$$\nH_{\\lambda\\left(\\tau+\\Lambda^\\mp(q,p,\\delta)\\right)}=\\lambda H_{\\tau+\\Lambda^\\mp(q,p,\\delta)}+\\left(\\tau+\\Lambda^\\mp(q,p,\\delta)\\right)H_\\lambda.\n$$\nSince the energy $\\tau+\\Lambda^\\mp(q,p,\\delta)$ is of order $R\\sqrt \\varepsilon$ in our zone of observation, these trajectories are those of $\\lambda H_{\\tau+\\Lambda^\\mp(q,p,\\delta)}$ up to some term of order $R\\sqrt\\varepsilon$. Since the function $\\lambda$ does not vanish in our zone of observation, the integral curves of $\\lambda H_{\\tau+\\Lambda^\\mp(q,p,\\delta)}$ are those of $ H_{\\tau+\\Lambda^\\mp(q,p,\\delta)}$ up to a change of parametrization of the curve. \nWe therefore consider the image of the Hamiltonian curves of the initial Markov process by the canonical transform $\\kappa_\\delta$ as being close to those of the functions \n$$\n\\widetilde\\Lambda^\\pm(s,z,\\sigma,\\zeta) = -\\sigma\\mp\\sqrt{s^2+\\delta^2\\check \\gamma(s,z,\\sigma,\\zeta)^2.}\n$$\n\n\n\\subsubsection{The relation of the energies}\nFor discussing the relation of the different energies occuring in our analysis, we also introduce the functions \n\\begin{equation}\\label{eigenvaluesbis}\n\\widetilde\\Lambda^\\pm_0(s,z,\\sigma,\\zeta) = -\\sigma\\mp\\sqrt{s^2+\\delta^2\\check \\gamma_0(z,\\zeta)^2}\n\\end{equation}\nthat belong to the Landau--Zener system \\eqref{eq:systreduitbis}. We observe that both energies $\\widetilde\\Lambda^\\pm$ and $\\widetilde\\Lambda^\\pm_0$ satisfy\n$$\n\\widetilde\\Lambda^\\pm, \\widetilde\\Lambda^\\pm_0 = -\\sigma \\mp |s| + O(\\delta^2|s|^{-1}),\n$$\nwhere we have used \n$$\n\\widetilde\\Lambda^\\pm_0(s,z,\\sigma,\\zeta) \\pm |s|= \\mp\\frac{\\delta^2\\check \\gamma_0(z,\\zeta)^2}{|s|+\\sqrt{s^2+\\delta^2\\check \\gamma_0(z,\\zeta)^2}}\n$$\nand a similar relation for $\\widetilde\\Lambda^\\pm$. Therefore, since $\\delta\\le R\\sqrt\\varepsilon$, we obtain\n$$\n\\widetilde\\Lambda^\\pm - \\widetilde\\Lambda_0^\\pm = O(R^2\\varepsilon|s|^{-1})\n$$\nand for any smooth cut-off function $\\widetilde\\theta\\in{\\mathcal C}^\\infty_c({\\mathbb R})$\n$$\n\\widetilde\\theta\\!\\left(\\frac{\\widetilde\\Lambda^\\pm(s,z,\\sigma,\\zeta)}{R\\sqrt\\varepsilon}\\right) = \\widetilde\\theta\\!\\left(\\frac{\\widetilde\\Lambda_0^\\pm(s,z,\\sigma,\\zeta)}{R\\sqrt\\varepsilon}\\right) + O(R\\sqrt\\varepsilon|s|^{-1}).\n$$\nHence a change in the energy localisation from $\\widetilde\\Lambda^\\pm$ to $\\widetilde\\Lambda^\\pm_0$ causes a deviation of the order $O(R^{-1}r^{-1})= O(R^{-1})$, when choosing $s\\sim r R^2\\sqrt\\varepsilon$ provided $r \\gg 1$.\n\n\n\\subsubsection{The numbering of the eigenvalues}\\label{sec:numbering}\nThe classical trajectories of the Landau--Zener system \\eqref{eq:systreduitbis} are generated by the eigenvalues\n$$\n-\\sigma \\pm\\sqrt{s^2 + \\delta^2 \\check\\gamma_0(z,\\zeta)^2}.\n$$\nFor enumerating these eigenvalues such that the classical trajectories in in the original and the new coordinates can be naturally linked, we consider the case $\\delta =0$ and use the vectors $H$ and $H'$ that have been defined in~(\\ref{def:H}) and~(\\ref{def:H'}). \n\n\\medskip\nWe recall that $H$ and $H'$ are associated with ingoing trajectories for $\\Lambda^+$ and $\\Lambda^-$, respectively, and satisfy $\\omega(H,H')<0$. Up to some perturbation term of order $R\\sqrt\\varepsilon$, the canonical transformation $\\kappa_0^{-1}$ sends $H$ and $H'$ on vectors that are collinear to $-\\partial_s-\\partial_\\sigma$ and $-\\partial_s+\\partial_\\sigma$ above the singular set $\\{s=0\\}$. Since\n$$\n\\omega(-\\partial_s-\\partial_\\sigma,-\\partial_s+\\partial_\\sigma)= 2>0,\n$$\nthe vector $H$ is collinear to $-\\partial_s+\\partial_\\sigma$ and $H'$ to $-\\partial_s-\\partial_\\sigma$ above $\\{s=0\\}$. Since\n$$\n-\\partial_s+\\partial_\\sigma=H_{-\\sigma-|s|},\\; -\\partial_s-\\partial_\\sigma=H_{-\\sigma+|s|}\\quad\\text{on}\\quad\\{s>0\\}\n$$\nand $s>0$ on ingoing trajectories, the vector field $H_{-\\sigma-|s|}$ corresponds to the plus mode, while $H_{-\\sigma+|s|}$ belongs to the minus mode. We \ntherefore number the eigenvalues in the new coordinates according to \\eqref{eigenvaluesbis}.\n\n\n\n\\subsubsection{The Hamiltonian trajectories}\nThe eigenvalues $\\widetilde\\Lambda^\\pm_0$ generate the Hamiltonian systems\n$$\n\\dot s = -1,\\quad \\dot z = \\mp \\frac{\\delta^2 \\check\\gamma_0\\partial_\\zeta \\check\\gamma_0}{\\sqrt{s^2+\\delta^2\\check \\gamma_0^2}},\\quad\n\\dot\\sigma = \\pm \\frac{s}{\\sqrt{s^2+\\delta^2\\check \\gamma_0^2}},\\quad \\dot\\zeta = \\pm \\frac{\\delta^2 \\check\\gamma_0\\partial_z \\check\\gamma_0}{\\sqrt{s^2+\\delta^2\\check \\gamma_0^2}},\n$$\nwith corresponding flow maps \n$$\n\\widetilde\\Phi_{0,\\pm}^{\\beth}:{\\mathbb R}^{2d+2}\\to{\\mathbb R}^{2d+2}.\n$$ \nUsing that $|\\dot z| + |\\dot\\zeta| = O(\\delta|s|^{-1})$ and in view of \\eqref{eigenvaluesbis}, we have\n\\begin{equation}\\label{eq:flow}\n\\widetilde\\Phi_{0,\\pm}^{\\beth}(s,z,\\sigma,\\zeta) = (s-\\beth,z,\\mp|s-\\beth|,\\zeta) + O(\\delta |s|^{-1})\n\\end{equation}\nfor all points $(s,z,\\sigma,\\zeta)$ in our zone of observation and propagation times $\\beth>0$. \nNote that for $s\\sim r R^2\\sqrt\\varepsilon$, we obtain an error of the order $R^{-1}r^{-1}$, which is smaller than $R^{-1}$.\n\n\n\\subsubsection{The non-adiabatic transitions}\nMonitoring the gap function \n$$\n\\widetilde g_0(s,z,\\zeta) = 2\\sqrt{s^2+\\delta^2\\check \\gamma_0(z,\\zeta)^2}\n$$\nalong the Hamiltonian trajectories associated with $\\widetilde\\Lambda^\\pm_0$ we look for points in ${\\mathbb R}^{2d+2}$, where a local minimum is attained. We obtain the condition \n$$\n0 = \\partial_s \\widetilde g_0 \\;\\partial_\\sigma\\widetilde\\Lambda^\\pm_0 + \\nabla_z\\widetilde g_0 \\cdot \\nabla_\\zeta \\widetilde\\Lambda^\\pm_0 \n- \\partial_\\sigma\\widetilde g_0\\; \\partial_s \\widetilde \\Lambda^\\pm_0 - \\nabla_\\zeta \\widetilde g_0\\cdot \\nabla_z \\widetilde\\Lambda^\\pm_0 = -\\partial_s \\widetilde g_0,\n$$\nthat is equivalent to $s=0$.  \nHence, the new jump manifold is the set\n$$\n\\widetilde\\Sigma_\\varepsilon = \\left\\{(s,z,\\sigma,\\zeta)\\in{\\mathbb R}^{2d+2}\\mid s=0,\\; 2\\delta|\\check\\gamma_0(z,\\zeta)|\\le R\\sqrt\\varepsilon\\right\\}.\n$$\nThe Landau--Zener formula of \\cite[Proposition~7]{FG03}, see \\S\\ref{sec:LZ_form}, suggests to perform non-adiabatic transitions with probability\n$$\n\\widetilde T_\\varepsilon(z,\\zeta) := \\exp\\!\\left(-\\frac{\\pi}{\\varepsilon} \\delta^2\\check\\gamma_0(z,\\zeta)^2\\right)\n$$\nwhen reaching the jump set $\\widetilde\\Sigma_\\varepsilon$. By the energy localization of our observables, we have $s=0$ and $\\sigma=O(R\\sqrt\\varepsilon)$ on the jump manifold $\\widetilde\\Sigma_\\varepsilon$. By Theorem~\\ref{prop:parametrization}, a Taylor expansion around $\\sigma=0$ reads\n$$\n\\check\\gamma(0,z,\\sigma,\\zeta) = \\check\\gamma_0(z,\\zeta) + \\sigma \\partial_\\sigma\\check\\gamma(0,z,0,\\zeta) + O(\\sigma^2\/\\delta_0).\n$$\nUsing that $\\delta\/\\delta_0$ is bounded, we obtain\n$$\n\\frac{\\delta^2}{\\varepsilon}\\left(\\check\\gamma(0,z,\\sigma,\\zeta) - \\check\\gamma_0(z,\\sigma)\\right) = O(R^3\\sqrt\\varepsilon)\n$$\nand\n$$\n\\widetilde T_\\varepsilon(z,\\zeta) = \\exp\\!\\left(-\\frac{\\pi}{\\varepsilon}\\delta^2\\check\\gamma(s,z,\\sigma,\\zeta)^2\\right) + O(R^3\\sqrt\\varepsilon).\n$$\nLemma~\\ref{lem:gamma} then provides\n\\begin{equation}\\label{eq:LZ}\n\\widetilde T_\\varepsilon = T_\\varepsilon\\circ\\kappa_\\delta + O(R^3\\sqrt\\varepsilon),\n\\end{equation}\nwhere the original transition rate $T_\\varepsilon(q,p,\\delta)$ has been defined in \\eqref{Tepsdelta}. Besides, the original jump condition $p\\cdot\\nabla_q g(q,\\delta) = 0$ is equivalent to\n$$\n\\widetilde\\beta(q,\\delta) (p\\cdot\\nabla_q\\widetilde\\beta(q,\\delta)) + \\delta^2 \\widetilde\\gamma(q) (p\\cdot\\nabla_q\\widetilde\\gamma(q)) = 0, \n$$\nthat is, $s=O(\\delta^2)$. Hence, the non-adiabatic transitions of the new and the original Markov process mostly differ due to the transition rate estimate \\eqref{eq:LZ}. \n\n\\subsubsection{The drift}\\label{subsubsec:drift}\nAt $s=0$,  we observe for all $\\delta\\ge0$  the energy relation\n\\begin{equation}\\label{driftunicity}\n\\widetilde\\Lambda^\\pm_0(0,z,\\sigma\\mp2 \\delta |\\check \\gamma_0(z,\\zeta)|,\\zeta)=\\widetilde\\Lambda^\\mp_0(0,z,\\sigma,\\zeta).\n\\end{equation}\nThe best drift is, of course, the exact one and is given by\n$$\n\\widetilde J_\\pm:\\sigma\\mapsto \\sigma\\mp2 \\delta |\\check \\gamma_0(z,\\zeta) |.\n$$ \nThis exact drift is performed in the direction of~$\\partial_\\sigma$ that is collinear to the difference of the two Hamiltonian vector fields in the particular case $\\delta=0$, motivating the geometric underpinning of the original drift construction, see Remark~\\ref{rem:energy}. \nWe also note  that the size of the drift $2\\delta|\\check\\gamma_0(z,\\zeta)|$ is precisely the gap size for points in the jump manifold.\n\n\n\n\n\n\\subsection{The semigroup for the normal form}\\label{sec:semigroup}\n\nThe Markov process described in the previous section \\S\\ref{sec:Markov} defines a semigroup $\\widetilde{\\mathcal L}_\\varepsilon$ acting on functions in the space \n$$\n\\widetilde{\\mathcal B} = \\left\\{f:{\\mathbb R}^{2d+2}\\times\\{-1,1\\}\\to{\\mathbb C}\\mid f\\;\\text{is measurable, bounded}\\right\\}.\n$$ \nFollowing the normal form transformation of the expectation values given in \\eqref{eq:exp_LZ}, we consider a symbol $c_{\\varepsilon,R}^{out}\\in\\widetilde{\\mathcal B}$ whose plus-minus-components are defined by  \n$$\nc_{\\varepsilon,R}^{\\pm,out}(s,z,\\sigma,\\zeta)=   b^\\pm (s,z,\\sigma,\\zeta)\\;\n\\widetilde\\theta\\!\\left(\\frac{\\widetilde\\Lambda^\\pm_0(s,z,\\sigma,\\zeta)}{R\\sqrt\\varepsilon}\\right),\n$$\nwhere the functions $b^\\pm$ and $\\widetilde\\theta$ have the following properties: $b^\\pm(s,z,\\sigma,\\zeta)$ are two smooth functions compactly supported in the outgoing region $\\{s<0\\}$ such that the random trajectories reaching their support have only one transition during the observation time that is of length $\\beth>0$. The functions $b^\\pm(s+\\beth,z,\\sigma,\\zeta)$ are supported in the incoming region $\\{s>0\\}$. The cut-off function $\\widetilde\\theta\\in{\\mathcal C}^\\infty_c({\\mathbb R})$ satisfies\n\\begin{align*}\n0\\leq \\widetilde\\theta\\leq 1,\\quad \n\\widetilde \\theta(u)=0\\;\\text{for}\\;|u|>1,\\quad\n\\widetilde \\theta(u)=1\\;\\text{for}\\;|u|<1\/2.\n\\end{align*}\nWe now analyse the pull back of the symbol $c_{\\varepsilon,R}^{out}\\in\\widetilde{\\mathcal B}$ by the semigroup for a suitably chosen time $\\beth>0$,  \n$$\nc_{\\varepsilon,R}^{in}:=\\widetilde{\\mathcal L}_{\\varepsilon}^\\beth\\, c_{\\varepsilon,R}^{out}.\n$$\nBy Assumption~$(A2)_\\delta$ of \\S\\ref{sec:delta}, we only consider transitions generated by one incoming mode. We assume that it is the plus mode and denote the two leading order contributions of $c^{+,in}_{\\varepsilon,R}$ by $c^{+,in}_{\\varepsilon,R,+}$ and $c^{+,in}_{\\varepsilon,R,-}$, where the subscript depends on the outgoing mode. Our aim is to relate\n$$\n\\left\\langle{\\rm op}_\\varepsilon(c_{\\varepsilon,R}^{\\pm,out})\\check v^\\varepsilon,\\check v^\\varepsilon\\right\\rangle \\qquad\\text{with}\\qquad\n\\left\\langle{\\rm op}_\\varepsilon(c_{\\varepsilon,R,\\pm}^{+,in})\\check v^\\varepsilon,\\check v^\\varepsilon\\right\\rangle.\n$$\nIf the incoming trajectories are associated with the minus mode, the arguments are analogous.\n\n \n\\subsubsection{Transport without transitions} \nThe component $c_{\\varepsilon,R,+}^{+,in}$ takes into account  the classical transport along the plus trajectories\nand the probability of staying on the same mode.\nBy conservation of energy along classical trajectories, we have   \n$$\n\\widetilde\\Lambda^+_0(\\widetilde \\Phi^{-\\beth}_{0,+}(s-\\beth,z,\\sigma,\\zeta))= \\widetilde\\Lambda^+_0(s-\\beth,z,\\sigma,\\zeta).\n$$\nBy \\eqref{eq:flow}, we therefore deduce\n\\begin{align*}\n& c_{\\varepsilon,R,+}^{+,in}(s-\\beth,z,\\sigma,\\zeta) = \\left(1- \\widetilde T _\\varepsilon(z,\\zeta)\\right) \n(c_{\\varepsilon,R}^{+,out}\\circ\\widetilde \\Phi^{-\\beth}_{0,+})(s-\\beth,z,\\sigma,\\zeta)\\\\\n& =   \\left(1- \\widetilde T _\\varepsilon(z,\\zeta)\\right) b^+(s,z,-|s|,\\zeta) \n\\;\\widetilde\\theta\\!\\left(\\frac{\\widetilde\\Lambda^+_0(s-\\beth,z,\\sigma,\\zeta)}{R\\sqrt\\varepsilon}\\right) +O(R\\sqrt\\varepsilon),\n\\end{align*}\nso that\n\\begin{align*}\n& c_{\\varepsilon,R,+}^{+,in} (s,z,\\sigma,\\zeta)\\\\\n&  = \\left(1- \\widetilde T _\\varepsilon(z,\\zeta)\\right)  b^+(s+\\beth,z,-|s+\\beth|,\\zeta)\\;\n\\widetilde \\theta\\!\\left(\\frac{\\widetilde\\Lambda^+_0(s,z,\\sigma,\\zeta)}{R\\sqrt\\varepsilon}\\right) +O(R\\sqrt\\varepsilon).\n \\end{align*}\n \n\n\\subsubsection{Transport and transitions with drift} \nThe component  $c_{\\varepsilon,R,-}^{+,in}$ is more intricate, since it incorporates classical transport through both modes, application of the transfer coefficient and of the drift. Indeed, the branches of minus trajectories which reach the support of $c_{\\varepsilon,R}^{-,out}$ result from plus trajectories that have been drifted.  More precisely, we have \n$$\n(\\widetilde \\Phi^{s}_{0,-}\\circ\\widetilde J_+\\circ\\widetilde\\Phi^{-s-\\beth}_{0,+})(s-\\beth,z,\\sigma,\\zeta) = \n\\left(s,z,|s|,\\zeta\\right) + O(R\\sqrt\\varepsilon).\n$$\nfor $(s,z,\\sigma,\\zeta)\\in{\\rm supp}(c_{\\varepsilon,R}^{-,out})$.  By conservation of energy and the drift relation~(\\ref{driftunicity}) we have\n$$\n\\widetilde \\Lambda^-_0\\!\\left(\\widetilde\\Phi^s_{0,-}\\circ \\widetilde J_+ \\circ \\widetilde\\Phi_{0,+}^{-s-\\beth}(s-\\beth,z,\\sigma,\\zeta)\\right)=\n\\widetilde \\Lambda^+_0(s-\\beth,z,\\sigma,\\zeta).\n$$\nApplying the transfer coefficient, we obtain \n \\begin{eqnarray*}\nc_{\\varepsilon,R,-}^{+,in}(s-\\beth,z,\\sigma,\\zeta)& = &  \\widetilde T_\\varepsilon(z,\\zeta)\\,\n(c_{\\varepsilon,R}^{-,out}\\circ\\widetilde \\Phi^{s}_{0,-}\\circ\\widetilde J_+\\circ\\widetilde\\Phi^{-s-\\beth}_{0,+})(s-\\beth,z,\\sigma,\\zeta)\\\\\n& = &  \\widetilde T_\\varepsilon(z,\\zeta)\\, b^-(s,z,|s|,\\zeta)\\,  \\widetilde\\theta\\!\\left(\\frac{\\widetilde\\Lambda^+_0(s-\\beth,z,\\sigma,\\zeta)}{R\\sqrt\\varepsilon}\\right) +O(R\\sqrt\\varepsilon),\n\\end{eqnarray*}\nthat is,\n\\begin{equation*}\nc_{\\varepsilon,R,-}^{+,in} (s,z,\\sigma,\\zeta)  =  \\widetilde T_\\varepsilon(z,\\zeta)   \\,b^-(s+\\beth, z,|s+\\beth|,\\zeta)\\,\n \\widetilde\\theta\\!\\left(\\frac{\\widetilde\\Lambda^+_0(s,z,\\sigma,\\zeta)}{R\\sqrt\\varepsilon}\\right)+O(R\\sqrt\\varepsilon).\n \\end{equation*}\n \n\\subsection{The transitions}\\label{sec:transitions}\n\nWe now prove that the semigroup $\\widetilde{\\mathcal L}_\\varepsilon$ effectively describes the dynamics of the Landau--Zener system \\eqref{eq:systreduitbis} in the sense that  \n\\begin{equation}\\label{claim}\n\\left\\langle{\\rm op}_\\varepsilon(c_{\\varepsilon,R}^{\\pm,out})\\check v^\\varepsilon,\\check v^\\varepsilon\\right\\rangle =\n\\left\\langle{\\rm op}_\\varepsilon(c_{\\varepsilon,R,\\pm}^{+,in})\\check v^\\varepsilon,\\check v^\\varepsilon\\right\\rangle + O(\\eta_\\varepsilon),\n\\end{equation}\nwhere the error term is obtained as\n$$\nO(\\eta_\\varepsilon) = O(r^{-1}) + O(R^{-1}) + O(\\varepsilon R^2 \\ln(r R)).\n$$\nBy Theorem~\\ref{prop:parametrization} the function $\\check\\gamma_0$ is a smooth function with bounded derivatives. Thus, we can \nfollow the argumentation developed in~\\cite[\\S5.3]{FL08}, crucially using the operator-valued Landau--Zener formula of~\\cite[Proposition~7]{FG03}. \n\n\n\\subsubsection{Using energy localization}\nWe work with the eigenprojectors\n$$\n\\widetilde\\Pi^\\pm_0(s,z,\\zeta) = \\frac12\\left( {\\rm Id} \\mp \\frac{1}{\\sqrt{s^2+\\delta^2\\check\\gamma_0(z,\\zeta)^2}}  \n\\begin{pmatrix}s & \\delta\\check\\gamma_0(z,\\zeta) \\\\ \\delta\\check\\gamma_0(z,\\zeta)& -s \\end{pmatrix}\\right)\n$$\nof the Landau--Zener system, numbered consistently with the eigenvalues in \\eqref{eigenvaluesbis}. We observe that for $|s| \\sim r R^2\\sqrt\\varepsilon$\n\\begin{eqnarray}\n\\nonumber\n\\widetilde\\Pi^+_0(s,z,\\zeta)&=&\n\\begin{pmatrix}0&0\\\\ 0&1\\end{pmatrix}+O(R^{-1})\n\\;\\;{\\rm in}\\;\\;\\{s>0\\},\\\\\n\\label{eq:Pitilde}\n\\widetilde\\Pi^+_0(s,z,\\zeta)&=&\\begin{pmatrix}1&0\\{\\bf 0}&0\\end{pmatrix}+O(R^{-1})\n\\;\\;{\\rm in}\\;\\;\\{s<0\\},\n\\end{eqnarray}\nand obtain similar asymptotics for $\\widetilde\\Pi^-_0$ since ${\\rm Id}=\\widetilde \\Pi^+_0+\\widetilde \\Pi^-_0$. \nBy Lemma~\\ref{lem:loc}, we then have \n$$\n\\left\\langle{\\rm op}_\\varepsilon( c_{\\varepsilon,R}^{\\pm,out}) \\check v^\\varepsilon,\\check v^\\varepsilon\\right\\rangle  =  \n\\left\\langle{\\rm op}_\\varepsilon(b^{\\pm}\\widetilde\\Pi^\\pm_0)\\check v^\\varepsilon,\\check v^\\varepsilon\\right\\rangle+O(R^{-2}) + O(R^{-1}\\sqrt\\varepsilon), \n$$\nand consequently, \n\\begin{eqnarray*}\n\\left\\langle{\\rm op}_\\varepsilon( c_{\\varepsilon,R}^{+,out})\\check v^\\varepsilon,\\check v^\\varepsilon\\right\\rangle &=&\\Big\\langle{\\rm op}_\\varepsilon(b^{+,out})\\check v^\\varepsilon_1,\\check v^\\varepsilon_1\\Big\\rangle+\nO(R^{-1}),\\\\\n\\left\\langle{\\rm op}_\\varepsilon( c_{\\varepsilon,R}^{-,out})\\check v^\\varepsilon,\\check v^\\varepsilon\\right\\rangle &=&\\Big\\langle{\\rm op}_\\varepsilon( b^{-,out}) \\check v^\\varepsilon_2,\\check v^\\varepsilon_2\\Big\\rangle+\nO(R^{-1})\n\\end{eqnarray*}\nwith\n$$\nb^{\\pm,out}(s,z,\\sigma,\\zeta)= b^\\pm (s,z,\\sigma,\\zeta)\n$$\nsupported in the outgoing region $\\{-r R^2\\sqrt\\varepsilon \\le s\\le -{r \\over 2}  R^2\\sqrt\\varepsilon\\}$. Similarly, we have \n\\begin{eqnarray*}\n\\left\\langle{\\rm op}_\\varepsilon(c_{\\varepsilon,R,+}^{+,in}) \\check v^\\varepsilon,\\check v^\\varepsilon\\right\\rangle &=& \n\\left\\langle{\\rm op}_\\varepsilon\\!\\left((1-\\widetilde T_\\varepsilon)  b^{+,in}\\right)\\check v^\\varepsilon_2,\\check v^\\varepsilon_2\\right\\rangle+O(R^{-2}) + O(R^{-1}\\sqrt\\varepsilon),\\\\\n\\left\\langle{\\rm op}_\\varepsilon(c_{\\varepsilon,R,-}^{+,in}) \\check v^\\varepsilon,\\check v^\\varepsilon\\right\\rangle &=&\n\\left\\langle{\\rm op}_\\varepsilon\\!\\left(\\widetilde T_\\varepsilon \\,b^{-,in}\\right)\\check v^\\varepsilon_2,\\check v^\\varepsilon_2\\right\\rangle+O(R^{-2}) + O(R^{-1}\\sqrt\\varepsilon)\n\\end{eqnarray*}\nwith \n$$\nb^{\\pm,in}(s,z,\\sigma,\\zeta)  =  b^\\pm(s+\\beth,z,\\mp|s+\\beth|,\\zeta).\n$$\nsupported in the incoming region $\\{{r \\over 2} R^2\\sqrt\\varepsilon \\le s\\le r R^2\\sqrt\\varepsilon\\}$.\n\n\n\\subsubsection{The Landau--Zener formula}\\label{sec:LZ_form}\nFollowing \\cite[Proposition~7]{FG03}, we rewrite the Landau--Zener system~\\eqref{eq:systreduitbis} as\n\\begin{equation}\\label{eq:LZ_G}\n\\frac{\\varepsilon}{i} \\partial_s \\check v^\\varepsilon = \\begin{pmatrix}s & \\sqrt\\varepsilon G\\\\ \\sqrt\\varepsilon G^* & -s\\end{pmatrix} \\check v^\\varepsilon\n\\quad\\text{with}\\quad G = \\frac{\\delta}{\\sqrt\\varepsilon}{\\rm op}_\\varepsilon(\\check\\gamma_0(z,\\zeta)).\n\\end{equation}\nThen there exist two vector-valued functions \n$k^{\\varepsilon,\\pm}\\in L^2({\\mathbb R}^d,{\\mathbb C}^2)$ such that  for any cut-off function $\\chi\\in{\\mathcal C}_c^\\infty([0,R^2])$ and for $\\pm s>0$\n\\begin{eqnarray*}\n\\chi(GG^*) \\check v^\\varepsilon_1(z,s) & = & \n\\chi(GG^*) {\\rm e}^{is^{2}\/(2\\varepsilon)} \n\\left|{\\tfrac{s}{\\sqrt \\varepsilon}}\\right|^{i\\frac{GG^*}{2}}\nk^{\\varepsilon,\\pm}_{1}(z)+O(R^2\\sqrt\\varepsilon\/s),\\\\\n\\chi(G^*G)\\check v^\\varepsilon_2(z,s) & = & \n\\chi(G^*G){\\rm e}^{-is^{2}\/(2\\varepsilon)}\n\\left|{\\tfrac{s}{\\sqrt \\varepsilon}}\\right|^{-i\\frac{G^*G}{2}}\nk^{\\varepsilon,\\pm}_{2}(z)+O(R^2\\sqrt\\varepsilon\/s),\n\\end{eqnarray*}\nwhere $k^{\\varepsilon,+}=S_\\varepsilon k^{\\varepsilon,-}$ with\n$$\nS_\\varepsilon=\n\\begin{pmatrix}a(GG^*) & -\n \\overline b(GG^*)G \\\\  b(G^*G)G^* & a(G^*G)\\end{pmatrix}.\n$$ \nThe functions defining the scattering matrix satisfy\n$$\na(\\lambda)={\\rm e}^{-\\pi\\lambda\/2},\\qquad a(\\lambda^2) + \\lambda |b(\\lambda)|^2 = 1,\\qquad \\lambda\\in{\\mathbb R}.\n$$\nMoreover, the asymptotics of \\cite[Lemma 8 \\& 9]{FG03} provide for any smooth and compactly supported symbol $\\phi\\in{\\mathcal C}^\\infty_c({\\mathbb R}^{2d+2})$\n\\begin{equation}\\label{eq:phase}\n\\left|{\\tfrac{s}{\\sqrt\\varepsilon}}\\right|^{\\pm\ni\\frac{G^*G}{2 }} \n{\\rm op}_\\varepsilon(\\phi)\n\\left|{\\tfrac{s}{\\sqrt\\varepsilon}}\\right|^{\\mp i\\frac{G^*G}{\n2}}=\n{\\rm op}_\\varepsilon(\\phi)+O(R^2\\varepsilon |\\ln(s\/\\sqrt\\varepsilon)|).\n\\end{equation}\nThese asymptotics yield an error of the order $1\/r$, which motivates to choose \n\\[\nr=r(\\varepsilon) = \\varepsilon^{-1\/32},\n\\] \nso that the error $r^3\\varepsilon ^{1\/8}$ in equations~(\\ref{eq:error}) and \\eqref{eq:exp_LZ} is of the same size as $1\/r$.\n\\begin{remark}\\label{rem:1\/8ter}\nIf $\\widetilde \\gamma(\\cdot,\\delta)$ had uniformly bounded derivatives, then, in view of Remarks~\\ref{rem:1\/8} and~\\ref{rem:1\/8bis}, we could choose $r =R$ and  obtain an overall remainder of the order $1\/R =\\varepsilon^{1\/8}$.\n\\end{remark}\n\n\n\n\\subsubsection{Applying the Landau--Zener formula}\nUsing the Landau--Zener formalism described in \\S\\ref{sec:LZ_form}, we now restrict ourselves to proving\n$$\n\\Big\\langle{\\rm op}_\\varepsilon(b^{+,out})\\check v^\\varepsilon_1,\\check v^\\varepsilon_1\\Big\\rangle = \\left\\langle{\\rm op}_\\varepsilon\\!\\left((1-\\widetilde T_\\varepsilon)  b^{+,in}\\right)\\check v^\\varepsilon_2,\\check v^\\varepsilon_2\\right\\rangle+O(\\eta_\\varepsilon),\n$$\nsince the proof for the second estimate in \\eqref{claim} is analogous. \n\n\\medskip\nWe first use the relation between $\\check v^\\varepsilon_1$ and $k^{\\varepsilon,-}_1$ on the outgoing region $\\{s<0\\}$ for $s\\le-rR^2\\sqrt\\varepsilon$ to obtain\n$$\n\\left\\langle{\\rm op}_\\varepsilon(b^{+,out})\\check v^\\varepsilon_1,\\check v^\\varepsilon_1\\right\\rangle \n= \\left\\langle{\\rm op}_\\varepsilon(b^{+,out})k^{\\varepsilon,-}_{1},k^{\\varepsilon,-}_{1}\\right\\rangle + O(r^{-1}) + O(\\varepsilon R^2 \\ln(rR)).\n$$\nThen, we perform the change of variable $s\\mapsto s-\\beth$, \n$$\n\\left\\langle{\\rm op}_\\varepsilon(b^{+,out})\\check v^\\varepsilon_1,\\check v^\\varepsilon_1\\right\\rangle  \n= \\left\\langle{\\rm op}_\\varepsilon(b^{+,in})k^{\\varepsilon,-}_{1},k^{\\varepsilon,-}_{1}\\right\\rangle + O(r^{-1}) + O(\\varepsilon R^2\\ln(rR)).\n$$\nSince we have assumed that the incoming minus contributions are negligibly small, we neglect the scattering contribution from $\\check v^\\varepsilon_1$ and consequently from $k^{\\varepsilon,+}_1$. We therefore deduce from the scattering relation $k^{\\varepsilon,-} = S_\\varepsilon^* k^{\\varepsilon,+}$ that \n$$\nk^{\\varepsilon,-}_1 = - G\\,\\overline b(GG^*) k^{\\varepsilon,+}_2 + O(\\eta_\\varepsilon)\n$$ \nand\n\\begin{align*}\n\\Big\\langle{\\rm op}_\\varepsilon(b^{+,out})\\check v^\\varepsilon_1,\\check v^\\varepsilon_1\\Big\\rangle  \n&= \n\\left\\langle{\\rm op}_\\varepsilon\\!\\left((1-\\widetilde T_\\varepsilon)b^{+,in}\\right)k^{\\varepsilon,+}_{2},k^{\\varepsilon,+}_{2}\\right\\rangle +O(\\eta_\\varepsilon)\\\\\n&=\n\\left\\langle{\\rm op}_\\varepsilon\\!\\left((1-\\widetilde T_\\varepsilon)b^{+,in}\\right)\\check v^\\varepsilon_{2},\\check v^\\varepsilon_{2}\\right\\rangle +O(\\eta_\\varepsilon).\n\\end{align*}\n\n\n\n\n\n\n\n\\section{Numerical simulations}\\label{sec:numerics}\n \nWe consider four specific examples of avoided crossings in one space dimension. The corresponding eigenvalues surfaces are plotted in Figure \\ref{fig:surfaces}, \nwhile the detailed definition of the four model systems is given in Tables~\\ref{tab:param1} and \\ref{tab:param2}. Three examples are taken from Tully's 1994 paper \\cite{Tu1} on the surface hopping algorithm of the fewest switches: the simple, the dual and the extended crossing. The arctangent crossing is included as an example, which meets the assumptions of our main Theorem~\\ref{theorem}. Its eigenvalues are defined by smooth functions, and the coefficients except for the minimum gap parameter $\\delta_0$ are of order one with respect to the semiclassical parameter~$\\varepsilon$. \nIn all simulations, the initial data\n$$\n\\psi_0(q) = (\\pi\\varepsilon)^{-1\/4} \\exp\\!\\left(-{\\textstyle\\frac{1}{2\\varepsilon}(q-q_0)^2 + \\frac{i}{\\varepsilon}} p_0(q-q_0)\\right) e^\\pm(q)\n$$\nare multiples of a Gaussian wave packet with phase space centers $(q_0,p_0)\\in{\\mathbb R}^2$ and a real-valued eigenvector $e^\\pm(q)$ of the matrix $V(q)$. Following \\cite{Tu1}, the three Tully examples have the  semiclassical parameter \n$$\n\\varepsilon=1\/\\sqrt{2000}\\approx 0.02,\n$$ \nwhich roughly corresponds to the mass of the hydrogen atom of $1836$ atomic units. For the arc\\-tangent crossing we have chosen $\\varepsilon=10^{-3}$. The time interval $[0,t_{\\rm fin}]$ of all simulations allows that the wave packet passes the crossing region once. \n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{surfaces}\n\\caption{The eigenvalue surfaces of the potentials considered for our numerical simulations. }\n\\label{fig:surfaces}\n\\end{figure}\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{c|c|c}\n& Simple & Arctangent \\\\\\hline\n$\\varepsilon$ & $2000^{-1\/2}$ & $10^{-3}$ \\\\\n$\\delta_0$ & $0.005$ & $10^{-3\/2}$ \\\\\\hline\ninitial level & minus & plus \\\\\n$(q_0,p_0)$ & $(-5,1)$ & $(-1,1)$ \\\\\n$t_{\\rm fin}$ & $10$ & $2$\\\\\\hline\n$\\beta(q)$ & $0.01\\,{\\rm sgn}(q)\\big(1-{\\rm e}^{-1.6|q|}\\big)$ & $\\arctan(q)$ \\\\\n$\\gamma(q)$ & $\\delta_0{\\rm e}^{-q^2}$ & $\\delta_0$ \\\\\n$\\alpha(q)$ & $0$ & $0$\n\\end{tabular}\n\\caption{\\label{tab:param1}\nFunctions and parameters defining the simple and the arctangent crossings. In both cases, the eigenvalue surfaces \nhave their minimal gap at $q=0$.}\n\\end{table}\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{c|c|c}\n& Dual & Extended\\\\\\hline\n$\\varepsilon$ & $2000^{-1\/2}$ & $2000^{-1\/2}$\\\\\n$\\delta_0$ & $0.015$ &  $6\\cdot 10^{-4}$\\\\\\hline\ninitial level & minus & plus\\\\\n$(q_0,p_0)$ & $(-5,1)$ & $(0,-1)$\\\\\n$t_{\\rm fin}$ & $10$ & $10$\\\\\\hline\n$\\beta(q)$ & $0.05 {\\rm e}^{-0.28 q^2} - 0.025$ & $\\delta_0$\\\\\n$\\gamma(q)$ & $\\delta_0{\\rm e}^{-0.06 q^2}$ & $0.1{\\rm sgn}(q)(1-{\\rm e}^{-0.9|q|})+0.1$\\\\\n$\\alpha(q)$ & $-\\beta(q)$ & $0$\\\\\n\\end{tabular}\n\\caption{\\label{tab:param2}\nFunctions and parameters defining the dual and the extended crossing. The dual crossing surfaces have their minimal gap at $q\\approx\\pm 1.6$. The surface gap of the extended crossing decreases monotonically as $q\\to-\\infty$.}\n\\end{table} \n\n\n\n\\subsection{A surface hopping algorithm}\n\nOur analysis of the dynamics through an avoided eigenvalue crossing suggests \na surface hopping algorithm formulated in terms of Wigner functions. Such an algorithm can either treat the effective \nLandau-Zener transitions by a deterministic branching scheme or by a probabilistic \naccept-reject mechanism. The probabilistic version, which will be discussed here, keeps the number of trajectories constant, \nwhich is to the best advantage for the memory requirements of the algorithm, see the simulations \nfor a model of pyrazine and of the  ammonia cation \\cite{LS,BDLT}.\n\n\\medskip\n\nFor notational simplicity, we restrict ourselves to the case, that the initial data are associated with the \nupper level. The same reasoning applies for initial data associated to the lower level with the obvious alterations. \nThe probabilistic surface hopping algorithm works as follows: \n\n\\subsubsection{Initial sampling}\nDraw $N\\in{\\mathbb N}$ pseudorandom phase space samples \n$$\n(q_1,p_1)^+,\\ldots,(q_N,p_N)^+,\n$$ \nwhich are independent and identically distributed according to $w^\\varepsilon_+(\\psi_0)$. If the initial data are a Gaussian wave packet, then the Wigner function is given by the explicit formula  \n$$\nw^\\varepsilon_+(\\psi_0)(q,p) = (\\pi\\varepsilon)^{-1} \\exp\\!\\left(-\\tfrac{1}{\\varepsilon}|(q,p)-(q_0,p_0)|^2\\right),\n$$\nthat is the densitiy function of a bivariate normal distribution.\n\n\\subsubsection{Transport}\nPropagate the sample points along the Hamiltonian curves of $\\Phi^t_+$.\n\n\\subsubsection{Non-adiabatic transitions}\nIf a trajectory $t\\mapsto(q_j^+(t),p_j^+(t))$ attains a local minimal gap of size smaller than $\\sqrt\\varepsilon$ at time $t^*$ in the phase space point $(q^*,p^*)$, then draw a pseudrandom number $\\zeta$ uniformly distributed in $[0,1]$. \nIf $T_\\varepsilon(q^*,p^*)>\\zeta$, then a hop occurs according to \n$$\n(q^*,p^*,+) \\longrightarrow (q^*,p_*+\\omega(q^*,p^*),-).\n$$\nOtherwise, the trajectory continues on the upper level. \n\n\\subsubsection{Computation of expectation values}\nIf at time $t$ there are $N^+$ trajectories on the upper and $N^-$ trajectories on the lower level, then the expectation values for observables \n$$\na(q,p)=a^+(q,p)\\Pi^+(q)+a^-(q,p)\\Pi^-(q)\n$$ \nare approximated as \n\\begin{equation}\n\\label{eq:exp}\n({\\rm op}_\\varepsilon(a)\\psi_t^\\varepsilon,\\psi_t^\\varepsilon)\\;\\approx\\; \\frac{1}{N^+} \\sum_{j=1}^{N^+}a^+(q_j^+(t),p_j^+(t)) + \\frac{1}{N^-} \\sum_{j=1}^{N^-}a^-(q_j^-(t),p_j^-(t)).\n\\end{equation}\n\n\\medskip\nThe overall accuracy of the approximation is then determined by the initial sampling, the discretization of the Hamiltonian flows, and the asymptotic accuracy of the surface hopping semigroup. If the flow discretization \nis a symplectic order $p$ method with time step $\\Delta_t$, then the error of the approximation (\\ref{eq:exp}) is\n$$\nO(1\/\\sqrt{N}) + O(\\Delta_t^p) + O(\\varepsilon^\\gamma),\n$$\n$\\gamma=1\/8$ according to Theorem~\\ref{theorem}.\nFor the numerical experiments presented here, the initial sampling and the discretized classical flows have been accurate enough, such that the asymptotic $\\varepsilon$-dependent error of the algorithm is dominant. \n\n\\medskip\n\nOur reference values have been obtained from numerically converged solutions of the Schr\\\"odinger equation (\\ref{eq:schro}), which have been computed by a Strang splitting scheme with Fourier collocation. \nAll figures show the reference values as solid lines, while the little stars and circles mark values computed by the surface hopping algorithm. We note, that the space grid for the Fourier collocation must \nresolve the oscillations of the wave function, which is easily achieved in one space dimension. For higher dimensional problems, however, such discretizations suffer from the curse of dimensionality, \nwhich is not the case for our surface hopping algorithm.    \n\n\n\n\\subsection{The simple and the arctangent crossing}\n\\label{sec:simple}\n\n\\begin{figure}\n\\subfigure[Simple: $\\varepsilon\\approx 0.02$, $\\delta_0=0.005$]{\\includegraphics[height=0.4\\textheight]{simple}}\n\\subfigure[Arctangent: $\\varepsilon=10^{-3}$, $\\delta_0 = \\sqrt\\varepsilon$]{\\includegraphics[height=0.4\\textheight]{atan}}\n\\caption{The simple and the arctangent crossing. The initial wave function is associated with the lower (a) and the upper level~(b). \nThe results of the surface hopping algorithm, marked with stars and circles, are in good agreement with the reference.} \n\\label{fig:simple}\n\\end{figure}\n\nFor both examples the surface hopping algorithm produces meaningful approximations of the dynamics even though the simple crossing has a non-smooth potential matrix and a surface gap just varying by a factor two.\nFigure~\\ref{fig:simple} shows population transfer away from the initial energy level and a corresponding change of the average momentum on the initial level. The final populations are approximated within an accuracy of \n$0.04$ to $0.05$. The error of the momentum expectation is even smaller.  \n\n\n\n\n\\subsection{The dual and the extended crossing}\n\\label{sec:dual}\n\n\\begin{figure}\n\\subfigure[Dual: $\\varepsilon\\approx 0.02$, $\\delta_0=0.015$]{\\includegraphics[height=0.4\\textheight]{dual}}\n\\subfigure[Extended: $\\varepsilon\\approx 0.02$, $\\delta_0=6\\cdot10^{-4}$]{\\includegraphics[height=0.4\\textheight]{extended}}\n\\caption{The dual and the extended crossing. The initial wave function is associated with the lower (a) and the upper eigenvector~(b). In both cases, the surface hopping algorithm expectedly fails to reproduce the dynamics.} \n\\label{fig:dual}\n\\end{figure}\n\n\nThese two examples illustrate the limitations of our approximation. Since the dual crossing model has two subsequent crossings at $q\\approx -1.6$ and $q\\approx 1.6$, one has to expect interferences between the upper and the lower level for the passage of the second crossing, which cannot be resolved by the surface hopping semigroup. The numerical simulations confirm this expectation. \nFigure~\\ref{fig:dual}(a) shows that the surface hopping algorithm correctly resolves the first passage, while the non-adiabatic transfer for the second passage is definitely wrong. Since the mean size of the eigenvalue gap and the mean Landau-Zener rate computed by the surface hopping approach qualitatively reflect the true dynamical situation also for the passage of the second crossing, the failure of the approximation must be due to unresolved interlevel interferences. \n\n\\medskip\n\nAlso the extended crossing case is not covered by our analysis, since the eigenvalue surfaces do not have a minimal gap but a distance which monotonically decreases as $q\\to-\\infty$. We have therefore also considered a modified surface hopping algorithm, which allows non-adiabatic transitions at any time step of the numerical simulation if the trajectory's Landau-Zener coefficient is larger than a random number uniformly distributed within the interval $[0,1]$. The outcome of this simulation is marked with circles. The reference dynamics show a monotonously decreasing population of the upper level. The surface hopping algorithm does not initiate any non-adiabatic transfer, since for the trajectories there is no  minimal surface gap. Consequently, it wrongly produces a constant upper level population. The modified surface hopping with unconstrained Landau-Zener transitions starts the non-adiabatic transfer much too early but finally arrives at an upper level population, which is rather close to the true solution.  \n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nLet $X$ be a topological space and let $U$, $V \\subseteq X$ be open subsets such \nthat $X=U\\cup V$. Frequently, natural invariants of $X$ can be determined by the \nrestriction of these invariants to $U$, $V$ and $U \\cap V$. The prototypical \nexamples are the Mayer--Vietoris exact sequences in algebraic topology. These \nresults have proved to be very useful for inductive arguments.\n\nNow let $X$ be a variety, scheme, algebraic space, or algebraic stack. It is \nstraightforward to adapt the topological results (e.g., Mayer--Vietoris with open coverings) to this situation. In algebraic geometry, however, open coverings are often too restrictive to use in inductive arguments. A consideration of the existing literature, motivated us to make the following definition.\n\\begin{definition}\n  Consider a cartesian diagram of algebraic stacks\n  \\begin{equation}\\label{E:MV-square}\n    \\vcenter{\\xymatrix{\n        U'\\ar[r]^{j'}\\ar[d]_{f_U} & X'\\ar[d]^f \\\\\n        U\\ar[r]^j & X,\\ar@{}[ul]|\\square\n      }}\n  \\end{equation}\n  where $j$ is an open immersion. If $W \\to X$ is a morphism of algebraic stacks, then we \n  let  $f_W \\colon W'=X'\\times_X W \\to W$ be the induced morphism. It will be convenient \n  to let $i\\colon Z \\hookrightarrow X$ denote a closed immersion with complement $U$.\n  \n  A \\emph{weak Mayer--Vietoris square} is a cartesian diagram of \n  algebraic stacks as in \\eqref{E:MV-square} such that for every morphism of algebraic \n  stacks $W \\to X$ with image disjoint from $U$, the induced morphism $f_W \\colon W' \\to W$ is an isomorphism.\n\\end{definition}\nThe condition of being a weak Mayer--Vietoris square is trying to capture that $X'$ contains \nall neighborhoods of the complement of $U$ in $X$. In particular, if $X$ and $X'$ are locally \nnoetherian, then being a weak Mayer--Vietoris square is equivalent to $f_Z$ being an isomorphism and $f$ being flat at all \npoints over $Z$ (Lemma \\ref{L:MV-different-notions}). \n\nFix a weak Mayer--Vietoris square as in \\eqref{E:MV-square}.  \nIf $f$ is \\'etale, then it is also known as an \\emph{\\'etale\nneighborhood}, or upper distinguished square, or Nisnevich square. These were recently treated in depth in~\\cite{MR2774654}. Some highlights of the theory are that \\'etale neighborhoods are pushouts in the $2$-category of algebraic stacks and that quasi-coherent sheaves (and many more things) can be glued along \\'etale neighborhoods. \\'Etale neighborhoods feature \nprominently in the interactions between algebraic geometry and topology. \n\n\nThere is another class of weak Mayer--Vietoris squares, which generalize \\'etale neighborhoods, that have been \nconsidered and applied to great effect in the past \\cite{MR0272779,MR1374653}. These are our main object of interest.\n\\begin{definition}\n  A \\emph{flat Mayer--Vietoris square} is a weak Mayer--Vietoris square as in \\eqref{E:MV-square} such that $f$ is flat.\n\\end{definition}\nOur first main result is the following.\n\\begin{maintheorem}\\label{MT:pushout_fmv}\n  Fix a flat Mayer--Vietoris square as in \\eqref{E:MV-square}. If $X$ is locally excellent, then the square \\eqref{E:MV-square} is a pushout in the \n  $2$-category of algebraic stacks.\n\\end{maintheorem}\n\nAn algebraic stack is locally excellent if it admits a smooth cover by an excellent scheme; in particular, algebraic stacks that are locally of finite type over the spectrum of a field, $\\mathbb{Z}$, or a complete local noetherian ring are locally excellent. \nTheorem \\ref{MT:pushout_fmv} is the key technical result used to establish Tannaka \nduality for algebraic stacks with non-separated diagonals \n\\cite{hallj_dary_coherent_tannakian_duality}---if the diagonals are separated, then \n\\cite[Cor.~6.5.1(g)]{MR1432058} is sufficient for the Tannakian application.\n\nIn general, Mayer--Vietoris squares are interesting since objects can be glued along\nthem. To formalize this, consider a $2$-presheaf $\\tfF\\colon\n  (\\STACKS{X})^\\circ\\to \\mathsf{Cat}$~\\cite[App.~D]{MR2774654}, e.g., the $2$-presheaf\n  $\\tfF(-)=\\mathsf{QCoh}(-)$ of quasi-coherent sheaves of modules. By pull-back, we\n  obtain a functor\n  \\[\n  \\Phi_\\tfF\\colon \\tfF(X)\\to  \\tfF(X')\\times_{\\tfF(U')} \\tfF(U) \n  \\]\n  where the right-hand side denotes triples $(W',\\theta,W_U)$ where\n  $W'\\in \\tfF(X')$, $W_U\\in \\tfF(U)$ and\n  $\\theta\\colon j'^*W'\\to f_U^*{W_U}$ is an\n  isomorphism. When the functor $\\Phi_\\tfF$ is an equivalence, we\n  say that we can glue $\\tfF$ along the square. \n\n  We do not have any general gluing results for $2$-\\emph{sheaves} as for\n  \\'etale neighborhoods~\\cite[Thm.~A]{MR2774654}---nor do we expect such---but we\n  will give gluing results for the following $2$-presheaves, where the values over an algebraic stack $Y$ are as follows:\n  \\begin{itemize}[leftmargin=2cm]\n    \\item[$\\mathsf{QCoh}(Y)$] the category of quasi-coherent sheaves of $\\Orb_Y$-modules.\n   \n    \\item[$\\AFF(Y)$] the category of affine morphisms to $Y$.\n    \\item[$\\mathsf{Qaff}(Y)$] the category of quasi-affine morphisms to $Y$.\n   \n    \\item[$\\AlgSp(Y)$] the category of representable morphisms $Y'\\to Y$.\n    \\item[$\\AlgSp_\\mathrm{lfp}(Y)$] the category of representable morphisms $Y'\\to Y$,\n      locally of finite presentation.\n    \\item[$\\Hom(Y,W)$] the category of morphisms to a fixed algebraic stack $W$.\n   \n    \\item[$\\mathrm{\\acute{E}t}(Y)$] the category of \\'etale representable morphisms $Y'\\to Y$,\n      or equivalently, the category of cartesian sheaves\n      of sets on the lisse-\\'etale site of $Y$.\n    \\item[$\\mathrm{\\acute{E}t}_c(Y)$] the category of finitely presented \\'etale representable\n      morphisms $Y'\\to Y$, or equivalently, the category of constructible\n      sheaves of sets.\n  \\end{itemize}\n  We prove the following gluing results. \n  \\begin{maintheorem}\\label{MT:glue_fmv}\n    Fix a flat Mayer--Vietoris square as in \\eqref{E:MV-square}. If $j$ is quasi-compact, \n    then\n    \\begin{enumerate}\n    \\item\\label{MTI:glue_fmv:qcoh+aff+qaff}\n      $\\Phi_{\\mathsf{QCoh}}$, $\\Phi_{\\AFF}$ and $\\Phi_{\\mathsf{Qaff}}$ are\n      equivalences of categories;\n    \\item\\label{MTI:glue_fmv:algsp-ff}\n      $\\Phi_{\\AlgSp}$ is fully faithful;\n    \\item\\label{MTI:glue_fmv:hom-ff}\n      $\\Phi_{\\Hom(-,W)}$ is fully faithful for every algebraic stack $W$ and an \n      equivalence if $W$ has quasi-affine diagonal; and\n    \\item\\label{MTI:glue_fmv:algsp-exc}\n      $\\Phi_{\\AlgSp_{\\mathrm{lfp}}}$ is an equivalence of categories if $X$ is\n      locally excellent.\n    \\end{enumerate}\n  \\end{maintheorem}\n  \\begin{maintheorem}\\label{MT:etale-gluing-for-wmv}\n    Fix a weak Mayer--Vietoris square as in \\eqref{E:MV-square}. \n    If $j$ is quasi-compact, then $\\Phi_{\\mathrm{\\acute{E}t}}$ and $\\Phi_{\\mathrm{\\acute{E}t}_c}$ are equivalences of categories.\n  \\end{maintheorem}\n  Theorem \\ref{MT:pushout_fmv} essentially follows from Theorem \\ref{MT:glue_fmv}\\itemref{MTI:glue_fmv:algsp-exc}.\n  Theorem \\ref{MT:glue_fmv}\\itemref{MTI:glue_fmv:algsp-ff}--\\itemref{MTI:glue_fmv:algsp-exc} relies upon Theorem~\\ref{MT:etale-gluing-for-wmv}\n  and general N\\'eron--Popescu desingularization \\cite{MR818160}.\n  Gabber's rigidity\n  theorem (Theorem~\\ref{T:rigidity-theorem}) features in the proof of \n  Theorem~\\ref{MT:etale-gluing-for-wmv}. Both results also depend upon some further \n  gluing results for quasi-coherent sheaves.\n\n  To prove Theorem \\ref{MT:glue_fmv}, we approximately follow the approach of \n  \\cite{MR1432058}. The main idea is pass to a square as in \n\\eqref{E:MV-square} where $f\\colon X' \\to X$ is replaced by its diagonal $\\Delta_f \n\\colon X' \\to X'\\times_X X'$ and $j\\colon U \\to X$ is replaced by $j'\\times j' \\colon \nU'\\times_{X} U' \\to X'\\times_X X'$. In particular, unless $f$ is unramified the resulting square will not be a flat Mayer--Vietoris square. Moreover, even if $X$ is a locally noetherian algebraic stack, then unless $f$ is locally of type, $X'\\times_X X'$ has no reason to be locally noetherian. For example, in applications one often takes $X=\\spec A$ and $X'=\\spec \\hat{A}$, where $\\hat{A}$ denotes the $I$-adic completion with respect to some ideal $I$ of $A$; in this situation, $X'\\times_X X'$ is only noetherian when $X'=X$.\n\nTo manage such squares, we have the following natural variant of what Moret-Bailly considered.\n\\begin{definition}\n  A \\emph{tor-independent Mayer--Vietoris square} is a weak Mayer--Vietoris square \n  as in \\eqref{E:MV-square} such that every morphism of algebraic stacks $W \\to X$ with image disjoint from $U$ is tor-independent of $f$ (Definition \\ref{D:f-flat}).\n\\end{definition}\nIf $f$ is affine and $U$ is the complement of a finitely presented closed immersion $i\\colon Z \\hookrightarrow X$, then a tor-independent Mayer--Vietoris square is the same as a triple $(X,Z,X')$ satisfying the (TI) condition in the terminology of~\\cite[0.2, 0.6]{MR1432058} (Lemma \\ref{L:MV-open-qc}\\itemref{LI:MV-open-qc:ti}). If $X'$ and $X$ are locally noetherian, then tor-independent Mayer--Vietoris squares are very similar to flat Mayer--Vietoris squares (Lemma \\ref{L:MV-different-notions}). \nWe now state our gluing result for tor-independent Mayer--Vietoris squares, which we can prove for $f$-flat objects (see Definition \\ref{D:f-flat}).\n\\begin{maintheorem}\\label{MT:glue_timv}\n  Fix a tor-independent Mayer--Vietoris square as in \\eqref{E:MV-square}. If $j$ is \n  quasi-compact, then \n  \\begin{enumerate}\n  \\item \\label{MTI:glue_timv:qcoh}$\\Phi_{\\mathsf{QCoh}_{f-\\mathrm{fl}}}$ is an equivalence and\n  \\item \\label{MTI:glue_timv:algsp}$\\Phi_{\\AlgSp_{f-\\mathrm{fl}}}$ is fully faithful.\n  \\end{enumerate}\n\\end{maintheorem}\nFor tor-independent Mayer--Vietoris squares, we prove the following non-noetherian variant of Theorem \\ref{MT:pushout_fmv}. \n\\begin{maintheorem}\\label{MT:timv_dm_push}\n  Fix a tor-independent Mayer--Vietoris square as in \\eqref{E:MV-square}. If $j$ \n  is quasi-compact, then it is a pushout in the $2$-category of Deligne--Mumford \n  stacks.\n\\end{maintheorem}\nSince we make no separation assumptions on our algebraic stacks, Theorem \n\\ref{MT:timv_dm_push} generalizes recent work of Bhatt \n\\cite{2014arXiv1404.7483B}. Note, however, that while Bhatt uses (derived) \nTannaka duality to prove a version of Theorem \\ref{MT:timv_dm_push} for quasi-compact and quasi-separated algebraic spaces, we work in the opposite direction (i.e., we use pushouts to prove Tannaka duality in~\\cite{hallj_dary_coherent_tannakian_duality}). \n\n\\begin{remark}\n  While it may appear that our results are weaker than the\n  corresponding \\'etale gluing results because we require $j$ to be\n  quasi-compact, this turns out to not be the case. Indeed, for\n  \\'etale neighborhoods, at least smooth-locally on $X$ there is always an \\'etale \n  neighborhood $X''$ of $Z'$ in $X'$ and an open $U_0 \\subseteq \n  U$ such that $U_0 \\to X$ is quasi-compact and the resulting square with $X'' \\to X$ is an \n  \\'etale neighborhood of $X\\setminus U_0$.\n\\end{remark}\n\n\\subsection*{Overview}\n  In Section~\\ref{S:prelim} we give some preliminaries on tor-independence. In\n  Section~\\ref{S:MV} we compare the different notions of Mayer--Vietoris\n  squares and give several examples.  In Section~\\ref{S:QCoh-gluing} we glue\n  quasi-coherent sheaves in tor-independent Mayer--Vietoris squares\n  (Theorem~\\ref{MT:glue_timv}\\itemref{MTI:glue_timv:qcoh} and\n  Theorem~\\ref{MT:glue_fmv}\\itemref{MTI:glue_fmv:qcoh+aff+qaff}).\n\n  In Section~\\ref{S:etale-sheaves} we prove some fundamental theorems for\n  \\'etale sheaves of sets on algebraic stacks. In particular, we prove that\n  every sheaf on a quasi-compact and quasi-separated algebraic stack is a\n  filtered colimit of constructible sheaves. We also discuss henselian pairs of stacks.\n\n  In Section~\\ref{S:etale-gluing} we prove Gabber's rigidity theorem and glue\n  \\'etale sheaves in weak Mayer--Vietoris squares\n  (Theorem~\\ref{MT:etale-gluing-for-wmv}).\n  In the noetherian case, Gabber's rigidity theorem follows immediately from\n  Ferrand--Raynaud~\\cite[App.]{MR0272779}. In the non-noetherian case, which is\n  essential for the applications in this paper, the previous\n  proof~\\cite[Exp.~20]{MR3309086} was much more involved. Using our results on\n  gluing of sheaves, we provide a self-contained proof (for $H^0$ but the methods\n  can be extended to $H^1$).\n\n  In Section~\\ref{S:algsp-gluing} we glue algebraic spaces and prove that\n  Mayer--Vietoris squares are pushouts (Theorems~\\ref{MT:pushout_fmv},\n  \\ref{MT:glue_fmv}, \\ref{MT:glue_timv}\\itemref{MTI:glue_timv:algsp},\n  and~\\ref{MT:timv_dm_push}).\n\n \n \n \n \n \n \n \n \n\\section{Preliminaries}\\label{S:prelim}\nHere we record some preliminary results that will be of use in subsequent sections. \nMost of these are globalizations of the affine results proved in \n\\cite[\\S2]{MR1432058}. We begin with the following definition.\n\\begin{definition}\\label{D:f-flat}\n  Let $f\\colon X' \\to X$ and $g\\colon W \\to X$ be morphisms of algebraic stacks.  Let $N \n  \\in \\mathsf{QCoh}(X')$ and $M \\in \\mathsf{QCoh}(W)$.\n  \\begin{enumerate}\n  \\item We say that $M$ and $N$ are \\emph{tor-independent} if \n    $\\mathcal{T}or_i^{X,f,g}(N,M) = 0$\n    for all $i>0$ \\cite[App.~C]{hallj_openness_coh}. Equivalently, for all smooth \n    morphisms $\\spec A \\to X$, $\\spec A' \\to \\spec A \\times_X X'$ and $\\spec B \\to \\spec \n    A\\times_X W$ we have \n    \\[\n    \\Tor^A_i(N(\\spec A' \\to X'),M(\\spec B \\to W)) = 0\n    \\]\n    for all $i>0$. \n  \\item We say that $M$ is \\emph{$f$-flat} if it is tor-independent of $\\Orb_{X'}$.  We let \n    $\\mathsf{QCoh}_{f-\\mathrm{fl}}(W) \\subseteq \\mathsf{QCoh}(W)$ denote the subcategory of $f$-flat \n    quasi-coherent sheaves on $W$.\n  \\item We say that $g$ is \\emph{$f$-flat} if $\\Orb_W$ is $f$-flat. Note that $g$ is $f$-flat \n    if and only if $f$ is $g$-flat. In particular, we may also say that $f$ and $g$ are \n    tor-independent.\n  \\end{enumerate}\n\\end{definition}\nThe following lemma is immediate from the definitions (we employ the notational conventions from \\cite{perfect_complexes_stacks}).\n\\begin{lemma}\\label{L:der_char_fflat}\n  Let $f\\colon X' \\to X$ be a morphism of algebraic stacks and let $M \\in \n  \\mathsf{QCoh}(X)$. Then $M$ is $f$-flat if and only if the natural map $\\mathsf{L} \\QCPBK{f}M \\to \n  f^*M$ is a quasi-isomorphism in $\\DQCOH(X')$.\n\\end{lemma}\nThe following notation will also be useful.\n\\begin{notation}\n  Let $i\\colon Z \\hookrightarrow X$ be a closed immersion of algebraic stacks, which is defined by \n  quasi-coherent ideal $I$. For each integer $n\\geq 0$, let $i^{[n]} \\colon Z^{[n]} \\hookrightarrow X$ be \n  the closed immersion defined by the quasi-coherent ideal $I^{n+1}$. Note that if $i$ is a \n  finitely presented closed immersion, then so too is $i^{[n]}$ for all $n\\geq 0$.\n\\end{notation}\nThe following lemma will eventually be improved (see Corollary \\ref{C:gluing_special}), but is for the meantime sufficient for our purposes.\n\\begin{lemma}\\label{L:flatness_conc_cl}\n  Consider a cartesian diagram of algebraic stacks:\n  \\[\n  \\xymatrix{V' \\ar[r]^{v'} \\ar[d]_{g_V} & W' \\ar[d]^g\\\\V \\ar[r]^v & W. }\n  \\]\n  Assume that $v$ is $g$-flat and a closed immersion.\n  \\begin{enumerate}\n  \\item \\label{LI:flatness_conc_cl:conc}  If $M \\in \n    \\mathsf{QCoh}_{g_v-\\mathrm{fl}}(V)$, then $v_*M \\in \\mathsf{QCoh}_{g-\\mathrm{fl}}(W)$.\n  \\item \\label{LI:flatness_conc_cl:iso} If $g_V$ is \n    an isomorphism, then $V^{[n]}$ is $g$-flat and $g_{V^{[n]}}  \\colon \n    W'\\times_W V^{[n]} \\to V^{[n]}$ is an isomorphism  for every \n    $n\\geq 0$.\n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  Both claims are smooth local on $W$, so we may assume that $W=\\spec A$ is an \n  affine scheme. Claim \\itemref{LI:flatness_conc_cl:conc} follows from tor-independent \n  base change \\cite[Cor.~4.13]{perfect_complexes_stacks}. Indeed, this provides \n  quasi-isomorphisms: \n  \\[\n  \\mathsf{L} \\QCPBK{g}(v_*M) \\simeq \\mathsf{L} \\QCPBK{g}\\mathsf{R} v_*M \\simeq \\mathsf{R} v'_*\n  \\mathsf{L}\\QCPBK{(g_V)}M \\simeq v'_*g_V^*M \\simeq g^*v_*M.\n  \\]  \n  Claim \\itemref{LI:flatness_conc_cl:iso} is essentially the local criterion for \n  flatness \\cite[$0_{\\mathrm{III}}$.10]{EGA}, but we will spell out the details. Assume that $V=\\spec (A\/I)$. \n  Fix an integer $n\\geq 1$. By induction we may also assume \n  that $g_{V^{[n-1]}} \\colon W'\\times_W \\spec (A\/I^n) \\to \\spec (A\/I^n)$ is an isomorphism\n  and $A\/I^n$ is $g$-flat. Now \\itemref{LI:flatness_conc_cl:conc} implies that\n  every \n  $A\/I^n$-module is $g$-flat. In particular, $I^n\/I^{n+1}$ is $g$-flat so the distinguished \n  triangle\n  \\[\n  \\mathsf{L}\\QCPBK{g}(I^n\/I^{n+1}) \\to \\mathsf{L} \\QCPBK{g}(A\/I^{n+1}) \\to \\mathsf{L} \\QCPBK{g}(A\/I^{n}) \\to   \\mathsf{L}\\QCPBK{g}(I^n\/I^{n+1})[1]\n  \\]\n  now implies that $A\/I^{n+1}$ is $g$-flat. By \\itemref{LI:flatness_conc_cl:conc}, we see \n  that every $A\/I^{n+1}$-module is $g$-flat. Clearly $g_{V^{[n]}}$ is a nil-immersion, so \n  to prove that it is an isomorphism it is sufficient to prove that it is also flat. This is \n  smooth local on $W'$, so we may assume that $W'=\\spec A'$. Let $N$ be an \n  $A\/I^{n+1}$-module; then $N$ is $g$-flat as seen above. It follows that there are \n  quasi-isomorphisms:\n  \\begin{align*}\n  (A'\/I^{n+1}A') \\otimes^{\\mathsf{L}}_{A\/I^{n+1}} N &\\simeq (A'\\otimes_A [A\/I^{n+1}]) \n  \\otimes_{A\/I^{n+1}}^{\\mathsf{L}} N \\\\\n    &\\simeq (A'\\otimes_A^{\\mathsf{L}} [A\/I^{n+1}]) \n  \\otimes_{A\/I^{n+1}}^{\\mathsf{L}} N \\\\\n    &\\simeq A'\\otimes_A^{\\mathsf{L}} N \\\\\n    &\\simeq (A'\\otimes_A N)[0].\n  \\end{align*}\n  Hence, $g_{V^{[n]}}$ is flat and we have the claim.\n\\end{proof}\nLet $X$ be an algebraic stack and let $i\\colon Z \\hookrightarrow X$ be a closed immersion with complement $j\\colon U \\to X$. Define\n\\[\n\\mathsf{QCoh}_{Z}(X) = \\{M \\in \\mathsf{QCoh}(X) \\,:\\, j^*M \\cong 0\\}.\n\\]\nNote that $\\mathsf{QCoh}_{Z}(X)$ only depends on the closed subset $|Z| \\subseteq |X|$.\n\\begin{lemma}\\label{L:fp_thickenings}\n  Let $X$ be a quasi-compact algebraic stack. Let $i\\colon Z \\hookrightarrow X$ be a finitely \n  presented closed immersion.\n  \\begin{enumerate}\n  \\item \\label{LI:fp_thickenings:ft} Let $M \\in \\mathsf{QCoh}_{Z}(X)$. If $M$ is of finite type, then there exists an $n\\gg 0$ such that the natural map \n      $M \\to i_*^{[n]}(i^{[n]})^*M$ is an isomorphism. \n  \\item\\label{LI:fp_thickenings:cl} If $W \\subset X$ is a closed substack with $|W| \\subseteq |Z|$, then $W \\subseteq Z^{[n]}$ for some~${n\\gg 0}$.\n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  For \\itemref{LI:fp_thickenings:ft}: we may assume that $X=\\spec A$ is an affine scheme \n  and $Z=\\spec (A\/I)$, where $I=(f_1,\\dots,f_r)$ is a finitely generated ideal of $A$. By \n  assumption, $M_{f_i} = 0$ for each $i=1$, $\\dots$, $r$. As $M$ is finitely generated, it \n  follows that there exists $n\\gg 0$ such that $f_i^nM = 0$ for all $i=1$, $\\dots$, $r$. The \n  claim follows.\n\n  For \\itemref{LI:fp_thickenings:cl}: let $W_0 = Z\\times_X W$. Then  $W_0 \\hookrightarrow W$ is a \n  surjective and finitely \n  presented closed immersion. From \\itemref{LI:fp_thickenings:ft}, it follows that $W \n  \\subseteq W_0^{[n]}$ for some $n\\gg 0$. But $W_0^{[n]} \\subseteq Z^{[n]}$ and we have \n  the claim. \n\\end{proof}\n\\section{Mayer--Vietoris squares}\\label{S:MV}\nIn this section, we compare various notions of Mayer--Vietoris squares. \n\\begin{lemma}\\label{L:mv_bc}\n  Fix a cartesian diagram as in \\eqref{E:MV-square}.\n  \\begin{enumerate}\n  \\item \\label{LI:mv_bc:w_f_mv} If the square is a weak (resp.~flat) Mayer--Vietoris \n    square, then it remains so after arbitrary base change on $X$.\n  \\item \\label{LI:mv_bc:mv} If the square is a tor-independent Mayer--Vietoris square, then it \n    remains so after $f$-flat base change on $X$.\n  \\item \\label{LI:mv_bc:verify}  The properties of being a flat, tor-independent, or weak \n    Mayer--Vietoris square are flat local on $X$.\n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  Claim \\itemref{LI:mv_bc:w_f_mv} is trivial. For \\itemref{LI:mv_bc:mv}: it is sufficient \n  to prove that if $w\\colon W \\to X$ is $f$-flat and $v\\colon V \\to X$ is $f$-flat, then \n  $W\\times_X V \\to W$ is $f_V$-flat, where {$f_V \\colon X'\\times_X V \\to V$}. This is local on $W$, $V$, $X$ and $X'$, so we may assume that they are all affine. Let $X=\\spec A$, $X'=\\spec A'$, $V=\\spec C$ and $W=\\spec B$. It is thus sufficient to prove that $(B\\otimes_A C) \\otimes^{\\mathsf{L}}_C (C\\otimes_A A') \\simeq (B\\otimes_A C) \\otimes_C (C\\otimes_A A')$. To see this, we observe that\n  \\[\n  (B\\otimes_A C) \\otimes^{\\mathsf{L}}_C (C\\otimes_A A') \\simeq   (B\\otimes_A C) \\otimes^{\\mathsf{L}}_C (C\\otimes_A^{\\mathsf{L}} A') \\simeq   B\\otimes_A (C \\otimes^{\\mathsf{L}}_A A') \\simeq B\\otimes_A (C \\otimes_A A'),\n  \\]\n  which gives the claim. The claim \\itemref{LI:mv_bc:verify} is immediate from flat \n  descent.\n\\end{proof}\nAs the following Lemma shows, the conditions for Mayer--Vietoris squares are\nmuch easier to check when a description of the complement is given.\n\\begin{lemma}\\label{L:MV-open-qc}\n  Fix a cartesian diagram as in \\eqref{E:MV-square}. Suppose that $U$ is the complement \n  of a finitely presented closed immersion $i\\colon Z \\hookrightarrow X$.\n  \\begin{enumerate}\n  \\item \\label{LI:MV-open-qc:weak} If $f_{Z^{[n]}} \\colon X'\\times_X Z^{[n]} \\to \n    Z^{[n]}$ is an isomorphism for all $n$, then the square is a weak \n    Mayer--Vietoris square.\n  \\item \\label{LI:MV-open-qc:ti} If $f_{Z} \\colon X'\\times_X Z \\to Z$ is an isomorphism and \n    $Z$ and $f$ are tor-independent, then the square is a tor-independent Mayer--Vietoris \n    square. \n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  We may assume that $X$ is affine (Lemma \\ref{L:mv_bc}\\itemref{LI:mv_bc:verify}). \n  Let $g\\colon W \\to X$ be a morphism of algebraic stacks with \n  image disjoint from $U$. We must prove that $f_W \\colon X'\\times_X W \\to W$ is an \n  isomorphism. This is smooth local on $W$, so we may also assume that $W$ is affine. \n  The morphism $g$ is now affine, so its schematic image $V$ exists and is disjoint from \n  $U$. In particular, $|V| \\subseteq |Z|$. By Lemma \n  \\ref{L:fp_thickenings}\\itemref{LI:fp_thickenings:cl}, $V \\subseteq Z^{[n]}$ for some $n\\gg \n  0$. Hence, $W \\to X$ factors through $Z^{[n]}$ for some $n\\gg 0$. The claim \n  \\itemref{LI:MV-open-qc:weak} is now immediate.  For \\itemref{LI:MV-open-qc:ti}, the \n  result follows from Lemma \\ref{L:flatness_conc_cl}\\itemref{LI:flatness_conc_cl:iso}.\n\\end{proof}\nNote that if $X$ is quasi-compact and quasi-separated and $j$ is quasi-compact, then $i\\colon Z \\hookrightarrow X$ as in Lemma \\ref{L:MV-open-qc} always exist \\cite[Prop.~8.2]{rydh-2014}.\n\nThe following lemma connects the various types of Mayer--Vietoris squares to each other.\n\\begin{lemma}\\label{L:MV-different-notions}\n  Fix a square as in \\eqref{E:MV-square}. Consider the following \n  conditions. \n  \\begin{enumerate}\n  \\item \\label{LI:MV-different-notions:fmv} The square is a flat Mayer--Vietoris \n    square.\n  \\item \\label{LI:MV-different-notions:mv_fz} The square is a weak Mayer--Vietoris square and $f$ is flat at every point of $Z'$.\n  \\item \\label{LI:MV-different-notions:mv} The square is a tor-independent \n    Mayer--Vietoris square.\n  \\item \\label{LI:MV-different-notions:wmv} The square is a weak Mayer--Vietoris square.\n  \\end{enumerate}\n  Then \n  \\itemref{LI:MV-different-notions:fmv}$\\implies$\\itemref{LI:MV-different-notions:mv_fz}$\\implies$\\itemref{LI:MV-different-notions:mv}$\\implies$\\itemref{LI:MV-different-notions:wmv}.\n  If $X$ and $X'$ are locally noetherian, then~\\itemref{LI:MV-different-notions:wmv}$\\implies$\\itemref{LI:MV-different-notions:mv_fz}. If there exists a Cartier divisor \n    $i\\colon Z \\hookrightarrow X$ with complement $U$ such that $f^{-1}(Z)\\to Z$ is an isomorphism\n    and $f^{-1}(Z) \\hookrightarrow X'$ is \n    also a Cartier divisor, then~\\itemref{LI:MV-different-notions:mv} holds.\n  \\end{lemma}\n  \\begin{proof}\n    That\n    \\itemref{LI:MV-different-notions:fmv}$\\implies$\\itemref{LI:MV-different-notions:mv_fz}$\\implies$\\itemref{LI:MV-different-notions:mv}$\\implies$\\itemref{LI:MV-different-notions:wmv}\n    is obvious. If $X$ and $X'$ are locally noetherian,\n    then~\\itemref{LI:MV-different-notions:mv_fz} follows from the local\n    criterion of flatness~\\cite[0$_{\\mathrm{III}}$.10.2.1--2]{EGA} (the\n    conditions are flat-local on $X$ and $X'$ so reduces to schemes).\n\n    For the last claim, it is sufficient \n    to prove that $\\Orb_Z$ is $f$-flat (Lemma \\ref{L:MV-open-qc}\\itemref{LI:MV-open-qc:ti}), which is local on $X$. So we may assume that \n    $Z=V(d)$ and obtain an exact sequence\n    \\[\n    0\\to \\Orb_X\\xrightarrow{d\\cdot} \\Orb_X\\to \\Orb_Z\\to 0.\n    \\]\n    Applying $\\mathsf{L} \\QCPBK{f}$ to this, we obtain a distinguished triangle in $\\DQCOH(X')$:\n    \\[\n    \\Orb_{X'} \\to \\Orb_{X'} \\to \\mathsf{L} \\QCPBK{f}\\Orb_Z \\to \\Orb_{X'}[1].\n    \\]\n    The resulting long exact exact of cohomology yields:\n    \\[\n    0\\to \\COHO{-1}(\\mathsf{L}\\QCPBK{f}\\Orb_Z) \\to \\Orb_{X'}\\xrightarrow{d\\cdot} \\Orb_{X'}\\to f^*\\Orb_{Z}\\to 0,\n    \\]\n    with all other terms $0$. Since $Z\\hookrightarrow X'$ is a Cartier divisor, $d$ is regular on \n$\\Orb_{X'}$ and so $\\COHO{-1}(\\mathsf{L}\\QCPBK{f}\\Orb_Z)=0$. \nHence, $\\mathsf{L} \\QCPBK{f}\\Orb_Z \\to f^*\\Orb_Z$ is a quasi-isomorphism and the \nsquare is a tor-independent Mayer--Vietoris square.\n  \\end{proof}\nAs the following lemma shows, blowing up provides a natural way to move from the weak Mayer--Vietoris setting to the tor-independent setting.\n  \\begin{lemma}\\label{L:blow-up-is-MV-square}\n    Fix a weak Mayer--Vietoris square as in \\eqref{E:MV-square}. If there is a finitely presented closed immersion $i\\colon Z \\hookrightarrow X$ with complement $U$, then\n  \\[\n  \\vcenter{\\xymatrix{\n      U'\\ar[r]^-{j'}\\ar[d]_{f_U} & \\Bl_{Z'}X'\\ar[d]^{\\tilde{f}} \\\\\n      U\\ar[r]^-{j} & \\Bl_Z X\\ar@{}[ul]|\\square\n    }}\n  \\] \n  is a tor-independent  Mayer--Vietoris square.\n  \\end{lemma}\n  \\begin{proof}\n    Since the exceptional divisors $E\\hookrightarrow \\Bl_Z X$ and $E'\\hookrightarrow \\Bl_Z X'$ are\n    Cartier divisors it is enough to verify that $E'\\to E$ is an isomorphism\n    (Lemma \\ref{L:MV-different-notions}). Let $I$ be the ideal defining $Z\\hookrightarrow X$ and $I'$\n    the ideal defining $Z\\hookrightarrow X'$. Then the inverse images of $Z$ in the\n    two blow-ups are\n    \\[\n    E=\\mathrm{Proj}_X(\\oplus_{k\\geq 0} I^k\/I^{k+1})\\quad\\text{and}\\quad\n    E'=\\mathrm{Proj}_{X'}(\\oplus_{k\\geq 0} I'^k\/I'^{k+1}).\n    \\]\n    Since $I'=I\\Orb_{X'}$ and $\\Orb_X\/I^m\\to \\Orb_{X'}\/I'^m$ is an isomorphism\n    for every $m$, these two graded rings are isomorphic\n    $\\Orb_X\/I=\\Orb_{X'}\/I'$-algebras. The result follows.\n  \\end{proof}\n  The following lemma is a key observation of Moret-Bailly and will be essential to the article.\n  \\begin{lemma}[{\\cite[Cor.~2.5.1]{MR1432058}}]\\label{L:sections-give-MV}\n    Fix a weak Mayer--Vietoris square as in \\eqref{E:MV-square}.\n    If $f$ admits a section $s\\colon X\\to X'$, then \n    \\[\n    \\vcenter{\\xymatrix{\n      U\\ar[r]^{j}\\ar[d]_{s_U} & X \\ar[d]^s\\\\\n      U'\\ar[r]^{j'} & X'\\ar@{}[ul]|\\square\n    }}\n    \\]\n    is a weak Mayer--Vietoris square. Moreover, if the square \\eqref{E:MV-square} is a \n    tor-independent Mayer--Vietoris square, then so too is the one above.\n  \\end{lemma}\n  \\begin{proof}\n    Let $w'\\colon W' \\to X'$ be a morphism with image disjoint from $U'$. It follows that the \n    composition $W' \\xrightarrow{w'} X' \\xrightarrow{f} X$ has image disjoint from $U$ \n    and so $W'\\times_{X} X' \\to \n    W'$ is an isomorphism. In particular, the following diagram is cartesian:\n    \\[\n    \\xymatrix{X \\ar[d]_s & \\ar[l]_{f\\circ w'} W' \\ar@{=}[d]\\\\X' \\ar[d]_f & \\ar[l]_{w'} W' \\ar@{=}[d]\\\\ X & \\ar[l]_{f\\circ w'} W' }\n    \\]\n    and so $W'\\times_{X'} X \\to W'$ is an isomorphism \n    as required. For the latter claim, it suffices to prove that $w'$ and $s$ are \n    tor-independent. But $f\\circ w'$ and $f$ are tor-independent and $f\\circ w'$ and $f\\circ \n    s = \\mathrm{id}$ are tor-independent, so $w'$ and $s$ are tor-independent. The claim \n    now follows from the cartesian diagram above.\n  \\end{proof}\n  %\n  \\begin{example}[{\\cite[Prop.~2.5.2]{MR1432058}}]\\label{E:diagonal-square-is-MV}\n    Fix a weak Mayer--Vietoris square as in \\eqref{E:MV-square}. Then\n     \\[\n    \\xymatrix@C+5mm{\n      U' \\ar[r]^{j'}\\ar[d]_{\\Delta_{f_U}} & X' \\ar[d]^{\\Delta_f}\\\\\n      U'\\times_U U' \\ar[r]^{j'\\times j'} & X'\\times_X X'\\ar@{}[ul]|\\square\n    }\n    \\]\n    is a weak Mayer--Vietoris square. Indeed, we can base change the square \n    \\eqref{E:MV-square} by $X' \\to X$ and the resulting square is still weak\n    (Lemma \\ref{L:mv_bc}). Taking the diagonal section to the \n    projection $X'\\times_X X' \\to X'$ and using Lemma~\\ref{L:sections-give-MV} gives the \n    claim. If the square is a tor-independent Mayer--Vietoris square and $f$ is $f$-flat (e.g., \n    flat), then the square above is a tor-independent Mayer--Vietoris square. This claim \n    follows from the same argument.\n  \\end{example}\n  In the next Proposition, we show that general Mayer--Vietoris squares can smooth-locally \n  be dominated by much simpler ones. \n  \\begin{proposition}\\label{P:qaff_dom}\n    Fix a weak Mayer--Vietoris square as in \\eqref{E:MV-square}. Then \n    smooth-locally on $X$, there is an \\'etale neighborhood $p\\colon X''\\to X'$\n    of $Z'$\n    such that the composition $f\\circ p\\colon X'' \\to X$ is quasi-affine. \n  \\end{proposition}\n  \\begin{proof}\n    By Lemma \\ref{L:mv_bc}, we may assume that $X$ is an affine scheme.\n    Observe that \n    the Deligne--Mumford locus of $X'$  is an open substack containing $Z'$. \n    In particular, there exists an affine scheme $V$ and an \\'etale morphism $V \\to X'$ whose image contains $Z'$. Let $Z_V = V\\times_{X'} \n  Z'$; then the composition $z\\colon Z_V \\to Z' \\simeq Z=X\\setminus U$ is affine and \\'etale. After passing to an \\'etale \n  cover of $X$, we may assume that the morphism $Z_V \\to Z' \\simeq Z$ has a section \n  $s\\colon Z \\to Z''$. Since $z$ is \\'etale and separated, $s$ is an open and closed \n  immersion; it follows that $X''=V \\setminus (Z_V \\setminus s(Z))$ is an open \n  subscheme of $V$. After replacing $X''$ with a quasi-compact open neighborhood\n  of $s(Z)$, we can assume that $X''$ is quasi-compact.\n  Thus, $X'' \\to X' \\to X$ is a quasi-affine morphism and $X'' \\to X'$ is an isomorphism over $Z'$.\n  \\end{proof}\n  %\n  The following is the last lemma of the section.\n  %\n  \\begin{lemma}\\label{L:supp_flat}\n    Fix a tor-independent Mayer--Vietoris square as in \\eqref{E:MV-square}. If $j$ is \n    quasi-compact, then $\\mathsf{QCoh}_Z(X) \\subseteq \\mathsf{QCoh}_{f-\\mathrm{fl}}(X)$. \n  \\end{lemma}\n  \\begin{proof}\n    We may assume that $X$ is affine and that $i\\colon Z\\to X$ is finitely presented. In this case if $N\\in \\mathsf{QCoh}_Z(X)$, then we may \n    write it as a limit of quasi-coherent subsheaves of finite type that also belong to \n    $\\mathsf{QCoh}_Z(X)$. Thus, it is sufficient to prove the result for such sheaves. By Lemma \n    \\ref{L:fp_thickenings}\\itemref{LI:fp_thickenings:ft}, there is an $n\\gg 0$ such that $N \n      \\to i_*^{[n]}(i^{[n]})^*N$ is an isomorphism. The result now follows from Lemma \n      \\ref{L:flatness_conc_cl}\\itemref{LI:flatness_conc_cl:conc}.\n  \\end{proof}\n\n\n  Flat Mayer--Vietoris squares and\n  weak Mayer--Vietoris squares are stable under arbitrary base change\n  but tor-independent Mayer--Vietoris squares are not. We now give six examples:\n  \\begin{itemize}\n    \\item Two examples of weak Mayer--Vietoris squares that are not\n      tor-independent Mayer--Vietoris squares.\n    \\item Two examples of closed immersions that give rise to non-flat\n      tor-independent Mayer--Vietoris squares.\n    \\item A tor-independent Mayer--Vietoris square that is not universally a tor-independent Mayer--Vietoris square.\n    \\item A flat Mayer--Vietoris as in \\eqref{E:MV-square} with $j$ not quasi-compact that \n      is not a pushout in the category of affine schemes.\n  \\end{itemize}\n  As we will see later, tor-independent Mayer--Vietoris squares satisfy gluing of\n  quasi-coherent sheaves. In particular,\n  $\\Gamma(X,\\Orb_X)=\\Gamma(X',\\Orb_{X'})\\times_{\\Gamma(U',\\Orb_{U'})}\\Gamma(U,\\Orb_U)$,\n  which does not always hold for weak Mayer--Vietoris squares.\n\n  \\begin{example}\\label{E:weak-but-not-MV}\n    Let $A=k[x]$, $B=A[z_1,z_2,\\dots]\/(xz_1,\\{z_k-xz_{k+1}\\}_{k\\geq 1})$\n    and $C=B\/(z_1)$. Then \n    $A\/(x^n) = B\/(x^n) = C\/(x^n) = k[x]\/(x^n)$ and $A_x = B_x = C_x = k[x]_x$.\n    Let $X=\\spec A$, $Z=\\spec A\/(x)$, $U=X\\setminus Z$,\n    $X'=\\spec B$, $U'=X'\\setminus Z$ and $X''=\\spec C$. Then the squares\n    \\[\n    \\vcenter{\\xymatrix{\n        U'\\ar[r]\\ar[d] & X'\\ar[d] & & U''\\ar[d] \\ar[r] & X'' \\ar[d] \\\\\n        U\\ar[r] & X\\ar@{}[ul]|\\square & & U' \\ar[r] & X' \\ar@{}[ul]|\\square\n      }}\n    \\]\n    are weak Mayer--Vietoris squares but not\n    tor-independent Mayer--Vietoris squares.\n    Indeed $A\\to B\\times_{B_x} A_x=B$ and\n    $B\\to C\\times_{C_x} B_x=C$ are not isomorphisms.\n\n    Note that $Z\\hookrightarrow X$ is a Cartier divisor but $Z\\hookrightarrow X'$ is not a Cartier\n    divisor.\n  \\end{example}\n\n  \\begin{example}\n    A diagonal Mayer--Vietoris square (Example~\\ref{E:diagonal-square-is-MV})\n    is typically not flat, e.g., let $A$ be a noetherian\n    ring, $I\\subseteq A$ an ideal and consider the $I$-adic completion\n    $\\widehat{A}_I$. Let $f\\colon X'=\\spec \\widehat{A}_I\\to X=\\spec A$ and\n    $j\\colon U=X\\setminus V(I)\\to X$. This gives rise to a flat Mayer--Vietoris square as in \n    \\eqref{E:MV-square} and the diagonal Mayer--Vietoris square is tor-independent. \n    However, $\\Delta_f$ is a non-flat closed immersion.\n  \\end{example}\n    \n  \\begin{example}\\label{EX:valuation-MV-square}\n    Let $V$ be a valuation ring with valuation $\\nu\\colon K(V)^\\times\\to\n    \\Gamma$ and let $x\\in V$. Then $V_x$ is also a valuation ring and $V_x=V_P$\n    where $P\\subseteq V$ is the maximal prime ideal properly contained in the\n    prime ideal $Q=\\sqrt{(x)}$. Explicitly:\n    \\begin{align*}\n    P &= \\{a\\in V\\;: \\forall n\\in \\mathbb{N}\\;:\\; \\nu(a)\\geq n\\nu(x)\\}\\\\\n    Q &= \\{a\\in V\\;: \\exists n\\in \\mathbb{N} \\;:\\; n\\nu(a)>\\nu(x)\\}.\n    \\end{align*}\n    Let $X=\\spec(V)$, $U=\\spec(V_P)$, $Z=\\spec(V\/x)$ and $X'=\\spec(V\/P)$.\n    The resulting square as in \\eqref{E:MV-square} is tor-independent. Indeed,\n    it is a weak Mayer--Vietoris square since $P\\subseteq (x^n)$ for all $n$.\n    It remains to verify that $\\Tor_i^A(V\/x,V\/P)=0$ for all $i>0$.\n    But $x\\notin P$ so $x$ is $V\/P$-regular, hence the Tors vanish.\n  \\end{example}\n\n  \\begin{example}\\label{E:mv-not-umv}\n    Consider the valuation $\\nu\\colon k(x,y)\\to \\mathbb{Z}^2$ with $\\nu(x)=(0,1)$\n    and $\\nu(y)=(1,0)$ where $\\mathbb{Z}^2$ is lexicographically ordered. Then\n    $Q=(x)$ is the maximal ideal and $P=(y,y\/x,y\/x^2,\\dots)$.\n    Then $X'=\\spec V\/P\\to X=\\spec V$, $U=\\spec V_P=\\spec V_x$ is\n    a tor-independent Mayer--Vietoris square as in the previous example.\n\n    Let $A=V\/y$ and let $z_n=y\/x^n$ denote the image of $y\/x^n$. Then\n    $A=k[x,z_1,...]\/(xz_1,z_k-xz_{k+1})_{(x,z_1,z_2,...)}$ and\n    $B=A\/PA=k[x]_{(x)}$. Let $Y'=\\spec B$, $Y=\\spec A$ and $U=\\spec A_x$.\n    As in Example~\\ref{E:weak-but-not-MV}, $A\/(x^n)=k[x]\/(x^n)$ and\n    $B\/(x^n)=k[x]\/(x^n)$ but $A\\to B\\times_{B_x} A_x=B$ is not an isomorphism.\n  \\end{example}\n\n  \\begin{example}\\label{E:fmv-nonqc-nonpo}\n    Let $X=\\spec A$ be the spectrum of an absolutely flat ring such that there exists\n    a non-discrete point $x\\in |X|$. Let $\\mathfrak{m}\\subseteq A$ be the corresponding\n    maximal ideal. For a concrete example, let $\\mathbb{P}$ be the set\n    of primes of $\\mathbb{Z}$, let $A=\\prod_{p\\in \\mathbb{P}} \\mathbb{F}_p$ and\n    let $\\mathfrak{m}$ be a maximal ideal containing the ideal $\\oplus_{p\\in \\mathbb{P}} \n    \\mathbb{F}_p$.\n   \n   \n    %\n    Let $X' = \\spec A\/\\mathfrak{m}$ and let $f\\colon X' \\to X$ be the induced\n    closed immersion. Let $j\\colon U=X\\setminus \\{x\\} \\to X$ be the open immersion of\n    the complement. Since $A$ is an absolutely flat ring, $f$ is also flat. Let \n    $U'=X'\\times_X U = \\emptyset$; then the resulting square is a flat Mayer--Vietoris \n    square but $j$ is not quasi-compact.\n\n    Note that the natural map $|X'|\\amalg |U|=|X'|\\amalg_{|U'|} |U|\\to |X|$ is\n    not a homeomorphism since $|X'|\\subset |X|$ is not open. In particular, the\n    functor $\\Phi_{\\mathrm{\\acute{E}t}}$ is not an equivalence, cf.\\ Corollary~\\ref{C:wMV-is-univ-submersive}.\n\n    Let $B = \\Gamma(U,\\Orb_U)$. If the square was a pushout in the category of \n    affine schemes, then corresponding to the  maps $X'\\to X'$ and $U \\to \n    \\spec B$, there would be a unique map\n    $g\\colon X \\to \\spec B\\amalg X'=\\spec (B\\times A\/\\mathfrak{m})$. Then\n    $g^{-1}(X')=X'$ which is a contradiction since $X'\\subseteq X$ is not open.\n    \n    This \n    example also shows that \n    \\[\n    \\Gamma(X,\\Orb_X) \\to \\Gamma(U,\\Orb_U) \\times \\Gamma(X',\\Orb_{X'})\n    \\]\n    is not an isomorphism. In particular, the functor $\\Phi_{\\mathsf{QCoh}}$ is not even fully faithful.\n  \\end{example}\n\\section{Gluing of modules in Mayer--Vietoris squares}\\label{S:QCoh-gluing}\nIn this section, we show that quasi-coherent sheaves of modules, and related\nobjects such as quasi-coherent sheaves of algebras, can be glued in tor-independent\nMayer--Vietoris squares. This generalizes previous results of Ferrand--Raynaud~\\cite[App.]{MR0272779} and Moret-Bailly \\cite{MR1432058}. We will prove this using some ideas from the theory\ndeveloped in \\cite[\\S5]{perfect_complexes_stacks} for triangulated categories that are perfectly suited to simultaneously deal with the non-flatness of $f$ and the non-affineness of $j$.  For quasi-compact and quasi-separated algebraic spaces and\nin the context of stable $\\infty$-categories, this was recently\naccomplished (independently) by Bhatt \\cite[Prop.~5.6]{2014arXiv1404.7483B}. Since we work with morphisms of algebraic stacks that may not have finite cohomological dimension, we do not expect gluing results to hold in this generality in the unbounded derived category. Before we get to gluing, we characterize the tor-independent squares in terms of derived categories.\n\\begin{notation}\n  Let $i\\colon Z \\hookrightarrow X$ be a closed immersion of algebraic stacks with complement $j\\colon \n  U \\to X$. Define\n  \\[\n  \\DQCOH[,Z](X) = \\{ M \\in \\DQCOH(X) \\,:\\, \\mathsf{L} \\QCPBK{j}M \\simeq 0\\}.\n  \\]\n\\end{notation}\n\\begin{proposition}\\label{P:MV_SQ_MB}\n  Fix a cartesian diagram as in \\eqref{E:MV-square} with $f$ concentrated, $j$ \n  quasi-compact and $X$ quasi-compact and quasi-separated. Consider the\n  following conditions:\n  \\begin{enumerate}\n    \\item \\label{PI:MV_SQ_MB:timv} the square is a tor-independent Mayer--Vietoris square;\n    \\item \\label{PI:MV_SQ_MB:dqc} $\\mathsf{R} f_*$ and $\\mathsf{L}\n      \\QCPBK{f}$ induce equivalences \n      $\\DQCOH[,Z](X)\\simeq\\DQCOH[,Z'](X')$;\n    \\item \\label{PI:MV_SQ_MB:tdqc} $\\mathsf{R} f_*$ and $\\mathsf{L}\n      \\QCPBK{f}$ induce $t$-exact equivalences\n      $\\DQCOH[,Z](X)\\simeq \\DQCOH[,Z'](X')$; and\n    \\item \\label{PI:MV_SQ_MB:tdqcb} $\\mathsf{R} f_*$ and $\\mathsf{L}\n      \\QCPBK{f}$ induce $t$-exact equivalences\n      $\\DQCOH[,Z]^b(X)\\simeq \\DQCOH[,Z']^b(X')$.\n  \\end{enumerate}\n  Then \\itemref{PI:MV_SQ_MB:timv}$\\implies$\\itemref{PI:MV_SQ_MB:dqc}$\\iff$\\itemref{PI:MV_SQ_MB:tdqc}$\\iff$\\itemref{PI:MV_SQ_MB:tdqcb}.\n  If either $f_Z$ is affine or $f_Z$ is representable and $Z$ has quasi-affine\n  diagonal, then all conditions\n  are equivalent.\n\\end{proposition}\nIn the application of Proposition \\ref{P:MV_SQ_MB} to the main result of this section (Theorem \\ref{T:MV-QCoh-gluing}), we will only need \\itemref{PI:MV_SQ_MB:timv}$\\implies$\\itemref{PI:MV_SQ_MB:dqc} when $X$ is an affine scheme and $X'$ is a quasi-affine scheme. We have included the general situation for independent interest. Note that condition \\itemref{PI:MV_SQ_MB:dqc} is the definition of a Mayer--Vietoris\n$\\DQCOH$-square~\\cite[Def.~5.4]{perfect_complexes_stacks} and Proposition \\ref{P:MV_SQ_MB} gives another proof\nof~\\cite[Ex.~5.5]{perfect_complexes_stacks}. \n\\begin{remark}\n  Taking $f\\colon \\spec k\\to BG$ as in\n  \\cite[Rem.~1.6]{hallj_dary_alg_groups_classifying} and $U=\\emptyset$\n  provides an example where \\itemref{PI:MV_SQ_MB:tdqcb} is satisfied,\n  but \\itemref{PI:MV_SQ_MB:timv} is not satisfied ($f$ is representable but\n  not affine\n  and its target does not have affine stabilizers).\n\\end{remark}\n\\begin{proof}[Proof of Proposition~\\ref{P:MV_SQ_MB}]\n  Trivially, \\itemref{PI:MV_SQ_MB:tdqc} implies \\itemref{PI:MV_SQ_MB:dqc} and \\itemref{PI:MV_SQ_MB:tdqcb}. Since $X$ is quasi-compact and quasi-separated, we may assume that there is a finitely presented complement $i\\colon Z \\hookrightarrow X$ of $U$.  If \\itemref{PI:MV_SQ_MB:tdqc} is satisfied, then\n  $\\mathsf{L} \\QCPBK{f}\\Orb_Z=f^*\\Orb_Z$ and the adjunction maps\n  $\\Orb_Z\\to f_*f^*\\Orb_Z$ and $f^*f_*\\Orb_{Z'}\\to \\Orb_{Z'}$ are\n  isomorphisms. If $f_Z$ is affine, then \\itemref{PI:MV_SQ_MB:timv} holds (Lemma \\ref{L:MV-open-qc}\\itemref{LI:MV-open-qc:ti}). Otherwise, if $Z$\n  has quasi-affine diagonal, then we start by noting that the\n  $t$-exactness of $\\mathsf{R}\n  f_*$ also shows that $\\mathsf{R} (f_Z)_*$ is $t$-exact.\n  By~\\cite[Lem.~2.2 (6)]{perfect_complexes_stacks}, it follows\n  that if $\\tilde{Z} \\to Z$ is a smooth morphism, where $\\tilde{Z}$ is\n  an affine scheme, then the pullback $\\tilde{f}_Z$ of $f_Z$ to\n  $\\tilde{Z}$ is such that $\\mathsf{R} (\\tilde{f}_Z)_*$ is\n  $t$-exact. Since $\\tilde{f}_Z$ is representable, we conclude that\n  $\\tilde{f}_Z$ is affine from Serre's Criterion \\cite[Thm.~8.7]{rydh-2009}.\n  By smooth descent, $f_Z$ is affine, and we\n  again see that \\itemref{PI:MV_SQ_MB:timv} holds.\n\n  Let us show that \\itemref{PI:MV_SQ_MB:dqc}$\\implies$\\itemref{PI:MV_SQ_MB:tdqc}. Since $f$ is concentrated and $X$ is quasi-compact and quasi-separated, there\n  exists an integer $n$ such that\n  \\[\n  \\trunc{\\geq k}\\mathsf{R} f_*N \\to \\trunc{\\geq k}\\mathsf{R}\n  \\QCPBK{f}(\\trunc{\\geq k-n}N)\n  \\]\n  for every integer $k$ and ${N} \\in \\DQCOH(X')$ \\cite[Thm.~2.6(1)]{perfect_complexes_stacks}. In particular, if\n  ${N}\\in \\DQCOH[,Z']^{\\leq 0}(X')$, then $\\mathsf{R} f_*\n  {N}$ is bounded above. Let $m$ denote the largest integer such that\n  $\\COHO{m}(\\mathsf{R} f_*{N})\\neq 0$. Then $\\COHO{m}({N})\\cong\n  \\COHO{m}(\\mathsf{L} \\QCPBK{f}\\mathsf{R} f_*{N})\\neq 0$ so $m\\leq 0$. Thus,\n  $\\mathsf{R} f_*$ is right $t$-exact and hence $t$-exact on\n  $\\DQCOH[,Z''](X')$. Similarly, if ${M}\\in\n  \\DQCOH[,Z]^{\\geq 0}(X)$, then we have isomorphisms\n  ${M}\\simeq \\mathsf{R} f_*\\trunc{\\geq -m}\\mathsf{L} \\QCPBK{f} {M}$\n  for every $m\\geq n$. It follows that $\\mathsf{L} \\QCPBK{f} {M}$ is supported\n  in degrees $\\geq 0$ so $\\mathsf{L} \\QCPBK{f}$ is $t$-exact on $\\DQCOH[,Z](X)$.\n\n  We will finish the proof by showing that \\itemref{PI:MV_SQ_MB:timv}$\\implies$\\itemref{PI:MV_SQ_MB:tdqcb}$\\implies$\\itemref{PI:MV_SQ_MB:tdqc}.\n  For every ${M} \\in \\DQCOH[,Z](X)$\n  and ${N} \\in \\DQCOH[,Z'](X')$ we have adjunction maps\n  \\[\n  \\eta_{{M}} \\colon {M} \\to \\mathsf{R} f_*\\mathsf{L}\n  \\QCPBK{f} {M} \\quad\\mbox{and}\\quad \\epsilon_{{N}} \\colon \\mathsf{L}\n  \\QCPBK{f} \\mathsf{R} f_* {N} \\to {N}.\n  \\]\n  We will show that these are quasi-isomorphism. \n\n  For \\itemref{PI:MV_SQ_MB:timv}$\\implies$\\itemref{PI:MV_SQ_MB:tdqcb} it is enough---by standard truncation\n  arguments---to prove:\n  \\begin{itemize}\n    \\item $\\eta_{{M}[0]}$ and $\\epsilon_{{N}[0]}$ are\n  quasi-isomorphisms; and\n  \\item $\\mathsf{L} f^*{M}[0]\\to (f^*{M})[0]$\n    and $(f_*{N})[0]\\to \\mathsf{R} f_*{N}[0]$ are\n    quasi-isomorphisms,\n  \\end{itemize}\n  where ${M}$ is a quasi-coherent $\\Orb_X$-module such\n  that $j^*{M} \\cong 0$ and ${N}$ is a quasi-coherent $\\Orb_{X'}$ such\n  that $j'^*{N} \\cong 0$. If $M$ and $N$ are of finite type, then there exists an integer \n  $n\\gg 0$ such\n  that $M \\to i^{[n]}_*(i^{[n]})^*M$ and $N \\to i'^{[n]}_*(i'^{[n]})^*N$ are isomorphisms \n  (Lemma \\ref{L:fp_thickenings}\\itemref{LI:fp_thickenings:ft}). Now Lemma \n  \\ref{L:flatness_conc_cl} informs us that ${f_{Z^{[n]}} \\colon Z'^{[n]} \\to Z^{[n]}}$ is an \n  isomorphism, $i^{[n]}$ and $f$ are tor-independent, and $M$ is $f$-flat. This immediately proves the claims \n  when $M$ and $N$ are of finite type. But every quasi-coherent sheaf on $X$ or \n  $X'$ is a directed limit of its quasi-coherent subsheaves of finite type \\cite{rydh-2014}, so we have the \n  claim in general.\n\n  To see\n  that \\itemref{PI:MV_SQ_MB:tdqcb}$\\implies$\\itemref{PI:MV_SQ_MB:tdqc} it is enough to prove that $\\mathsf{L} \\QCPBK{f}$ is left\n  $t$-exact on $\\DQCOH[,Z](X)$ and that $\\mathsf{R} f_*$ is right\n  $t$-exact on $\\DQCOH[,Z'](X')$. For the first claim, let\n  ${M}$ be a complex in $\\DQCOH[,Z]^{\\geq 0}(X)$. We may write\n  ${M}$ as a homotopy colimit of its truncations $\\trunc{\\leq\n    n}{M}$. Since $\\mathsf{L} \\QCPBK{f}$ commutes with coproducts and is\n  $t$-exact on $\\DQCOH[,Z]^b(X)$, it follows that $\\mathsf{L} \\QCPBK{f}\n  {M}\\in \\DQCOH[,Z']^{\\geq 0}(X')$ so $\\mathsf{L} \\QCPBK{f}$ is\n  $t$-exact. Also, if ${N}$ is a complex in $\\DQCOH[,Z']^{\\leq\n    0}(X')$, then since $f$ is concentrated and $X$ is quasi-compact and quasi-separated, there exists an integer $n$ such that $\\trunc{>0}\\mathsf{R} f_*{N} \\to \\trunc{>0}\\mathsf{R}\n  f_*(\\trunc{\\geq -n}{N}) \\simeq 0$. Hence, $\\mathsf{R} f_*$ is $t$-exact. \n\\end{proof}\nThe following theorem generalizes~\\cite[Thm.~3.1]{MR1432058} ($f$ affine)\nand \\cite[App.]{MR0272779} ($f$ affine and flat) and is Theorem \n\\ref{MT:glue_timv}\\itemref{MTI:glue_timv:qcoh}. \n\\begin{theorem}\\label{T:MV-QCoh-gluing}\nFix a tor-independent Mayer--Vietoris square as in \\eqref{E:MV-square} with $j$ quasi-compact. \nThe functors \n\\[\n\\Phi_{\\MOD} \\colon \\MOD(X) \\rightleftarrows \\MOD(X') \\times_{\\MOD(U')} \\MOD(U) \\colon \\Psi\n\\]\nwhere\n\\[\n\\Phi_{\\MOD}(N) = (f^*N,j^*N,\\delta) \\quad \\mbox{and} \\quad \\Psi(N',N_U,\\alpha) = f_*N'\\times_\\alpha j_*N_U\n\\]\nand $\\delta$ is the canonical isomorphism $j'^*f^*N\\cong f_U^*j^*N$, are adjoint. Also, $\\Phi_{\\MOD}$ preserves tensor products and the restriction of $\\Phi_{\\MOD}$ to $\\mathsf{QCoh}_{f-\\mathrm{fl}}(X)$ induces an equivalence of categories\n\\begin{equation}\n\\Phi_{\\mathsf{QCoh},f-\\mathrm{fl}}\\colon \\mathsf{QCoh}_{f-\\mathrm{fl}}(X)\\to \\mathsf{QCoh}(X')\\times_{\\mathsf{QCoh}(U')} \\mathsf{QCoh}_{f_U-\\mathrm{fl}}(U)\\label{TE:MV-QCoh-gluing}\n\\end{equation}\nthat preserves short exact sequences. Moreover,\n\\begin{enumerate}\n\\item \\label{TI:MV-QCoh-gluing:supp}$f^* \\colon \\mathsf{QCoh}_Z(X) \\to \\mathsf{QCoh}_{Z'}(X')$ is an equivalence;\n\\item $\\Phi_{\\mathsf{QCoh}}$ preserves and reflects\n  \\begin{enumerate}\n  \\item \\label{TI:MV-QCoh-gluing:zero} zero objects,\n  \\item \\label{TI:MV-QCoh-gluing:surj} surjective homomorphisms and \n  \\item \\label{TI:MV-QCoh-gluing:ft} modules of finite type; and\n  \\end{enumerate}\n\\item $\\Phi_{\\mathsf{QCoh},f-\\mathrm{fl}}$ preserves and reflects\n  \\begin{enumerate}\n  \\item \\label{TI:MV-QCoh-gluing:fp} modules of finite presentation and\n  \\item \\label{TI:MV-QCoh-gluing:flat} flat modules.\n  \\end{enumerate}\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n  Let $k=j\\circ f_U$. That $\\Phi_{\\MOD}$ and $\\Psi$ are adjoints is \n  clear. Hence, to prove the equivalence \\eqref{TE:MV-QCoh-gluing} it is \n  enough to show that the unit $N\\to \\Psi(\\Phi_{\\MOD}(N))$ and the counit \n  $\\Phi_{\\MOD}(\\Psi(N',N_U,\\delta))\\to\n  (N',N_U,\\delta)$ of the adjunction are isomorphisms when restricted to the relevant \n  subcategories. This is smooth local on $X$, so we may assume that $X$ is an affine \n  scheme.\n\n  Until further notice, we will assume that $X'$ is a quasi-compact and quasi-separated \n  algebraic space (even quasi-affine scheme is sufficient). Now the functor\n  \\[\n  \\mathsf{L} f^* \\colon \\DQCOH[,Z](X) \\to \\DQCOH[,Z'](X')\n  \\]\n  is an equivalence of categories (Proposition \\ref{P:MV_SQ_MB}). Thus, we have a \n  Mayer--Vietoris $\\DQCOH$-square in the sense of \n  \\cite[Def.~5.4]{perfect_complexes_stacks}, which provides some natural \n  distinguished triangles~\\cite[Lem.~5.6]{perfect_complexes_stacks} that we now describe.\n  \\begin{enumerate}\n  \\item[(i)] For every $N\\in\\DQCOH(X)$, there is a distinguished triangle:\n    \\[ \n    \\xymatrix@C2pc{%\n      N \\ar[r]\n      & \\mathsf{R} j_* \\mathsf{L} j^* N \\oplus \\mathsf{R} f_*\\mathsf{L} f^*N \\ar[r]\n      & \\mathsf{R} f_*\\mathsf{L} f^*\\mathsf{R} j_*\\mathsf{L} j^*N\\ar[r]\n      & N[1]. }\n    \\]\n  \\item[(ii)] Conversely, given $N_U \\in \\DQCOH(U)$, $N' \\in\n    \\DQCOH(X')$ and an isomorphism $\\delta \\colon \\mathsf{L} j'^*N' \\to\n    \\mathsf{L} f_U^*N_U$, we define $N$ by the following distinguished triangle in\n    $\\DQCOH(X)$: \n    \\[\n    \\xymatrix{%\n      N \\ar[r]\n       & \\mathsf{R} j_*N_U \\oplus \\mathsf{R} f_*N' \\ar[rr]^{\\begin{psmallmatrix}\\eta^f_{\\mathsf{R} j_*N_U} & -\\alpha\\end{psmallmatrix}}\n       && \\mathsf{R} f_*\\mathsf{L} f^*\\mathsf{R} j_*N_U\\ar[r] & N[1],}\n    \\]\n    where $\\alpha \\colon \\mathsf{R} f_*N' \\to \\mathsf{R} f_*\\mathsf{L} f^*\\mathsf{R} j_*N_U$ is the composition:\n    \\begin{align*}\n    \\mathsf{R} f_*N' \\xrightarrow{\\mathsf{R} f_*\\eta^{j'}_{N'}}& \\mathsf{R} f_*\\mathsf{R} j'_*\\mathsf{L}  j'^*N'\\\\\n    \\xrightarrow{\\mathsf{R} f_*\\mathsf{R} j'_*\\delta}& \\mathsf{R} f_*\\mathsf{R} j'_*\\mathsf{L} f_U^*N_U \\cong \\mathsf{R} f_*\\mathsf{L} f^*\\mathsf{R} j_*N_U.\n    \\end{align*}    \n    Then the induced maps $\\mathsf{L} j^*N\n    \\to N_U$ and $\\mathsf{L} f^*N \\to N'$ are isomorphisms. \n  \\end{enumerate}\n  Now let $N \\in \\mathsf{QCoh}_{f-\\mathrm{fl}}(X)$; then the distinguished triangle from (i) reduces \n  to the following distinguished triangle:\n  \\[\n  \\xymatrix@C2pc{%\n      N \\ar[r]\n      & \\mathsf{R} j_* j^*N \\oplus \\mathsf{R} f_*f^*N \\ar[r]\n      & \\mathsf{R} f_*\\mathsf{L} f^*\\mathsf{R} j_*j^*N\\ar[r]\n      & N[1]. }\n  \\]\n  Observe that tor-independent base change \\cite[Cor.~4.13]{perfect_complexes_stacks} \n  implies that:\n  \\[\n  \\mathsf{R} f_*\\mathsf{L} f^*\\mathsf{R} j_*j^*N \\simeq \\mathsf{R} f_*\\mathsf{R} j'_* \\mathsf{L} f^*_U \\mathsf{L} j^*N \\simeq \\mathsf{R} k_*\\mathsf{L} j'^*\\mathsf{L} f^*N \\simeq \\mathsf{R} k_*k^*N.\n  \\]\n  Hence, taking the long exact cohomology sequence of the distinguished triangle above, we obtain the following exact sequence:\n  \\[\n  0 \\to N \\to j_*j^*N \\oplus f_*f^*N \\to k_*k^*N \\to 0.\n  \\]\n  So the natural map $N \\to \\Psi(\\Phi_{\\MOD}(N))$ is an \n  isomorphism when $N$ is $f$-flat.\n\n  Conversely, given a triple $(N',N_U,\\delta)$, where $N' \\in \\mathsf{QCoh}(X')$ and $N_U \\in \\mathsf{QCoh}_{f_U-\\mathrm{fl}}$, (ii) provides a distinguished\n  triangle:\n  \\[\n  \\xymatrix{%\n    N \\ar[r] & \\mathsf{R} j_* N_U \\oplus \\mathsf{R}  f_* N' \\ar[r] & \\mathsf{R} k_*N'_U\\ar[r] & N[1],}\n  \\]\n  such that the induced maps $\\mathsf{L} j^*N \\to N_U$ and $\\mathsf{L} f^*N \\to N'$\n  are isomorphisms. Since $\\Phi(N',N_U,\\delta)=\\COHO{0}(N)$, it is enough to\n  show that $N$ is concentrated in degree $0$. To see this, we have a distinguished triangle:\n  \\[\n  \\xymatrix{%\n    \\COHO{0}(N)[0] \\ar[r] & N \\ar[r]\n    & \\trunc{\\geq 1}(N) \\ar[r] & \\COHO{0}(N)[1].}\n  \\]\n  If we apply the $t$-exact functor $\\mathsf{L} j^*$ to this triangle, then the third term\n  vanishes so $\\trunc{\\geq 1}(N)\\in \\DQCOH[,Z](X)$. If we instead apply the\n  right $t$-exact functor $\\mathsf{L} f^*$ to this triangle, we obtain the\n  triangle:\n  \\[\n  \\xymatrix{%\n    \\mathsf{L} f^*\\COHO{0}(N)[0] \\ar[r] & N'[0] \\ar[r]\n    & \\mathsf{L} f^*\\trunc{\\geq 1}(N) \\ar[r] & \\mathsf{L} f^*\\COHO{0}(N)[1].}\n  \\]\n  The first two terms are concentrated in degrees $\\leq 0$ and the third is\n  concentrated in degrees $\\geq 1$ since $\\DQCOH[,Z](X)\\to\n  \\DQCOH[,Z'](X')$ is a $t$-exact equivalence. It follows that\n  $\\trunc{\\geq 1}(N)\\simeq 0$. \n\n  Hence, we have proven the equivalence \\eqref{TE:MV-QCoh-gluing} when $f\\colon X' \\to X$ is \n  quasi-compact, quasi-separated and representable. We now address the general case. By \n  Proposition \\ref{P:qaff_dom}, smooth-locally on $X$ there is an \\'etale neighborhood \n  $X''$ of $Z'$ in $X'$ such that the induced composition $w\\colon X'' \\to X$ is quasi-affine. Let $U''=X''\\times_X U$.\n  It now follows \n  from the case considered already, as well as \\cite[Ex.~1.2]{MR2774654}, that we have \n  equivalences:\n  \\begin{align*}\n  \\mathsf{QCoh}_{w-\\mathrm{fl}}(X) &\\simeq \\mathsf{QCoh}(X'') \\times_{\\mathsf{QCoh}(U'')} \\mathsf{QCoh}_{w_U-\\mathrm{fl}}(U)\\\\\n    &\\simeq \\left(\\mathsf{QCoh}(X'') \\times_{\\mathsf{QCoh}(U'')} \\mathsf{QCoh}(U') \\right)\n      \\times_{\\mathsf{QCoh}(U')}  \\mathsf{QCoh}_{w_U-\\mathrm{fl}}(U)\\\\\n    &\\simeq \\mathsf{QCoh}(X')\\times_{\\mathsf{QCoh}(U')} \\mathsf{QCoh}_{w_U-\\mathrm{fl}}(U). \n  \\end{align*}\n  Note that $\\mathsf{QCoh}_{f-\\mathrm{fl}}(X)\\subseteq \\mathsf{QCoh}_{w-\\mathrm{fl}}(X)$ is\n  a full subcategory and that we have an equivalence \n  \\[\n  \\mathsf{QCoh}_{f-\\mathrm{fl}}(X)\\to \\mathsf{QCoh}_{w-\\mathrm{fl}}(X)\\times_{\\mathsf{QCoh}_{w_U-\\mathrm{fl}}(U)} \\mathsf{QCoh}_{f_U-\\mathrm{fl}}(U).\n  \\]\n  It follows that $\\Phi_{\\mathsf{QCoh},f-\\mathrm{fl}}$ is an equivalence \n  and it preserves short exact sequences.\n \n  Now for \\itemref{TI:MV-QCoh-gluing:supp}: if $N \\in \\mathsf{QCoh}_Z(X)$, then $N$ is $f$-flat (Lemma \\ref{L:supp_flat}). Hence,\n  \\[\n  N \\to \\Psi(f^*N,0,0)=f_*f^*N\n  \\]\n  is an isomorphism. Also if $N' \\in \\mathsf{QCoh}_{Z'}(X')$, then\n  \\[\n  (f^*f_*N',0,0)=\\Phi_{\\MOD}(f_*N') \\to (N',0,0)\n  \\]\n  is an isomorphism and the claim follows.\n  \n  For \\itemref{TI:MV-QCoh-gluing:zero}: the preservation is obvious. For the reflection: if \n  $N \\in \\mathsf{QCoh}(X)$ and $j^*N\\cong 0$, then $N \\in \\mathsf{QCoh}_Z(X)$. But if $f^*N \\cong 0$ \n  too, then $N \\cong 0$ by \\itemref{TI:MV-QCoh-gluing:supp}.\n\n  For \\itemref{TI:MV-QCoh-gluing:surj}: the preservation is because $\\Phi_{\\MOD}$ \n  admits a right adjoint $\\Psi$ and so is right exact. For the reflection: if $u\\colon N \\to M$ \n  is a morphism in $\\mathsf{QCoh}(X)$ and $j^*u$ and $f^*u$ are surjective, then $j^*\\coker(u) = \n  0$ and $f^*\\coker(u) = \\coker(f^*u) = 0$. It follows from \n  \\itemref{TI:MV-QCoh-gluing:zero} that $\\coker(u) = 0$ and $u$ is surjective.\n  \n   For \\itemref{TI:MV-QCoh-gluing:ft}: the preservation is clear. For the reflection: we may \n   assume that $X$ is affine. Write $M$ as a filtered union of quasi-coherent subsheaves \n   $M_\\lambda$ of finite type. For sufficiently large $\\lambda$ we see that \n   $\\Phi_{\\MOD}(M_\\lambda) \\to \\Phi(M)$ is surjective. By \\itemref{TI:MV-QCoh-gluing:surj}, \n   we see that $M_\\lambda = M$ and so $M$ is of finite type.\n\n   For \\itemref{TI:MV-QCoh-gluing:fp}: the preservation is clear. For the reflection:   Let $M$ be a $f$-flat quasi-coherent $\\Orb_X$-module such that $\\Phi(M)$ is\n  of finite presentation. By  \\itemref{TI:MV-QCoh-gluing:ft} we know that $M$ is of finite \n  type. Since we are free to assume that $X$ is affine, there is an exact sequence $0\\to \n  K\\to \\Orb_X^{\\oplus n}\\to M\\to 0$. But $M$ is $f$-flat, so the sequence remains exact after \n  applying $f^*$. Since $\\Phi(K)$ is of finite type, so is $K$ and hence $M$ is of finite\n  presentation.\n\n  For \\itemref{TI:MV-QCoh-gluing:flat}: the preservation is clear. For the reflection: as before, we \n  may assume that $X$ is affine and $f$ is quasi-affine. Let $N \\in \\mathsf{QCoh}_{f-\\mathrm{fl}}(X)$ and let $M \\in \\mathsf{QCoh}(X)$. It is sufficient to prove that\n  $\\trunc{<0}(M\\otimes^{\\mathsf{L}}_{\\Orb_X} N) \\simeq 0$. We begin with the following \n  distinguished triangle:\n  \\[\n\\xymatrix{  C  \\ar[r] & M \\ar[r] & \\mathsf{R} j_*j^*M \\ar[r] & C[1]}.\n  \\]\n  Observe that the derived projection formula \\cite[Cor.~4.12]{perfect_complexes_stacks} \n  implies that\n  \\[\n  (\\mathsf{R} j_*j^*M) \\otimes^{\\mathsf{L}}_{\\Orb_X} N \\simeq \\mathsf{R} j_*((j^*M)\\otimes^{\\mathsf{L}}_{\\Orb_{U}} j^*N).\n  \\]\n  But $j^*N$ is flat and so we conclude immediately that $\\trunc{<0}((\\mathsf{R} j_*j^*M) \n  \\otimes^{\\mathsf{L}}_{\\Orb_X} N) \\simeq 0$. It remains to prove that $\\trunc{<0}(C\\otimes^{\\mathsf{L}}_{\\Orb_X} N) \\simeq 0$. To this end, we first note that $\\trunc{<0}C \\simeq 0$ and $j^*C \\simeq 0$. Moreover, \n  \\[\n  \\mathsf{L} f^*(C\\otimes^{\\mathsf{L}}_{\\Orb_X} N) \\simeq (\\mathsf{L} f^*C) \\otimes^{\\mathsf{L}}_{\\Orb_{X'}} \\mathsf{L} f^*N \\simeq (\\mathsf{L} f^*C) \\otimes^{\\mathsf{L}}_{\\Orb_{X'}} f^*N.\n  \\]\n  By assumption, $f^*N$ is flat and so for all integers $k$ there are isomorphisms:\n  \\[\n  \\COHO{k}\\bigl((\\mathsf{L} f^*C) \\otimes^{\\mathsf{L}}_{\\Orb_{X'}} f^*N\\bigr) \\cong \\COHO{k}(\\mathsf{L} f^*C) \\otimes_{\\Orb_{X'}} f^*N.\n  \\]\n  But $C \\in \\DQCOH[,Z](X)$, so $\\trunc{<0}C \\simeq 0$ implies $\\trunc{<0}(\\mathsf{L} f^*C) \\simeq 0$ (Proposition \\ref{P:MV_SQ_MB}). Putting this all together, we see that $\\trunc{<0}(\\mathsf{L} f^*(C \\otimes_{\\Orb_X}^{\\mathsf{L}} N)) \\simeq 0$. But $j^*(C\\otimes^{\\mathsf{L}}_{\\Orb_X} N) \\simeq 0$, which implies that $\\trunc{<0}(C\\otimes_{\\Orb_X}^{\\mathsf{L}} N) \\simeq 0$ (Proposition \\ref{P:MV_SQ_MB} again). \n\\end{proof}\nNote that \\ref{T:MV-QCoh-gluing}\\itemref{TI:MV-QCoh-gluing:flat} gives a vast generalization of \\cite[Prop.~4.1(iii)]{MR1432058}, where only the descent of \\'etaleness is proved. \n\\begin{remark}\n  Assume that we are in the situation of Theorem \\ref{T:MV-QCoh-gluing}. If $f$ is \n  concentrated, then the Mayer--Vietoris triangle shows\n  that the functor $\\DQCOH(X)\\to \\DQCOH(X')\\times_{\\DQCOH(U')} \\DQCOH(U)$ is\n  essentially surjective. It is, however, not fully faithful. The reason is\n  a well-known fault of the derived category: whereas cones are unique up\n  to isomorphism, morphisms between cones are not unique. One way to fix this\n  problem is to work with $\\infty$-categories. Then one obtains the expected\n  equivalence, cf.~\\cite[Prop.~5.6]{2014arXiv1404.7483B}.\n\\end{remark}\n\nWe now have a number of corollaries.\n\\begin{corollary}\\label{C:MV-QCoh-gluing:ff}\n  Assume that we are in the situation of Theorem \\ref{T:MV-QCoh-gluing}.   If\n  ${M\\in \\MOD(X)}$ and $N\\in \\mathsf{QCoh}_{f-\\mathrm{fl}}(X)$, then the natural\n  map:\n  \\[\n  \\Hom(M,N)\\to \\Hom(f^*M,f^*N)\\times_{\\Hom(j'^*f^*M,j'^*f^*N)}\n  \\Hom(j^*M,j^*N)\n  \\]\n  is bijective.\n\\end{corollary}\n\\begin{proof}\nFollows from the unit $N\\to \\Psi(\\Phi_{\\MOD}(N))$ being an isomorphism.\n\\end{proof}\n\\begin{corollary}\\label{C:MV-QCoh-gluing:flatness}\n  Assume that we are in the situation of Theorem \\ref{T:MV-QCoh-gluing}. If $f$ is flat, \n  then $\\Phi_{\\mathsf{QCoh}}$ is an equivalence of abelian categories and preserves and reflects flatness.\n\\end{corollary}\n\\begin{corollary}[{\\cite[Cor.~3.4.3]{MR1432058}}]\\label{C:gluing_special}\n  Assume that we are in the situation of Theorem \\ref{T:MV-QCoh-gluing}.\n  Then $N \\in \\mathsf{QCoh}(X)$ is $f$-flat if and only if $j^*N$ is \n    $f_U$-flat.\n\\end{corollary}\n\\begin{proof}\n  The necessity is clear. For the sufficiency: if $j^*N$ \n  is $f_U$-flat, then $\\tilde{N}=\\Psi(f^*N,j^*N,\\delta)$ is an $f$-flat quasi-coherent sheaf and \n  there is a natural map $\\eta \\colon N \\to \\tilde{N}$. Now $j^*\\eta$ and $f^*\\eta$ are \n  isomorphisms, so $\\ker(\\eta)$ is $f$-flat (Lemma \\ref{L:supp_flat}) and $\\eta$ is \n  surjective (Theorem \\ref{T:MV-QCoh-gluing}\\itemref{TI:MV-QCoh-gluing:surj}). So we have an exact sequence:\n  \\[\n  0 \\to \\ker(\\eta) \\to N \\to \\tilde{N} \\to 0\n  \\]\n  and $\\ker(\\eta)$ and $\\tilde{N}$ are $f$-flat. It follows that $N$ is $f$-flat, which gives \n  the sufficiency.\n\\end{proof}\n\nWe now consider Theorem \\ref{T:MV-QCoh-gluing} in the context of algebras, or equivalently, affine schemes.\n\\begin{corollary}\\label{C:gluing-of-algebras}\n   Assume that we are in the situation of Theorem \\ref{T:MV-QCoh-gluing}. \n  The natural functor\n  \\[\n    \\Phi_{\\AFF,f-\\mathrm{fl}}\\colon\n    \\AFF_{f-\\mathrm{fl}}(X)\\to\n    \\AFF(X')\\times_{\\AFF(U')} \\AFF_{f_U-\\mathrm{fl}}(U),\n  \\]\n  is an equivalence of categories. Moreover, the functor $\\Phi_{\\AFF}$ preserves and reflects\n  \\begin{enumerate}\n    \\item\\label{CI:reflects:alg:closed-imm} closed immersions;\n    \\item\\label{CI:reflects:alg:finite} finite morphisms;\n    \\item\\label{CI:reflects:alg:integral} integral morphisms; and\n    \\item\\label{CI:reflects:alg:fin-type} morphisms of finite type.\n  \\end{enumerate}\n  and $\\Phi_{\\AFF,f-\\mathrm{fl}}$ preserves and reflects\n  \\begin{enumerate}[resume]\n    \\item\\label{CI:reflects:alg:fin-pres} morphisms of finite presentation.\n  \\end{enumerate}\n\\end{corollary}\n\\begin{proof}\n  An $\\Orb_X$-algebra structure on a $\\Orb_X$-module $M$ is given by\n  homomorphisms $\\Orb_X\\to M$ and $M\\otimes_{\\Orb_X} M\\to M$ satisfying various\n  compatibility conditions. If $M$ is $f$-flat, then an algebra structure on\n  $\\Phi_{\\MOD}(M)$ descends to a\n  unique algebra structure on $M$ by Corollary \\ref{C:MV-QCoh-gluing:ff}.\n\n  That $\\Phi_{\\AFF}$ preserves all the properties follows by definition.\n  To see that $\\Phi_{\\AFF}$ reflects the properties, we may work fppf-locally on\n  $X$ and assume that $X$ is affine and work with the categories of algebras. We let $\\Phi \n  = \\Phi_{\\MOD}$ for the remainder of the proof.\n\n  \\itemref{CI:reflects:alg:closed-imm}\n  This follows from Theorem~\\ref{T:MV-QCoh-gluing}\\itemref{TI:MV-QCoh-gluing:surj}.\n\n  \\itemref{CI:reflects:alg:finite}\n  Let $A\\to B$ be a homomorphism of $\\Orb_X$-algebras. Write $B$ as a filtered\n  union of finitely generated $A$-submodules $B_\\lambda$. If $\\Phi(B)$ is a\n  finite $\\Phi(A)$-algebra, then $\\Phi(B_{\\lambda_0}) \\to \\Phi(B)$ is\n  surjective for sufficiently large $\\lambda$. We conclude that \n  $B=B_\\lambda$ is a finite $A$-algebra from\n  Theorem~\\ref{T:MV-QCoh-gluing}\\itemref{TI:MV-QCoh-gluing:surj}.\n\n  \\itemref{CI:reflects:alg:integral}\n  Let $A\\to B$ be a homomorphism of $\\Orb_X$-algebras. If $\\Phi(A)\\to \\Phi(B)$\n  is integral, then $j^*A\\to j^*B$ is\n  integral. Thus, if $B_0$ is the integral closure of $A$ in $B$, then\n  $j^*B_0=j^*B$. Write $B$ as the filtered union of finitely generated\n  $B_0$-subalgebras $B_\\lambda\\subseteq B$. Since $j^*B_0=j^*(B_\\lambda)=j^*B$,\n  we have that $B\/B_\\lambda$ is $f$-flat;\n  it follows that $\\Phi(B_\\lambda)\\subseteq \\Phi(B)$\n  is a $\\Phi(B_0)$-subalgebra of finite type. Thus $\\Phi(B_\\lambda)$ is a finite\n  $\\Phi(B_0)$-algebra, so $B_\\lambda$ is a finite $B_0$-algebra. It follows that\n  $B=\\bigcup_\\lambda B_\\lambda$ is integral over $A$.\n\n  \\itemref{CI:reflects:alg:fin-type}\n  If $A\\to B$ is a homomorphism of $\\Orb_X$-algebras such that\n  $\\Phi(A)\\to \\Phi(B)$ is of finite type, then write $B$ as\n  a filtered union of finitely generated $A$-subalgebras $B_\\lambda$. For\n  sufficiently large $\\lambda$, we have that $\\Phi(B_\\lambda)\\to \\Phi(B)$\n  is surjective, hence so is $B_\\lambda\\to B$ so $A\\to B$ is of finite type.\n\n  \\itemref{CI:reflects:alg:fin-pres}\n  If $A\\to B$ is a homomorphism of $\\Orb_X$-algebras such that $\\Phi(A)\\to\n  \\Phi(B)$ is of finite presentation, then we\n  have already seen that $A\\to B$ is of finite type. There is an exact sequence\n  $0\\to I\\to A[x_1,x_2,\\dots,x_n]\\to B\\to 0$ and if $B$ is $f$-flat, then this\n  sequence remains exact after applying $f^*$. We conclude that $I$ is\n  a finitely generated ideal, hence that $A\\to B$ is of finite presentation.\n\\end{proof}\n\n\\begin{corollary}\\label{C:fmv-qaff_gluing}\n  Assume that we are in the situation of Theorem \\ref{T:MV-QCoh-gluing}. If $f$ is flat, \n  then $\\Phi_{\\AFF}$ and $\\Phi_{\\mathsf{Qaff}}$ are equivalences of categories.\n\\end{corollary}\n\\begin{proof}\n  The equivalence of $\\Phi_{\\AFF}$ follows immediately from Corollary \n  \\ref{C:gluing-of-algebras}. For $\\mathsf{Qaff}$, we must work a little more. Some notation will \n  be useful: if $W \\to Y$ is quasi-affine, then let $\\bar{W} \\to Y$ denote its affine hull. \n  Note that the formation of $\\bar{W} \\to Y$ commutes with flat base change on $Y$. \n  Similarly, for a morphism $\\alpha\\colon W_1 \\to W_2$ of quasi-affine schemes over $Y$ \n  we let $\\bar{\\alpha}$ denote the induced morphism between the affine hulls. \n\n  Now for the faithfulness: let $\\alpha$, $\\beta \\colon W_1 \\to W_2$ be morphisms in \n  $\\mathsf{Qaff}(X)$ such that $\\Phi_{\\mathsf{Qaff}}(\\alpha) = \\Phi_{\\mathsf{Qaff}}(\\beta)$. By the result for \n  $\\AFF$, we see that $\\bar{\\alpha}=\\bar{\\beta}$ and the claim follows.\n\n  Next for the fullness: consider quasi-affine $X$-schemes $W_1$ and $W_2$ and a \n  morphism $(\\alpha',\\alpha_U) \\colon \\Phi_{\\mathsf{Qaff}}(W_1) \\to \\Phi_{\\mathsf{Qaff}}(W_2)$. The result \n  for $\\AFF$ implies that there is a morphism $\\bar{\\alpha} \\colon \\bar{W}_1 \\to \n  \\bar{W}_2$ such that $\\Phi_{\\AFF}(\\bar{\\alpha}) = (\\bar{\\alpha'},\\bar{\\alpha_U})$. It is \n  sufficient to prove that $W_1 \\subseteq \\bar{\\alpha}^{-1}(W_2)$. But this may be \n  checked on points and $X'\\amalg U \\to X$ is surjective. The claim follows.\n\n  Finally, for the essential surjectivity. Now fix \n  a triple $(W',W_U,\\theta)$ in the codomain for \n  $\\Phi_{\\mathsf{Qaff}}$. This leads to a triple $(\\bar{W'},\\bar{W_U},\\bar{\\theta})$ in the \n  codomain of $\\Phi_{\\AFF}$ that may be glued to an affine $X$-scheme $\\bar{W}$. Since \n  $U \\subseteq X$ is quasi-compact and $f$ is flat and an isomorphism over $Z$, it is \n  easily verified that $X'\\amalg U \\to X$ is universally submersive. In particular, by base \n  changing along $\\bar{W} \\to X$ we may glue the quasi-compact open subsets $W' \n  \\subseteq \\bar{W}'$ and $W_U \\subseteq \\bar{W_U}$ to a quasi-compact open subset $W \n  \\subseteq \\bar{W}$. This proves the claim.\n\\end{proof}\nWe conclude this section with the following generalization of~\\cite[Cor.~4.3]{MR0272779}.\n\n\\begin{corollary}\\label{C:normalization}\n  Assume that we are in the situation of Theorem \\ref{T:MV-QCoh-gluing}. Let $\\eta\\colon \\Orb_X\\to j_*\\Orb_U$ and \n  let $\\eta'\\colon \\Orb_{X'}\\to j'_*\\Orb_{U'}$ denote the unit maps.\n  \\begin{enumerate}\n  \\item $\\eta$ is injective if and only if $\\eta'$ is injective.\n  \\item $\\eta$ is integrally closed if and only if $\\eta'$ is integrally closed.\n  \\item If $\\overline{X}$ denotes the integral closure of $X$ in $U$, i.e.,\n    $\\spec_X(\\mathcal{A})$ where $\\mathcal{A}$ is the integral closure of\n    $\\Orb_X$ with respect to $\\eta$, then $\\overline{X}':=\\overline{X}\\times_X\n    X'$ is the integral closure of $X'$ in $U'$ and the square\n    of $U\\to \\overline{X}$ and $\\overline{X}'\\to \\overline{X}$\n    is a tor-independent Mayer--Vietoris square.\n  \\end{enumerate}\n\\end{corollary}\n\\begin{proof}\n  By Corollary \\ref{C:gluing-of-algebras}\\itemref{CI:reflects:alg:closed-imm},\n  there is a bijection of partially ordered sets\n  \\[\n    \\Phi\\colon \\Cl_{f-\\mathrm{fl}}(X)\\to \\Cl(X')\\times_{\\Cl(U')} \\Cl_{f_U-\\mathrm{fl}}(U),\n  \\]\n  where $\\Cl(X)$ denotes the set of closed substacks $V\\hookrightarrow X$ and\n  $\\Cl_{f-\\mathrm{fl}}$ denotes the subset of closed substacks such that\n  $\\Orb_V$ is $f$-flat. If we let $\\overline{U}$ and\n  $\\overline{U'}$ denote the schematic closures of $U$ and $U'$ in $X$ and\n  $X'$ respectively, then $\\overline{U}$ is $f$-flat (Corollary \\ref{C:gluing_special}) and $\\overline{U}$\n  corresponds to a triple $(\\overline{U}\\times_X X',U',U)$ on the right hand\n  side. But $\\overline{U}$ is minimal among the closed substacks of $X$ that\n  contains $U$ and $\\overline{U'}$ is minimal among the closed substacks\n  of $X'$ that contains $U'$. It follows that \n  $\\Phi(\\overline{U})=(\\overline{U'},U',U)$. Thus, $X=\\overline{U}$ if and only\n  if $X'=\\overline{U'}$. Equivalently, $\\eta$ is injective if and only if\n  $\\eta'$ is injective.\n\n  Similarly, Corollary \\ref{C:gluing-of-algebras}\\itemref{CI:reflects:alg:integral} induces an equivalence of categories of\n  integral morphisms\n  \\[\n    \\Phi\\colon \\Int_{f-\\mathrm{fl}}(X)\\to \\Int(X')\\times_{\\Int(U')} \\Int_{f_U-\\mathrm{fl}}(U).\n  \\]\n  If we let $\\Int(X,U)$ denote the integral morphisms $W\\to X$ such that\n  $W|_U\\to U$ is an isomorphism and $U$ is schematically dense in $W$, then\n  $\\Int(X,U)$ is equivalent to the bounded lattice of sub-$\\Orb_X$-algebras of\n  $j_*\\Orb_U$ that are integral over $\\Orb_X$. These extensions are\n  automatically $f$-flat, since they are $f_U$-flat after restricting to\n  $\\Orb_U$ (Corollary \\ref{C:gluing_special}). We thus obtain a bijection of bounded lattices:\n  \\[\n    \\Phi\\colon \\Int(X,U)\\to \\Int(X',U').\n  \\]\n  Indeed, the only non-obvious detail is that $U'$ is schematically dense\n  in $\\Phi(W)=W\\times_X X'$ and that $U$ is schematically dense in\n  $\\Phi^{-1}(W',U',U)$. This follows from the\n  previous part since the square \n  \\[\n  \\vcenter{\\xymatrix{\n      W\\times_X U'\\ar[r]\\ar[d] & W\\times_X X'\\ar[d] \\\\\n      W\\times_X U\\ar[r]& W\\ar@{}[ul]|\\square\n    }}\n  \\]\n  is a tor-independent Mayer--Vietoris square (Lemma \\ref{L:mv_bc}\\itemref{LI:mv_bc:mv}).\n  Moreover, the minimal elements of these lattices are $\\overline{U}$ and $\\overline{U'}$.\n  and the maximal elements are $\\overline{X}$ and $\\overline{X'}$. The result\n  follows.\n\\end{proof}\n\n\n\\section{\\'Etale sheaves of sets on stacks}\\label{S:etale-sheaves}\nIn this section we generalize some fundamental results on constructible sheaves\nin SGA4 from schemes to algebraic stacks.\n\nLet $X$ be an algebraic stack. We let $\\mathrm{\\acute{E}t}(X)$ denote the category of \\'etale\nrepresentable morphisms $E\\to X$. We identify $\\mathrm{\\acute{E}t}(X)$ with the category of\ncartesian lisse-\\'etale sheaves of sets. Under this identification\nfinitely presented \\'etale morphisms correspond to constructible sheaves\nof sets.\n\nIf $X$ is a quasi-compact and quasi-separated algebraic space or\nDeligne--Mumford stack, then there is an \\'etale presentation by an affine\nscheme. Using this presentation it is easily seen that every \\'etale sheaf on\n$X$ is a filtered colimit of constructible sheaves. We will now extend this\nresult to every quasi-compact and quasi-separated algebraic stack.\n\nRecall that if $f\\colon X\\to Y$ is flat of finite presentation with\ngeometrically reduced fibers, then there exists a factorization $X\\to\n\\pi_0(X\/Y)\\to Y$ where the first map has connected fibers and the second\nis representable and \\'etale~\\cite[Thm.~2.5.2]{MR2820394}. This construction\ncommutes with arbitrary base change on $Y$ and is functorial in $X$.\nThe following result is due to J.\\ Wise.\n\n\\begin{proposition}[{\\cite[Thm.~4.5]{wise_logmaps}}]\nLet $f\\colon X\\to Y$ be flat of finite presentation with geometrically reduced\nfibers (e.g., $f$ smooth, quasi-compact and quasi-separated). If every\n\\'etale sheaf on $X$ is a filtered colimit of constructible sheaves\n(e.g., $X$ is\na quasi-compact and quasi-separated algebraic space), then $f^*\\colon\n\\mathrm{\\acute{E}t}(Y)\\to \\mathrm{\\acute{E}t}(X)$ admits a left-adjoint $f_!\\colon \\mathrm{\\acute{E}t}(X)\\to \\mathrm{\\acute{E}t}(Y)$ with\nthe following properties:\n\\begin{enumerate}\n\\item If $(E\\to X)\\in \\mathrm{\\acute{E}t}(X)$, then the unit induces a $X$-morphism\n  $E\\to f^*f_!E$.\n  This gives a factorization $E\\to f_!E\\to Y$ of the morphism $E\\to X\\to Y$\n  such that $E\\to f_!E$ has geometrically connected fibers.\n\\item $f_!$ preserves constructible sheaves.\n\\item $f_!$ commutes with pull-back: $g^*f_!=f'_!g'^*$ for any morphism\n  $g\\colon Y'\\to Y$, where $f'\\colon X':=X\\times_Y Y'\\to Y'$ and\n  $g'\\colon X'\\to X$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nFor constructible sheaves, it is readily seen that $f_!(E\\to X):=(\\pi_0(E\/Y)\\to\nY)$ is a left adjoint of $f^*$ and it commutes with arbitrary base change.  It\nremains to extend the construction to non-constructible \\'etale sheaves $E\\to\nX$. If $E=\\varinjlim E_\\lambda$ is a filtered colimit of constructible sheaves,\nthen necessarily $f_!E=\\varinjlim f_!E_\\lambda$.\n\\end{proof}\n\nWe may now generalize \\cite[Exp.~IX, Cor.~2.7.2, Prop.~2.14]{MR0354654}\nand~\\cite[Exp.~XII, Prop.~6.5 (i)]{MR0354654} to quasi-compact and\nquasi-separated algebraic stacks.\n\n\\begin{proposition}\\label{P:qcqs-cons-colimit}\nLet $X$ be a quasi-compact and quasi-separated algebraic stack. Then\nevery \\'etale sheaf of sets is a filtered colimit of constructible\nsheaves.\n\\end{proposition}\n\\begin{proof}\nThe result is known for affine schemes (and quasi-compact and quasi-separated\nschemes). Pick a smooth presentation $p\\colon U\\to X$ with $U$ affine. Let\n$F\\to X$ be an \\'etale sheaf. Choose an epimorphism $\\coprod_{i\\in I} G'_i\\to\np^*F$ where the $G'_i$ are constructible. Let $G_i=p_!G'_i$ which is a\nconstructible sheaf. Then $\\coprod G_i=p_!(\\coprod_i G'_i)\\to F$ is an\nepimorphism since $\\coprod_i G'_i\\to p^*p_!(\\coprod_i G'_i)\\to p^*F$ is an\nepimorphism.\n\nThe remainder of the proof is standard, cf.~\\cite[Exp.~IX,\n  Cor.~2.7.2]{MR0354654}. For every finite subset $J\\subseteq I$, the coproduct\n$G_J=\\coprod_{i\\in J} G_i$ is constructible. The fiber product\n$H_J:=G_J\\times_F G_J$ is not constructible but at least quasi-separated since\nit is a subsheaf of the constructible sheaf $G_J\\times_X G_J$. Consider the\nset $\\Lambda$ of pairs $(J,H')$ where $J\\subseteq I$ is finite and $H'\\subseteq\nH_J$ is quasi-compact, and hence constructible. For $\\lambda=(J,H')\\in\n\\Lambda$, let $F_\\lambda=\\coker(\\equalizer{H'}{G_J})$ which is a constructible sheaf. We\norder $\\Lambda$ by $(J_1,H'_1)\\leq (J_2,H'_2)$ if $J_1\\subseteq J_2$ and\n$g(H'_1)\\subseteq H'_2$ where $g\\colon H_{J_1}\\to H_{J_2}$. Then\n$F=\\varinjlim_{\\lambda\\in\\Lambda} F_\\lambda$ is a filtered colimit of\nconstructible sheaves.\n\\end{proof}\n\n\\begin{proposition}\\label{P:mono-decomposed}\nLet $X$ be a quasi-compact and quasi-separated algebraic stack. Let\n$F\\in \\mathrm{\\acute{E}t}(X)$ be a constructible sheaf of sets. Then there exists finite morphisms\n$p_i\\colon X'_i\\to X$, $i=1,2,\\dots,n$ and finite sets $A_1,A_2,\\dots,A_n$ and a\nmonomorphism $F\\hookrightarrow \\prod (p_i)_* \\underline{A_i}_{X_i}$.\n\\end{proposition}\n\\begin{proof}\nThere exists a stratification of $X$ into locally closed constructible\nsubstacks $Y_i$ such that $F|_{Y_i}$ is locally\nconstant~\\cite[Prop.~4.4]{MR2774654}. If $u_i\\colon Y_i\\to X$ denotes the\ncorresponding quasi-compact immersion, then $F\\to \\prod\n(u_i)_*(u_i)^*F$ is a monomorphism. After refining the stratification, we can\nassume that the cardinality of $F|_{Y_i}$ is constant. Let $q_i\\colon Y'_i\\to\nY_i$ be a finite \\'etale surjective morphism such that $q_i^*u_i^*F$ is a\nconstant sheaf with value $A_i$.\n\nLet $X_i$ be the closure of $Y_i$ and let $p_i\\colon X'_i\\to X$ be the\nintegral closure of $X$ with respect to $Y'_i\\to Y_i\\to X_i\\to X$. Then $p_i$ is\nintegral and $p_i|_{Y_i}=q_i$.  If $v_i\\colon Y'_i\\to X'_i$ denotes the open\nimmersion, then $(v_i)_*\\underline{A_i}_{Y_i'}=\\underline{A_i}_{X_i'}$ is\nconstant. Thus,\n\\[\nF\\to \\prod (u_i)_*(u_i)^*F\n  \\hookrightarrow \\prod (u_i)_*(q_i)_*(q_i)^*(u_i)^*F\n  = \\prod (p_i)_*\\underline{A_i}_{X'_i}\n\\]\nis a monomorphism.\n\nFinally, write $X'_i\\to X_i$ as an inverse limit of finite\nmorphisms~\\cite{rydh-2014}. By an easy limit argument, we can replace\n$p_i$ by a finite morphism.\n\\end{proof}\n\nFor an algebraic stack $X$, we let $\\clopen(X)$ denote the boolean algebra of\nclosed and open substacks.\n\n\\begin{proposition}\\label{P:H0-bij-vs-clopen}\nLet $h\\colon Y\\to X$ be a morphism of\nalgebraic stacks. If $X$ is quasi-compact and quasi-separated, then the following conditions are equivalent.\n\\begin{enumerate}\n\\item\\label{PI:H0-bij}\nFor every sheaf of sets $F\\in \\mathrm{\\acute{E}t}(X)$, the canonical map\n\\[\nH^0(X,F)\\to H^0(Y,h^*F)\n\\]\nis bijective.\n\\item\\label{PI:H0-bij:cons}\nCondition~\\itemref{PI:H0-bij} for constructible sheaves.\n\\item\\label{PI:clopen-bij}\nFor every finite morphism $f\\colon X'\\to X$, the canonical map\n\\[\n\\clopen(X')\\to \\clopen(Y\\times_X X')\n\\]\nis bijective.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nThe equivalence between~\\itemref{PI:H0-bij} and~\\itemref{PI:H0-bij:cons}\nfollows by\nProposition~\\ref{P:qcqs-cons-colimit}. That~\\itemref{PI:H0-bij} implies~\\itemref{PI:clopen-bij} follows by the\nfollowing two observations: (a) if $A$ is a two-point set, then $H^0(X,f_*\\underline{A}_{X'})=\\clopen(X')$, and (b) by finite base change $h^*f_*\\underline{A}_{X'}=(f_Y)_*\\underline{A}_{Y\\times_X X'}$.\n\nTo see that~\\itemref{PI:clopen-bij} implies~\\itemref{PI:H0-bij}, take a monomorphism $F\\hookrightarrow G$ as in\nProposition~\\ref{P:mono-decomposed}. Then by~\\itemref{PI:clopen-bij}, $H^0(X,G)\\to H^0(Y,h^*G)$\nis bijective. It follows that $H^0(X,F)\\to H^0(Y,h^*F)$ is injective.\nFinally, take $H=G\\amalg_F G$. Then we have a diagram\n\\[\n\\xymatrix{\nH^0(X,F)\\ar[r]\\ar[d] & H^0(X,G)\\ar@<.5ex>[r]\\ar@<-.5ex>[r]\\ar[d] & H^0(X,H)\\ar[d]\\\\\nH^0(Y,h^*F)\\ar[r] & H^0(Y,h^*G)\\ar@<.5ex>[r]\\ar@<-.5ex>[r] & H^0(Y,h^*H)\n}\n\\]\nwith exact rows and injective vertical maps and bijective middle map.\nIt follows that the left map is bijective.\n\\end{proof}\nWe recall the following well-known definition.\n\\begin{definition}[Henselian pairs]\\label{D:henselian}\nA pair of algebraic stacks $(X,X_0)$ is\na \\emph{henselian pair} if $i\\colon X_0\\hookrightarrow X$ is a closed immersion and for\nevery finite morphism $X'\\to X$, the natural map\n\\[\n\\clopen(X')\\to\\clopen(X'\\times_X X_0)\n\\]\nis bijective.\n\\end{definition}\nWe have the following simple lemma.\n\\begin{lemma}\\label{L:hens_p_int}\nLet $(X,X_0)$ be a henselian pair. Let $X'\\to X$ be an integral morphism. If $X$ is quasi-compact and quasi-separated, then $(X',X'\\times_X X_0)$ is a henselian pair.\n\\end{lemma}\n\\begin{proof}\n  Since $X'\\to X$ is a limit of\nfinite morphisms~\\cite{rydh-2014}, the result follows from a simple approximation argument.\n\\end{proof}\n\n\\begin{remark}[Proper base change]\\label{R:proper_bc_et}\n  Let $(X,X_0)$ be a henselian pair, where $X$ is quasi-compact and quasi-separated. If \n  $g\\colon X'\\to X$ is \\emph{proper and representable},\n  then $(X',X'\\times_X X_0)$ is a henselian pair (see\n  \\cite[Cor.~B.4]{MR3148551} and \\cite[Cor.~1]{MR1286833}). This\n  follows from the existence of the Stein factorization\n  $X'\\to \\spec_{\\Orb_X} g_*\\Orb_{X'}\\to X$ where the first map is\n  proper with geometrically connected fibers and the second map is\n  integral \\cite[\\spref{0A1C}]{stacks-project}. This is a baby case of\n  the proper base change theorem in \\'etale cohomology.\n\\end{remark}\n\n\n\n\\section{Mayer--Vietoris squares in \\'etale cohomology}\\label{S:etale-gluing}\nWe will now glue \\'etale morphisms, or equivalently, \\'etale sheaves of\nsets. It is thus natural to introduce the following squares which are\nanalogous to Mayer--Vietoris $\\DQCOH$-squares.\n  \\begin{definition}\n    Fix a cartesian square as in \\eqref{E:MV-square}. It is a Mayer--Vietoris $\\mathrm{\\acute{E}t}$-square if \n    the following conditions are satisfied:\n    \\begin{enumerate}\n    \\item the natural transformation $f^*j_*\\to j'_*f_U^*$\n      is an isomorphism for every cartesian sheaf of sets $F\\in\\mathrm{\\acute{E}t}(U)$; and\n    \\item $f^*\\colon \\mathrm{\\acute{E}t}_{Z}(X)\\to\\mathrm{\\acute{E}t}_{Z'}(X')$ is \n      an equivalence of categories, where $\\mathrm{\\acute{E}t}_Z(X) = \\{F \\in \\mathrm{\\acute{E}t}(X) \\,:\\, j^*F = 0\\}$ and similarly \n      for $\\mathrm{\\acute{E}t}_{Z'}(X')$.\n    \\end{enumerate}\n  \\end{definition}\n  Note that $\\mathrm{\\acute{E}t}_Z(X)$ does not depend on the choice of $Z$ and that\n  $i_*\\colon \\mathrm{\\acute{E}t}(Z)\\to \\mathrm{\\acute{E}t}_Z(X)$ is an isomorphism.\n\n  For Mayer--Vietoris $\\mathrm{\\acute{E}t}$-squares, gluing is immediate from recollement.\n\n  \\begin{theorem}\\label{T:gluing-of-etale}\n    Consider a Mayer--Vietoris $\\mathrm{\\acute{E}t}$-square. Then the functor\n    \\[\n    \\Phi_\\mathrm{\\acute{E}t}\\colon \\mathrm{\\acute{E}t}(X)\\to \\mathrm{\\acute{E}t}(X')\\times_{\\mathrm{\\acute{E}t}(U')} \\mathrm{\\acute{E}t}(U)\n    \\]\n    is an equivalence of categories.\n   \n   \n  \\end{theorem}\n  \\begin{proof}\n    By recollement~\\cite[Exp.~IV, Thm.~9.5.4]{MR0354652},\n    \\[\n    \\mathrm{\\acute{E}t}(X)\\cong (\\mathrm{\\acute{E}t}(Z),\\mathrm{\\acute{E}t}(U),i^*j_*),\n    \\]\n    that is,\n    the category $\\mathrm{\\acute{E}t}(X)$ is equivalent to the category of triples\n    $E_Z\\in \\mathrm{\\acute{E}t}(Z)$, $E_U\\in \\mathrm{\\acute{E}t}(U)$, $\\psi\\colon E_Z\\to i^*j_*E_U$.\n    Similarly,\n    \\[\n    \\mathrm{\\acute{E}t}(X')\\cong (\\mathrm{\\acute{E}t}(Z'),\\mathrm{\\acute{E}t}(U'),i'^*j'_*)\n    \\]\n    and\n    \\begin{align*}\n    \\mathrm{\\acute{E}t}(X')\\times_{\\mathrm{\\acute{E}t}(U')} \\mathrm{\\acute{E}t}(U) &\\cong (\\mathrm{\\acute{E}t}(Z'),\\mathrm{\\acute{E}t}(U),i'^*j'_*f_U^*) \\\\\n     &\\cong (\\mathrm{\\acute{E}t}(Z),\\mathrm{\\acute{E}t}(U),(f_Z)_*i'^*j'_*f_U^*)\n    \\end{align*}\n    where we have used that $(f_Z)_*$ is an equivalence of categories.\n    Since $(f_Z)_*i'^*j'_*f_U^*=(f_Z)_*i'^*f^*j_*=i^*j_*$ the result follows.\n   \n   \n   \n   \n  \\end{proof}\n\n  We will now proceed to show that weak Mayer--Vietoris squares are\n  Mayer--Vietoris $\\mathrm{\\acute{E}t}$-squares. We begin with the following result that\n  generalizes~\\cite[Cor.~4.4]{MR0272779}.\n  \\begin{proposition}\\label{P:wMV-clopen}\n    Fix a weak Mayer--Vietoris square as in \\eqref{E:MV-square}. Assume that \n    $(X,Z)$ and $(X',Z')$ are henselian pairs. If $X$, $X'$, $U$ and $U'$ are all quasi-compact and \n    quasi-separated, then the natural map\n    \\[\n    f_U^*\\colon \\clopen(U)\\to \\clopen(U')\n    \\]\n    is bijective.\n  \\end{proposition}\n  \\begin{proof}\n    Since $X$ and $U$ are quasi-compact and quasi-separated, we may assume that the complement $i\\colon Z \\hookrightarrow X$ is finitely presented \\cite[Prop.~8.2]{rydh-2014}. Thus, we may replace the square with its blow-up so that it becomes a tor-independent \n    Mayer--Vietoris square (Lemma~\\ref{L:blow-up-is-MV-square}). Note\n    that $(X',Z')$ and $(X,Z)$ remain henselian pairs (Remark \\ref{R:proper_bc_et}).\n    \n  By Corollary~\\ref{C:normalization}, we may replace $X$ and $X'$ by\n  $\\overline{X}$ and $\\overline{X'}$ and assume that $X$ and $X'$ are\n  integrally closed with respect to $U$ and $U'$ respectively. Since the open\n  and closed subsets of an algebraic stack $W$ are in bijection with\n  idempotents of $\\Gamma(W,\\Orb_W)$, it follows that $\\clopen(X)\\to \\clopen(U)$\n  and $\\clopen(X')\\to \\clopen(U')$ are bijections. The corollary thus follows from\n  the commutativity of the following diagram:\n  \\[\n  \\begin{gathered}[b]\n  \\xymatrix{\n  \\clopen(Z)\\ar@{=}[d] & \\clopen(X)\\ar[l]_{\\cong}\\ar[r]^{\\cong}\\ar[d] & \\clopen(U)\\ar[d] \\\\\n  \\clopen(Z) & \\clopen(X')\\ar[l]_{\\cong}\\ar[r]^{\\cong} & \\clopen(U').}\\\\[-\\dp\\strutbox]\n  \\end{gathered}\n  \\qedhere\n  \\]\n  \\end{proof}\n\n  We can now prove Gabber's rigidity theorem. For affine henselian\n  pairs, this is proven in~\\cite[Exp.~20, Thm.~2.1.1]{MR3309086}. See\n  Remark~\\ref{R:rigidity} for some history of this result.\n\n  \\begin{theorem}[Rigidity theorem]\\label{T:rigidity-theorem}\n    Fix a weak Mayer--Vietoris square as in \\eqref{E:MV-square}. Assume that \n    $(X,Z)$ and $(X',Z')$ are henselian pairs. If $X$, $X'$, $U$ and $U'$ are all \n    quasi-compact and quasi-separated, then the natural map:\n    \\[\n    H^0(U,F)\\to H^0(U',F)\n    \\]\n    is a bijection for all sheaves of sets $F\\in\\mathrm{\\acute{E}t}(U)$.\n  \\end{theorem}\n  \\begin{proof}\n    It is enough to prove that $\\clopen(V)\\to \\clopen(U'\\times_U V)$ is\n    bijective for every finite morphism $V\\to U$\n    (Proposition~\\ref{P:H0-bij-vs-clopen}). By\n    Zariski's main theorem~\\cite[Thm.~8.1]{rydh-2014}, we can extend the finite\n    morphism $V\\to U$ to a finite morphism $\\overline{V}\\to X$. Since weak\n    Mayer--Vietoris squares are stable under arbitrary base change (Lemma~    \\ref{L:mv_bc}\\itemref{LI:mv_bc:w_f_mv}),\n    it is enough to prove\n    that $\\clopen(U)\\to \\clopen(U')$ is bijective, which is Proposition~\\ref{P:wMV-clopen}.\n  \\end{proof}\n\n  \\begin{corollary}\\label{C:wMV-is-MV-Et}\n    Fix a weak Mayer--Vietoris square as in \\eqref{E:MV-square}. If $j$ is quasi-compact, \n    then it is a Mayer--Vietoris $\\mathrm{\\acute{E}t}$-square. \n  \\end{corollary}\n  \\begin{proof}\n    We need to verify that the natural morphism $f^*j_*\\to j'_*f_U^*$ is\n    an isomorphism. This equality certainly holds\n    over $U'$ since the counits of the adjunctions $(j^*,j_*)$ and $(j'^*,j'_*)$\n    are isomorphisms and hence $j'^*j'_*f_U^*=f_U^*$ and\n    $j'^*f^*j_*=f_U^*j^*j_*=f_U^*$. It is thus enough to verify the equality\n    over points of $Z$. We can first assume that $X$ is affine by working\n    smooth-locally on $X$ and then replace $X$ with the henselization at a\n    point $z\\in Z$. Then $X'$ is Deligne--Mumford in a neighborhood of $Z$ and\n    we can thus replace $X'$ with the henselization at $z\\in Z$; in particular, $X$ and $X'$ \n    are quasi-compact and quasi-separated. Then\n    the equality $f^*j_*=j'_*f_U^*$ becomes $H^0(U,F)=H^0(U',f_U^*F)$, \n    which follows by the rigidity theorem.\n  \\end{proof}\n  We can now prove Theorem \\ref{MT:etale-gluing-for-wmv}.\n  \\begin{proof}[Proof of Theorem \\ref{MT:etale-gluing-for-wmv}]\n    Combine Corollary \\ref{C:wMV-is-MV-Et} with Theorem \\ref{T:gluing-of-etale}.\n  \\end{proof}\n\n  \\begin{corollary}\\label{C:wMV-is-univ-submersive}\n    Fix a weak Mayer--Vietoris square as in \\eqref{E:MV-square}. If $j$ is quasi-compact,  \n    then $X'\\amalg U\\to X$ is universally submersive and\n    $|X|=|X'|\\amalg_{|U'|} |U|$ is a pushout of topological spaces.\n  \\end{corollary}\n  \\begin{proof}\n    Since weak Mayer--Vietoris squares are preserved under arbitrary base\n    change it is enough to prove the latter statement. Set-theoretically,\n    $|X|=|X'|\\amalg_{|U'|} |U|$ holds since $f_Z \\colon Z' \\to Z$ is an isomorphism.\n    It is thus enough to prove that $|X|$ has the correct topology. Now a morphism of stacks is an open immersion if and only if it is an \\'etale\n    monomorphism. That an \\'etale morphism is a monomorphism can be checked\n    pointwise; thus, we have a bijection\n    \\[\n    \\Phi_{\\Op}\\colon \\Op(X)\\to \\Op(X')\\times_{\\Op(U')} \\Op(U).\n    \\]\n    It follows that a subset $W\\subseteq |X|$ is open if and only if\n    $j^{-1}(W)$ and $f^{-1}(W)$ are open.\n  \\end{proof}\n\n  \\begin{remark}\\label{R:rigidity}\n    The rigidity theorem holds more generally for cohomology as well. Fix a weak Mayer--Vietoris square as in \\eqref{E:MV-square}\n    and assume\n    that $(X,Z)$ and $(X',Z')$ are affine henselian pairs. If $n=0$\n    (resp.\\ $n\\leq 1$, resp.\\ $n$ an integer), then\n    \\[\n    H^n(U,F)\\to H^n(U',F)\n    \\]\n    is a bijection for all sheaves of sets $F\\in\\mathrm{\\acute{E}t}(U)$ (resp.\\ sheaves of\n    ind-finite groups, resp.\\ sheaves of torsion abelian groups). When $X$ is\n    noetherian, this is Gabber--Fujiwara's rigidity\n    theorem~\\cite[Cor.~6.6.4]{MR1360610}. For $n=0,1$, this was extended\n    to non-noetherian schemes by Gabber~\\cite[Thm.~7.1]{gabber-2005a},\n    cf.\\ \\cite[Exp.~20, Thm.~2.1.1]{MR3309086}. For $n\\geq 2$, the\n    non-noetherian case is\n    sketched by Gabber in~\\cite[Exp.~20, \\S4.4, CTC]{MR3309086}.  Note that\n    the general case reduces to the case where $X'$ is the completion of $X$\n    in $Z$. Indeed, such a completion is a weak Mayer--Vietoris square and\n    the completions\n    of $X$ in $Z$ and $X'$ in $Z$ are equal by definition.\n  \\end{remark}  \n\n\\section{Gluing of algebraic spaces along Mayer--Vietoris squares}\\label{S:algsp-gluing}\nIn this section, we prove the main theorems of the article. We begin with a slight strengthening of Theorem \\ref{MT:timv_dm_push}.\n\n\\begin{proposition}\\label{P:cocartesian-in-cat-of-quasi-DM}\nFix an algebraic stack $S$ and a tor-independent Mayer--Vietoris square as in \\eqref{E:MV-square} over $S$ with $j$ quasi-compact. Let $W\\to S$ be an algebraic stack. Then\n\\[\n\\Phi_{\\Hom_S(-,W)}\\colon \\Hom_S(X,W)\\to \\Hom_S(X',W)\\times_{\\Hom_S(U',W)} \\Hom_S(U,W)\n\\]\nis fully faithful. If either\n\\begin{enumerate}\n\\item $W\\to S$ is Deligne--Mumford; or\n\\item $\\Delta_{W\/S}$ is quasi-finite and $\\Delta_{\\Delta_{W\/S}}$ is a quasi-compact\n  immersion (e.g., $\\Delta_{W\/S}$ separated);\n\\end{enumerate}\nthen $\\Phi_{\\Hom_S(-,W)}$ is an equivalence of groupoids.\nIn particular, the square is cocartesian in the category\nof Deligne--Mumford stacks.\n\\end{proposition}\n\\begin{proof}\nThe question is fppf-local on $X$, so we may assume that $X$ is affine. We\nmay also replace $S$ and $W$ with $X$ and $W\\times_S X\\to X$ and assume that\n$X=S$. Further, we may replace $X'$ with a quasi-compact open neighborhood\nof $Z$.\nThen, we may also assume that $W$ is quasi-compact.\n\nIf $W\\to X$ is arbitrary (resp.\\ representable, resp.\\ a\nmonomorphism), then $\\Delta_{W\/X}$ is representable (resp.\\ a monomorphism,\nresp.\\ an isomorphism). Fully faithfulness of $\\Phi_{\\Hom_X(-,W)}$ follows if\n$\\Phi_{\\Hom_X(-,W\\times_{W\\times_X W} X)}$ is an equivalence for\nevery morphism $X\\to W\\times_X W$. By induction on the diagonal,\nwe may thus assume that $\\Phi_{\\Hom_X(-,W)}$ is fully faithful.\n\nIf $W$ is Deligne--Mumford, then there exists an \\'etale presentation\n$W'\\to W$. If $W$ is as in the second case, then by~\\cite[Thm.~7.2]{MR2774654}\nthere exists an \\'etale representable morphism $W'\\to W$ and a finite\nfaithfully flat morphism $V\\to W'$ such that $V$ is affine.\n\nGiven maps $U\\to W$ and $X'\\to W$ that agree on $U'$, we obtain, by pulling\nback $W'\\to W$, an element of $\\mathrm{\\acute{E}t}(X')\\times_{\\mathrm{\\acute{E}t}(U')} \\mathrm{\\acute{E}t}(U)$, hence a unique element of $(E\\to\nX)\\in \\mathrm{\\acute{E}t}(X)$ by Corollary~\\ref{C:wMV-is-MV-Et} and\nTheorem~\\ref{T:gluing-of-etale}. Pulling-back the square along\n$E\\to X$, we may replace $X$ by $E$ and assume that $W=W'$.\n\nIn the second case, we pull-back $V\\to Z'=Z$ to finite flat morphisms\nover $X'$, $U'$ and $U$. These glue to a unique finite faithfully flat morphism\n$F\\to X$ (Corollary \\ref{C:gluing-of-algebras} and Theorem \\ref{T:MV-QCoh-gluing}\\itemref{TI:MV-QCoh-gluing:flat}). We may thus replace $X$ with $F$ and assume that $V=W=W'$.\n\nNow $W$ is affine. Let\n$A_W=\\Gamma(W,\\Orb_W)$ for any algebraic stack $W$. The\nmap $\\Phi_{\\Hom(-,W)}$ then becomes\n\\begin{align*}\n\\Hom(A_W,A_X) &\\to\n  \\Hom(A_W,A_{X'})\\times_{\\Hom(A_W,A_{U'})} \\Hom(A_W,A_U) \\\\\n&= \\Hom(A_W,A_{X'}\\times_{A_{U'}}A_U).\n\\end{align*}\nThis is an isomorphism since $A_X\\to A_{X'}\\times_{A_{U'}}A_{U}$ is an\nisomorphism by Theorem~\\ref{T:MV-QCoh-gluing} applied to the structure sheaf\n$\\Orb_X$.\n\\end{proof}\nWe can now prove Theorem~\\ref{MT:timv_dm_push}.\n\\begin{proof}[Proof of Theorem~\\ref{MT:timv_dm_push}]\nThis is the last statement of Proposition~\\ref{P:cocartesian-in-cat-of-quasi-DM}.\n\\end{proof}\nWe can now also generalize Corollary \\ref{C:MV-QCoh-gluing:ff}\nfrom quasi-coherent sheaves to algebraic spaces.\n\\begin{corollary}\\label{C:gluing-of-maps-of-alg-spaces}\nFix a tor-independent Mayer--Vietoris square as in \\eqref{E:MV-square} with $j$ quasi-compact. Let $Y\\to X$ and $Z\\to X$ be algebraic spaces. If $Y\\to X$ is $f$-flat,\nthen\n\\[\n\\Hom_X(Y,Z)\\to \\Hom_X(Y\\times_X X',Z)\\times_{\\Hom_X(Y\\times_X U',Z)} \\Hom_X(Y\\times_X U,Z)\n\\]\nis bijective.\n\\end{corollary}\n\\begin{proof}\nSince $Y\\to X$ is $f$-flat, the pull-back of the square along $Y\\to X$ is\na tor-independent Mayer--Vietoris square (Lemma \\ref{L:mv_bc}\\itemref{LI:mv_bc:mv}).\nThe result thus follows from\nProposition~\\ref{P:cocartesian-in-cat-of-quasi-DM}.\n\\end{proof}\nWe now have proved Theorem \\ref{MT:glue_timv} in its entirety.\n\\begin{proof}[Proof of Theorem \\ref{MT:glue_timv}]\n  Claim \\itemref{MTI:glue_timv:qcoh} is Theorem \\ref{T:MV-QCoh-gluing} and claim \n  \\itemref{MTI:glue_timv:algsp} is Corollary \\ref{C:gluing-of-maps-of-alg-spaces}.\n\\end{proof}\n\n\n\\begin{remark}\nThe map in Corollary~\\ref{C:gluing-of-maps-of-alg-spaces} need not be\ninjective if the square is a weak Mayer--Vietoris square. Indeed,\nExample~\\ref{E:weak-but-not-MV} is an example of a weak Mayer--Vietoris square\nsuch that $\\Gamma(X)\\to \\Gamma(X')\\times_{\\Gamma(U')}\\Gamma(U)$ is not\ninjective. If $f,g\\in \\Gamma(X)$ are two element that have equal images, then\nthe corresponding maps $f,g\\colon X\\to \\mathbb{A}^1$ become equal after restricting\nto $X'$ and $U$.\n\\end{remark}\nWe can also now prove Theorem \\ref{MT:glue_fmv}.\n\\begin{proof}[Proof of Theorem \\ref{MT:glue_fmv}]\nThat $\\Phi_{\\mathsf{QCoh}}$ is an equivalence is Corollary \\ref{C:MV-QCoh-gluing:flatness}. That $\\Phi_{\\AFF}$ and $\\Phi_{\\mathsf{Qaff}}$ are equivalences is Corollary \\ref{C:fmv-qaff_gluing}. That $\\Phi_{\\AlgSp}$ is fully faithful is a special case of\nCorollary~\\ref{C:gluing-of-maps-of-alg-spaces}. That $\\Phi_{\\Hom(-,W)}$ is fully faithful for every algebraic stack $W$ is Proposition \\ref{P:cocartesian-in-cat-of-quasi-DM}. That $\\Phi_{\\Hom(-,W)}$ is an equivalence when $W$ has quasi-affine diagonal follows from Corollary \\ref{C:fmv-qaff_gluing} and an identical argument to \\cite[Cor.~6.5.1(a)]{MR1432058}.\n\nIt remains to prove \\itemref{MTI:glue_fmv:algsp-exc}: $\\Phi_{\\AlgSp_{\\mathrm{lfp}}}$\nis an equivalence when $X$ is locally excellent.\nFor quasi-separated algebraic\nspaces the essential surjectivity of $\\Phi_{\\AlgSp_{\\mathrm{lfp},\\qs}}$\nfollows as in~\\cite[Thm.~5.2 (ii), Cor.~5.6 (iii), Thm.~5.7]{MR1432058} but\nsince we are working in a slightly more general setting let us write out the\ndetails. For brevity, we let $\\Phi=\\Phi_{\\AlgSp_{\\mathrm{lfp}}}$.\n\nSince algebraic spaces satisfy descent for the fppf topology, we may use Proposition \\ref{P:qaff_dom} and the \\'etale gluing result \\cite[Thm.~A]{MR2774654}, and so assume that $X$ is affine and $X'$ is quasi-affine.\n\nIf $P$ is a property of morphisms of algebraic spaces, then we say that a\nmorphism of triples is $P$ if the three components are $P$. Since $X'\\amalg\nU\\to X$ is faithfully flat and quasi-compact, a morphism $f\\colon W\\to Z$ in\n$\\AlgSp_{\\mathrm{lfp}}(X)$ is quasi-compact (resp.\\ quasi-separated, resp.\\ \\'etale,\nresp.\\ open, resp.\\ a monomorphism) if and only if $\\Phi(f)$ has the same\nproperty~\\cite[IV.2.7.1, IV.17.7.3 (ii)]{EGA}.\n\nWe now prove essential surjectivity of $\\Phi$. Thus, consider a triple $W'\\to X'$, $W_U\\to\n U$, $W_{U'}\\to U'$ on the right-hand side. \n\nWe will begin by showing that it\nis enough to prove essential surjectivity of $\\Phi$ for the subcategories of\n\\emph{quasi-compact} algebraic spaces\n(cf.~\\cite[Thm.~5.7]{MR1432058}). Write $W'$ and $W_U$ as filtered\nunions of\nquasi-compact open subspaces $W'_\\lambda$ and $W_{U,\\mu}$ respectively. Since $j'\\colon U' \\to X'$ is quasi-compact, for\nevery $\\lambda$, the open subspace $W'_\\lambda\\cap W_{U'}$ is\nquasi-compact. Hence, for sufficiently large $\\mu=\\mu(\\lambda)$, the inverse\nimage $W_{U',\\mu}:=f_U^{-1}(W_{U,\\mu})$ contains $W'_\\lambda\\cap W_{U'}$. We\nmay thus form the triple $(W'_\\lambda\\cup W_{U',\\mu},W_{U,\\mu},W_{U',\\mu})$ of\nquasi-compact algebraic spaces. By assumption, this triple is in the\nessential image of $\\Phi$ and descends to an algebraic space $W_{\\lambda,\\mu}$.\nWe then let $W=\\bigcup_{\\lambda,\\mu} W_{\\lambda,\\mu}$ where the union runs over\nall $\\lambda$ and $\\mu\\geq \\mu(\\lambda)$.\n\nWe next assume that these morphisms are also quasi-compact and\nquasi-separated. In this case, we claim that we are free to replace $X$ with\nany flat covering\n$(X_i\\to X)$ such that every $X_i\\to X$ is a filtered limit of flat and finitely\npresented morphisms $X_{i,\\lambda}\\to X$. Indeed, assume that the result holds\nfor the $X_i$, that is, there exists an algebraic space $W_i\\to X_i$ of finite\npresentation such that $\\Phi(W_i)\\cong (W',W_{U'},W_U)\\times_X X_i$.  Then, by\nstandard limit arguments, there is for every $i$ and every sufficiently large\n$\\lambda=\\lambda(i)$ an algebraic space $W_{i,\\lambda}\\to X_{i,\\lambda}$ of\nfinite presentation such that $\\Phi_{X_{i,\\lambda}}(W_{i,\\lambda})\\cong\n(W',W_{U'},W_U)\\times_X X_{i,\\lambda}$. Since $\\Phi$ is fully faithful over\n$X_{i,\\lambda}\\times_X X_{i,\\lambda}$ and $X_{i,\\lambda}\\times_X\nX_{i,\\lambda}\\times_X X_{i,\\lambda}$ there is a canonical gluing datum for\n$W_{i,\\lambda}\\to X_{i,\\lambda}$ along $X_{i,\\lambda}\\to X$ which is flat and\nof finite presentation.  So by fppf descent, $W_{i,\\lambda}\\to X_{i,\\lambda}$\ndescends to an algebraic space over the open image of $X_{i,\\lambda}\\to\nX$. Since we can find a finite number of such $X_{i,\\lambda}$ that cover $X$,\nthe claim follows.\n\nSince $X$ is excellent the completion map $\\widehat{X}_x\\to X$ is a regular\nmorphism. Hence, by Popescu's theorem \\cite{MR818160}, it is a limit of smooth morphisms. Since\n$(\\widehat{X}_x\\to X)_{x\\in X}$ is a flat cover, we may replace $X$ with\n$\\widehat{X}_x$ for some $x$ and assume that $X$ is the spectrum of a complete local ring. The completion of $X'$ at $z$ equals the completion of $X$ at $z$; hence,\n$X'\\to X$ has a section $s\\colon X\\to X'$. By Lemma~\\ref{L:sections-give-MV},\nthis gives rise to a new tor-independent Mayer--Vietoris square. Corollary~\\ref{C:gluing-of-maps-of-alg-spaces} for this square implies that\n\\begin{align*}\n\\Hom&(X',W')\\times_{\\Hom(U',W_{U'})} \\Hom(U,W_U)\\\\\n&=(\\Hom(X,s^*W')\\times_{\\Hom(U,W_{U})} \\Hom(U',W_{U'}))\\times_{\\Hom(U',W_{U'})} \\Hom(U,W_U)\\\\\n&=\\Hom(X,s^*W'),\n\\end{align*}\nwhich implies that $\\Phi(s^*W')\\cong (W',W_{U'},W_U)$. Thus, $\\Phi$ is essentially\nsurjective for finitely presented algebraic spaces. In fact, by the initial reduction to the \nquasi-compact case, we have proved that $\\Phi$ is essentially surjective for triples of \nquasi-separated algebraic spaces.\n\nLet us finally prove that $\\Phi$ is also essentially surjective for algebraic\nspaces that are not quasi-separated. It is enough to prove that it is\nessentially surjective for quasi-compact algebraic spaces. By the previous\nargument, it is enough to prove that if $\\bar{X}=\\varprojlim_\\lambda\nX_{\\lambda}$ is a limit of affine schemes, and\n$(\\bar{W}',\\bar{W}_{U'},\\bar{W}_U):=(W',W_{U'},W_U)\\times_X \\bar{X}$ is in the\nessential image of $\\Phi$, then so is $(W',W_{U'},W_U)\\times_X X_\\lambda$ for\nsufficiently large $\\lambda$.\n\nThus, let $\\bar{W}\\to \\bar{X}$ be an algebraic space such that\n$\\Phi(\\bar{W})=(\\bar{W}',\\bar{W}_{U'},\\bar{W}_U)$ and pick an affine\npresentation $\\bar{V}\\to \\bar{W}$. Note that $\\bar{V}\\to \\bar{W}\\to \\bar{X}$ is\nfinitely presented. This induces morphisms of triples\n\\[\n\\Phi(\\bar{V})=\n(\\bar{V}',\\bar{V}_{U'},\\bar{V}_U)\\to\n(\\bar{W}',\\bar{W}_{U'},\\bar{W}_U)\\to\n(\\bar{X}',\\bar{U'},\\bar{U})\n\\]\nwhere the first map is surjective and \\'etale and the composition is of finite\npresentation. For sufficiently large $\\lambda$, we may thus descend this to a\nmorphism of triples\n\\[\n(V'_\\lambda,V_{U',\\lambda},V_{U,\\lambda})\\to\n(W'_\\lambda,W_{U',\\lambda},W_{U,\\lambda})\\to\n(X'_\\lambda,U'_\\lambda,U_\\lambda)\n\\]\nover $X_\\lambda$ where the first map is \\'etale and the composition is of\nfinite presentation. Thus, there exists an algebraic space $V_\\lambda\\to\nX_\\lambda$, unique up to unique isomorphism, such that $\\Phi(V_\\lambda)\\cong\n(V'_\\lambda,V_{U',\\lambda},V_{U,\\lambda})$.\n\nLet $R'_\\lambda=V'_\\lambda\\times_{W'_\\lambda} V'_\\lambda$ and similarly over\n$U'_\\lambda$ and $U_\\lambda$. Then the triple\n$(R'_\\lambda,R_{U',\\lambda},R_{U,\\lambda})$ is locally of finite presentation\nand quasi-separated over $(X'_\\lambda,U'_\\lambda,U_\\lambda)$ and hence\nisomorphic to $\\Phi(R_\\lambda)$ for an essentially unique $R_\\lambda\\to\nX_\\lambda$. By fully faithfulness, we obtain an \\'etale equivalence relation\n$\\equalizer{R_\\lambda}{V_\\lambda}$ and we let $W_\\lambda$ be its quotient\nalgebraic space. By fully faithfulness, $\\Phi(W_\\lambda)$ is isomorphic to\n$(W'_\\lambda,W_{U',\\lambda},W_{U,\\lambda})$ and the theorem follows.\n\\end{proof}\nFinally, we prove Theorem \\ref{MT:pushout_fmv}.\n\\begin{proof}[Proof of Theorem \\ref{MT:pushout_fmv}]\n  We must show that for every algebraic stack $W$, the functor\n  \\[\n  \\Phi_{\\Hom(-,W)}\\colon \\Hom(X,W)\\to \\Hom(X',W)\\times_{\\Hom(U',W)} \\Hom(U,W)\n\\]\nis an equivalence of groupoids. Using Proposition \\ref{P:qaff_dom} and the \\'etale gluing \nresult \\cite[Thm.~A]{MR2774654}, we may assume that $X$ is affine and $X'$ is quasi-affine. In particular, we are free to assume that $W$ is quasi-compact. We have already seen that $\\Phi_{\\Hom(-,W)}$ is fully faithful in\nProposition~\\ref{P:cocartesian-in-cat-of-quasi-DM}. To see that it is\nessentially surjective, pick a smooth presentation $W_0\\to W$ where $W_0$ is an\naffine scheme. Pulling back, we obtain a triple in $(X'_0,U'_0,U_0)\\in\n\\AlgSp_\\mathrm{lfp}(X')\\times_{\\AlgSp_{\\mathrm{lfp}}(U')} \\AlgSp_{\\mathrm{lfp}}(U)$; hence, a representable morphism $X_0\\to X$ by\nTheorem~\\ref{MT:glue_fmv}. Since $X'\\amalg U'\\to X$ is\nfaithfully flat and quasi-compact, it follows that $X_0\\to X$ is smooth.  Also,\n\\[\n\\vcenter{\\xymatrix{\n        U'_0\\ar[r]\\ar[d] & X'_0\\ar[d] \\\\\n        U_0\\ar[r] & X_0,\\ar@{}[ul]|\\square\n      }}\n\\]\nis a flat Mayer--Vietoris square.  Since $W_0$ is\naffine, we obtain a unique morphism $X_0\\to W_0$ compatible with $X'_0\\to W_0$\nand $U_0\\to W_0$ (Proposition~\\ref{P:cocartesian-in-cat-of-quasi-DM}). By the full\nfaithfulness of $\\Phi_{\\Hom(-,Z)}$, the induced morphisms $\\equalizer{X_0\\times_X X_0}{X_0} \\to W_0 \\to W$ coincide, so there is a unique morphism $X \\to W$ and the result follows.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe $S=1$ spin antiferromagnetic Heisenberg chain has received much\nattention, both experimentally and theoretically, since\nHaldane\\cite{haldane}  conjectured that its low-energy properties\nare qualitatively different from that of the exactly solved $S=1\/2$\nmodel. The $S=1$ chain (together with all other integer spin chains) has\na finite gap in the excitation spectrum and hidden topological order\nin the ground state, which is characterized by the string correlation\nfunction.\\cite{denN89} On the other hand, the bulk spin-spin correlations of the\nmodel are short ranged, having a finite correlation length,\n$\\xi$. In an open chain of length $L$,\nthere are spin $S=1\/2$ degrees of freedom at each edge and the\nend-to-end correlations approach a finite value in an exponential\nfashion, having the same characteristic length scale, $\\xi$, as bulk\ncorrelations.\\cite{sorensenaffleck94}\n\nQuenched disorder, which is realized by random couplings, also has different\neffects for $S=1\/2$ and $S=1$. In the former case any amount of disorder is\nenough to drive the system into a new type of fixed point, \\cite{fisherxx} whereas for the\n$S=1$ chain, weak disorder is irrelevant and the properties of the weakly\nrandom chain are the same as that of the pure one.\\cite{hyman-dimer} For stronger disorder, however,\nthe low-energy properties of the system are changed and detailed analytical and\nnumerical investigations were devoted to clarify the properties of the new random fixed\npoints. \n\nThe analytical studies of the random chain are made by variants of the strong\ndisorder renormalization group\n(RG) method, which has been introduced for the $S=1\/2$ chain by Ma, Dasgupta,\nand Hu \\cite{mdh} and\nhas been analyzed in great details by Fisher.\\cite{fisherxx} This strong disorder RG\nmethod has been used afterwards for a large variety of random quantum\nand classical systems, (for a review, see Ref. \\onlinecite{review}). For the $S=1$ chain, extensions of\nthe original Ma-Dasgupta rules are necessary\\cite{Hyma97,monthus} to describe the disorder induced phases\nin the system, which include a gapless Haldane (GH) phase, for intermediate disorder, and a\nrandom singlet (RS) phase, for stronger disorder.\n\nNumerical studies of the random $S=1$ chain have been made by exact\ndiagonalization\\cite{nishiyama} of\nthe density matrix renormalization\\cite{Hida}\n(DMRG), by quantum Monte Carlo (QMC) methods,\\cite{todo,bergkvist} and by numerical implementations of the strong\ndisorder RG method. \\cite{saguia02} Despite considerable numerical effort, several aspects of the\nlow-energy properties of the random $S=1$ chain are still unclear and some numerical\nresults are conflicting.\nIn the numerical calculations mainly boxlike  distribution of disorder is considered,\nwhich, as noted in Ref. \\onlinecite{comment}, represent only a limited strength of randomness.\nIn numerical RG studies both the GH and the RS phases are identified; however, the transition\npoint between these phases is rather approximate.\nIn DMRG calculation (see also Ref. \\onlinecite{nishiyama}) Hida \\cite{Hida} has identified only\nthe GH phase, and conjectured that the\nRS phase is not accessible for any finite strength  of disorder. In a comment to Hida's work,\\cite{Hida}\nYang and Hyman\\cite{comment} have predicted the appearance of the RS phase for some type of\npower-law distribution of the disorder.\nAnother numerical work by QMC simulations \\cite{todo} has shown the existence of the RS phase even for\nthe boxlike distribution and these results are confirmed by independent QMC simulations.\\cite{bergkvist}\nIn the QMC calculations, some properties of the RS phase are verified (cf. scaling relation between length\nand time, decay of the string correlation function), but results about\nthe spin-spin correlation function are different from the RG predictions. At the critical point no\nnumerical estimates are available to check analytical RG predictions. We note on recent studies\nof Griffiths effects\\cite{griffiths} in the system with enforced dimerization\\cite{Daml02,arakawa}\nand related work on the random $S=3\/2$ and higher spin chains\\cite{Refa02,clri04,segu03}.\n\nIn this paper, our aim is to study the low-energy properties of the random $S=1$ chain by the\nDMRG method.\\cite{DMRGbook} \nThe features of our study are the following: (i) We consider a more general (power-law) distribution\nof disorder, which allows us to enter more deeply into the RS phase, thus to obtain convincing evidence\nof its existence. (ii) We calculate a different physical quantity, the end-to-end correlation function, which\ncarries important information about the phases of the system. The average end-to-end correlation\nfunction has a finite limiting value in the GH phase and vanishes in the RS phase. Furthermore,\nin the GH phase from the low-value tail of its distribution, independent estimates about the dynamical\nexponent are obtained. (iii) We try to perform a comparative analysis between the properties of the\nsystem at the critical point and in the RS phase and to check the available RG predictions.\n\nThe structure of our paper is the following. The model, the basic ingredients of the strong\ndisorder RG methods, and the conjectured phases are given in Sec. \\ref{Sec:2}. Results of our\nDMRG studies are presented in Sec. \\ref{Sec:3} and discussed in Sec. \\ref{Sec:4}.\n\n\\section{The model and the strong disorder RG results}\n\\label{Sec:2}\n\\subsection{Model}\nWe consider the spin $S=1$\nrandom antiferromagnetic Heisenberg chain with the Hamiltonian\n\\begin{equation}\nH = \\sum_i J_{i} \\vec{S}_i \\cdot \\vec{S}_{i+1},\n\\label{hamilton}\n\\end{equation}\nwhere the $J_i > 0$ are independent and identically distributed random variables.\nHere, we use the following power-law distribution\n\\begin{equation}\np_\\delta (J) = \\delta^{-1} J^{-1+1\/\\delta}\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n{\\rm for} \\ 0 \\leq J \\leq 1,\n\\label{distribut}\n\\end{equation}\nwhere $\\delta^2 ={\\rm var}[ \\ln J]$ measures the strength of disorder. In previous numerical\nwork, a boxlike distribution was used,\n\\begin{equation}\nP_W(J)=\\left\\{\n\\begin{aligned}\n&1\/W& &{\\rm for}\\quad 1-W\/2<J<1+W\/2\\\\\n&0& & {\\rm otherwise,}\n\\end{aligned}\n\\right.\n\\label{box}\n\\end{equation}\nin which the strength of disorder grows with $W$. Note that the possible maximal value, $W=2$, corresponds\nto the uniform distribution, which can be obtained from  Eq. (\\ref{distribut}) with $\\delta=1$ and having\na prefactor, $1\/2$, and a range $0 \\le J \\le 2$. Consequently, the\npower-law distribution for $\\delta>1$ represents a disorder, which is stronger than any boxlike\ndisorder.\n\nThe low-energy behavior of the system of size, $L$, is encoded in the distribution of the\nlowest gap, $\\Delta$, denoted by $P_L(\\Delta)$. We note that for an open chain the first gap\ncorresponds to the localized edge states; therefore, one should study the second (not localized)\ngap. The average spin-spin correlation function is denoted by\n\\begin{equation}\nC(i,j)=[\\langle S^{z}_i S^{z}_j \\rangle]_{\\rm av},\n\\label{cij}\n\\end{equation}\nwhere $[\\cdots ]_{\\rm av}$ stands for averaging over quenched\ndisorder. For bulk correlations with $|j-i| \\ll i,j=O(L)$, we have $C(i,j)=C_b(|j-i|)$,\nwhereas for end-to-end correlations, $C(1,L) \\equiv C_1(L)$.\nThe string correlation function of the model is defined by\\cite{denN89}\n\\begin{equation}\nO^z(r)=- \\langle S_l^z \\exp\\left[ i \\pi \\left( S_{l+1}^z+S_{l+2}^z+\\dots+S_{l+r-1}^z \\right)\\right] S_{l+r}^z \\rangle\\;,\n\\label{string}\n\\end{equation}\nand its large $r$ limiting value is the string order parameter. For several quantities it turned\nout useful to consider the average of its inverse. More\nprecisely, for a physical observable, $f$, we denote by $f^{\\rm iv}$ the following quantity:\n\\begin{equation}\nf^{\\rm iv}=\\frac{1}{[ f^{-1} ]_{\\rm av}}\\;,\n\\label{inverse}\n\\end{equation}\nwhat we shall call as inverse average.\n\n\\subsection{Weak disorder limit--Haldane phase}\n\\label{limiting}\n\nIn absence of randomness ($J_{i} = J$) the spectrum is gapped,\\cite{haldane} and bulk\nspin-spin correlations are short ranged, $C_b(r) \\sim \\exp(-r\/\\xi)$ with\n$\\xi=6.03$\\cite{sorensenaffleck94}. On the contrary, end-to-end spin-spin correlations\nand the string correlation function have a finite limiting value.\nFor weak disorder, when the distribution of $J$ is sufficiently narrow,\nthe Haldane gap is robust and the system stays in\nthe Haldane phase.\\cite{hyman-dimer} The border of the Haldane phase can be estimated by\nnoting that the Haldane gap is robust against enforced dimerization,\\cite{ah87}\nwhen even and odd couplings are different, so that  \n\\begin{equation}\nJ_i=J(1+D (-1)^i)\\exp(\\delta \\zeta_i)\\;,\n\\label{J_S}\n\\end{equation}\nwhere $\\zeta_i$ are random numbers of mean zero and variance unity.\nThe pure system ($\\delta=0$) for $D<0.25$ stays in the Haldane\nphase\\cite{singh} and at the phase transition point the coupling at an\nodd bond, $J_o$, and that at an even bond, $J_e$, are related as $J_o\n= 0.6 J_e$. We expect that in the presence of disorder the Haldane gap\nstays finite, if the maximum ($J_{max}$) and the minimum ($J_{min}$)\nvalues of the couplings satisfy $J_{min}\/J_{max} > 0.6$. From this\nargument we obtain for the border of the Haldane phase for the box\ndistribution $W_G \\approx 0.5$. On the other hand, for the power-law\ndistribution in Eq. (\\ref{distribut}) $J_{min}=0$; therefore, for any\n$\\delta>0$ the Haldane phase is expected to be destroyed.\n\n\\subsection{Strong disorder limit--RG approach}\n\nFor strong disorder the low-energy properties of the system are\nexplored by variants of the strong disorder RG approach.  In the\nstandard Ma-Dasgupta--type RG approach, the couplings of the random\nantiferromagnetic Heisenberg chain are put in descending order and the\nlargest coupling defines the energy scale, $\\Omega$, in the system.\nDuring renormalization the pair of spins with the largest coupling,\nsay $J_i=\\Omega$, are replaced by a singlet and decimated out. At the\nsame time a new coupling is generated between the spins at the two\nsides of the singlet, which is given in a perturbation calculation as\n\\begin{equation}\n\\tilde{J}=\\frac{4}{3} \\frac{J_{i-1} J_{i+1}}{J_i}\\;.\n\\label{J_pert}\n\\end{equation}\nAs noticed by Boechat, Saguia, and Continentino\\cite{saguia96} for weak disorder some of\nthe generated new couplings can be larger than the energy scale,\n$\\Omega$. Therefore, the standard strong disorder RG approach works\nonly for strong enough disorder and describes only the RS phase of\nthe system.\n\nTo cure this problem, different types of RG approaches are proposed.\nMonthus  \\it et al. \\rm \\cite{monthus}  suggested to replace the pair of spin\n$S=1$ connected by the strongest bond by a pair of $S=1\/2$. In this\ncase the renormalized system consists of a set of spin $S=1$ and\n$S=1\/2$ degrees of freedom, having both antiferromagnetic and\nferromagnetic couplings. The renormalized couplings, which are\ncalculated perturbatively, are all smaller than $\\Omega$. This RG\napproach, during which no spin larger than $S=1$ is generated, can be\nused to describe both the gapless Haldane and the RS phases and\nprovides precise numerical estimates about the critical exponents.\n\nIn another modified RG approach, Saguia \\it et al. \\rm \\cite{saguia02} use the\nstandard perturbative approach in Eq.~(\\ref{J_pert}), provided\n$max(J_{i-1},J_{i+1})<3\\Omega\/4$.  Otherwise the triplet of spins with\ncouplings, $max(J_{i-1},J_{i+1})$, and $\\Omega$ is replaced by a single\nspin. Also in this method the variation of the energy scale is\nmonotonic: the generated two new couplings are both smaller than\n$\\Omega$.\n\nRecently, a variant of the strong disorder RG method was proposed by one of\nus,\\cite{Peter} in\nwhich the pair of spin with the strongest coupling is decimated out,\nbut--and this is a feature of our current method--the new coupling\nbetween the remaining spins is calculated nonperturbatively. The four\nspins with couplings $J_{i-1},~J_{i}$, and $J_{i+1}$ are replaced by\ntwo spins and during decimation the lowest gap in the two systems\nremains the same. It is easy to see that the rule we use is somewhat similar\nto the approach by Saguia \\it et al. \\rm  \\cite{saguia02} However, this method has no\ndiscontinuity in the approximation, which could be important in the\nvicinity of the critical point, where a crossover takes place between\nthe different decimation regimes.\n\n\\subsection{Disorder-induced phases}\n\nBased on a modified strong disorder RG approach\\cite{Hyma97,monthus,saguia02} and different\nnumerical calculations,\\cite{Hida,todo,bergkvist} the following scenario of the\nphase transition in the model is conjectured with increasing strength\nof disorder.\n\n\\subsubsection{Gapless Haldane phase}\n\nFor sufficiently strong disorder ($\\delta>\\delta_G$ or $W > W_G$), the\ngap in the Haldane phase is closed and one arrives to the gapless\nHaldane phase. As we have argued in Sec. \\ref{limiting}, $\\delta_G=0$ and\n$W_G \\approx 0.5$. The GH phase is a quantum Griffiths\nphase,\\cite{griffiths} in which the correlation length, $\\xi(\\delta)$,\nis finite, whereas the typical time scale, $t_r \\sim \\Delta^{-1}$, is\ndivergent. Relation between the size of the system, $L$, and the\nsmallest gap is given by\n\\begin{equation}\n\\Delta \\sim L^{-z}\\;,\n\\label{z'}\n\\end{equation}\nwhere $z$ is the disorder induced dynamical exponent. The\ndistribution of the lowest gap is given by\n\\begin{equation}\nP_L(\\Delta){\\rm d} \\Delta= L^{-z}\\tilde{P}\\left[\\frac{\\Delta}{L^z}\\right] {\\rm d} \\Delta\\;,\n\\label{eq:scaling-conv}\n\\end{equation}\nand $\\tilde{P}(x) \\sim x^{-1+1\/z}$ for small $x$, so that from the\nlow-energy tail $z$ can be calculated.  Similarly the distribution of\nthe end-to-end correlation function has a vanishing tail, which\nbehaves as\\cite{ijl01} $P(C_1) \\sim C_1^{-1+1\/z}$, which gives an independent\nway to calculate the\ndynamical exponent.  Some thermodynamical\nquantities such as the local susceptibility, $\\chi$, and the specific\nheat, $c_v$, are singular at low temperature,\\cite{review}\n\\begin{equation}\n\\chi(T) \\sim T^{-1+1\/z},\\quad c_v(T) \\sim T^{1\/z}\\;.\n\\label{T}\n\\end{equation}\nThe limit of divergence of $\\chi(T)$ is signaled by $z=1$, and the corresponding disorder\nis denoted by $\\delta_1$ ($W_1$). The separation of the two parts of the GH phase with\n$z<1$ and $z>1$ can\nbe located by considering the inverse average of the gap, $\\Delta^{\\rm iv}$, and\nthe inverse average of the end-to-end correlation function, $C_1^{\\rm iv}$. In the\nnonsingular region, $z<1$, both $\\Delta^{\\rm iv}$ and $C_1^{\\rm iv}$ are finite,\nwhereas in the singular region, $z>1$, both are vanishing.\n\nTo see this, we consider the inverse average of the gap\n\\begin{equation}\n\\Delta^{\\rm iv} \\sim \\left[ \\int_{\\Delta_{min}}^{\\Delta_{max}} \\Delta^{-2+1\/z} {\\rm d} \\Delta\\right]^{-1}\n\\sim \\frac{z-1}{\\Delta_{min}^{-1+1\/z}-\\Delta_{max}^{-1+1\/z}}\\;,\n\\label{inv}\n\\end{equation}\nwhich indeed tends to zero, if $z>1$ and $\\Delta_{min} \\to 0$. On the other hand for the vanishing of\nthe average gap one needs $\\Delta_{max} \\to 0$. One can use a similar reasoning for the end-to-end\ncorrelation function, for which the upper limit of the distribution, $C_1^{max}>0$, thus\n$[C_1]_{\\rm iv}>0$, in the whole region, $\\delta<\\delta_1$.\n\nIn a static sense, the gapless Haldane phase is noncritical: the average end-to-end correlation\nfunction, as well as the string order parameter, is finite in the complete GH phase.\n\n\\subsubsection{Critical point}\n\nIncreasing the strength of disorder over a critical value ($\\delta_C$ or $W_C$), the system arrives\nat the random singlet phase. As the critical strength of disorder is approached, the correlation\nlength diverges: $\\xi \\sim (\\delta_C-\\delta)^{-\\nu}$, with $\\nu=(1+\\sqrt{13})\/2$ and the string\norder parameter vanishes as\\cite{Hyma97,monthus} $O^z(\\delta) \\sim\n(\\delta_C-\\delta)^{2 \\beta}$, with $\\beta=[2(3-\\sqrt 5)\/\n(\\sqrt13 -1)]$. At the critical point the string order-parameter decays\nalgebraically, $O^z(r) \\sim r^{-\\eta^{st}}$, with $\\eta^{st}=2\\beta\/\\nu$.\nThe end-to-end correlation function goes to zero algebraically, too,\n$C_1(L) \\sim L^{-\\eta_1}$, similarly to the\nbulk spin-spin correlation function, $C_b(r) \\sim r^{-\\eta}$. Here, however, there are no theoretical conjectures\nabout the exponents $\\eta_1$ and $\\eta$.\nThe relation at the critical point between the correlation length and the relaxation time is strongly\nanisotropic,\n\\begin{equation}\n\\ln \\, t_r \\sim \\xi^{\\psi}\\;,\n\\label{psi}\n\\end{equation}\nwith $\\psi=1\/3$; thus, the dynamical exponent, $z$, is formally infinity. This type of infinite disorder\nscaling is seen in the distribution of the gaps, which is given by\n\\begin{equation}\nP_L(\\Delta){\\rm d} \\Delta=\nL^{-\\psi}\\tilde{P}\\left[\\frac{\\ln \\, {\\Delta}}{L^\\psi}\\right] {\\rm d} \\, \\ln \\, \\Delta\\;.\n\\label{eq:scaling-psi}\n\\end{equation}\nIn the space of variables, dimerization ($D$) and disorder ($\\delta$),\nthe critical point of the system represents a multicritical point in which three\nGriffiths phases with different symmetry meet.\\cite{Daml02} The corresponding\nexponents follow by permutation symmetry and the calculation can be generalized\nfor higher values of $S$.\\cite{Daml02}\n\n\\subsubsection{Random singlet phase}\n\nFor a disorder $\\delta>\\delta_C$ ($W>W_C$), the low-energy behavior of\nthe system is controlled by an infinite disorder fixed point and the\nsystem is in the RS phase. The RS phase is a critical phase, both\n$\\xi$ and $t_r$ are divergent, and its properties are assumed to be\nidentical to the RS phase of the random $S=1\/2$ chain. This latter\nsystem is studied in great detail by Fisher\\cite{fisherxx} with the\nasymptotically exact strong disorder RG method and these results have\nbeen confronted with detailed numerical\ninvestigations.\\cite{henelius,ijr00,stolze} Here we repeat that\nin the RS phase there is infinite disorder scaling, so that relations in\nEqs. (\\ref{psi}) and (\\ref{eq:scaling-psi}) are valid with an exponent,\n$\\psi=1\/2$.  The RS phase is instable against enforced dimerization,\nas given in Eq. (\\ref{J_S}), and the correlation length behaves as\n$\\xi(D) \\sim D^{-\\nu}$, with $\\nu=2$. In the RS phase the bulk and\nend-to-end correlation functions decay algebraically.  In Table\n\\ref{table1} we collected the conjectured values of the critical\nexponents both in the random singlet phase and at the critical point and\ncompared these values with the estimates obtained in this paper.\n\n\\begin{table}\n\\caption{Theoretical predictions for the critical exponents in the random singlet (RS) phase and at\nthe critical point (CP). Values obtained in this paper are given in square brackets.\n\\label{table1}}\n \\begin{tabular}{|c|c|c|c|c|c|c|}  \\hline\n   & $\\eta^{st}$ & $\\beta$  & $\\nu$     & $\\psi$   & $~\\eta~$ & $~\\eta_1~$\\\\ \\hline\nRS & $0.382$[0.41(4)] & - &      2     &     1\/2 [0.45(5)]    &    2  &  1 [0.86(6)]   \\\\\nCP & $0.509[0.39(3)]$ & $0.586$  & $2.30$     &    1\/3 [0.35(5)]  & - & - [0.69(5)]     \\\\  \\hline\n\n  \\end{tabular}\n  \\end{table}\n\n\\subsection{Summary of the existing numerical results}\n\nIn previous numerical studies the box distribution in Eq. (\\ref{box})\nhas been used. In Table \\ref{table2} we present the estimates of the\nborders of the different phases obtained by different numerical\nmethods, such as by numerical implementation of the strong disorder\nRG, by DMRG, and by QMC. We note that for the power-law disorder in\nEq. (\\ref{distribut}) the critical disorder is estimated\\cite{comment}\nas $\\delta_C \\approx 1.5$. Using variants of the strong disorder RG\nmethod,\\cite{monthus,saguia02} the calculated critical exponents in the RS phase--within\nnumerical precision--correspond to the predicted, analytical\nvalues. To reach the asymptotic region,\nhowever, one often needs to treat very long chains of length\n$L=10^4-10^6$, see Ref. \\onlinecite{monthus}. The DMRG and QMC investigations have led\nto different conclusions for the strongest box disorder, with $W=2$.\nNo RS phase is found by DMRG, \\cite{Hida} whereas by QMC infinite\ndisorder scaling is detected.\\cite{todo,bergkvist} The average string correlation function\nwas shown to decay algebraically with\\cite{bergkvist} $\\eta_{st}=0.378(6)$, close to\nthe theoretical result in Table \\ref{table1}. On the other hand, the\naverage spin-spin correlation function was found to have an exponent,\\cite{bergkvist}\n$\\eta=1$, which greatly differs from the theoretical value of\n$\\eta=2$.\n\n\\begin{table}\n\\caption{Numerical estimates for the borders of the different phases\nof the random $S=1$ chain with boxlike disorder, see Eq. (\\ref{box}). The different phases are\ndefined by\n$W<W_G$, Haldane phase; $W_G < W < W_1$, GH phase with nondivergent\nlocal susceptibility; $W_1 < W < W_C$, GH phase with divergent\nlocal susceptibility; and $W>W_C$, RS phase.\n\\label{table2}}\n \\begin{tabular}{|c|c|c|c|}  \\hline\n   & $W_G$   & $W_1$     & $W_C$  \\\\ \\hline\nRG[\\onlinecite{monthus}] & & & 1.48   \\\\\\hline\nRG[\\onlinecite{saguia02}] & &0.76 & 2.   \\\\\\hline\nDMRG[\\onlinecite{Hida}] & &1.8 & no   \\\\\\hline\nQMC[\\onlinecite{todo}] & 1.37 & 1.7 & 1.8   \\\\\\hline\nQMC[\\onlinecite{bergkvist}] &  &  & $<2$   \\\\  \\hline\n\n  \\end{tabular}\n  \\end{table}\n\n\\section{Numerical investigations}\n\\label{Sec:3}\n\\subsection{The DMRG method}\n\nMost of our numerical results are based on DMRG calculations. In this\ncase we used open chains up to length $L=64$, for weak disorder and up to\n$L=32$ for strong disorder and calculated the\nlowest two gaps, the end-to-end correlation function and the string\norder parameter. This latter quantity is calculated for open chains between\npoints $i=L\/4$ and $j=3L\/4$. Note that for an open chain the first gap is related to\nthe surface degrees of freedom and goes to zero exponentially with $L$. The\ncharacteristic bulk excitations are given by the second gap and we studied\nthis quantity. In the numerical calculation we have retained up to $m=180$\nstates in the DMRG and checked that convergence of the numerical results is reached. We used\nthe power-law distribution of disorder in Eq. \\!\\!(\\ref{distribut}) in the canonical ensemble, i.e., there was\nno constraint to the value of the sum of the odd and even couplings. In this way there is a\nnonzero residual dimerization, which could be the source of some error for small systems.\nHowever, using the microcanonical ensemble, in which the sum of the odd couplings is the same\nas that of the even couplings, could leave to different finite-size exponents for the end-to-end\ncorrelation function, which is known for the random transverse-field Ising chain.\\cite{bigpaper,young,cecile}\nWe have calculated typically 10 000 independent disorder realizations in each case.\n\n\n\\subsection{Gapless Haldane phase}\n\n\\subsubsection{Nonsingular region: $z<1$ }\n\nWe have calculated the distribution of the (second) gap and determined\nits inverse average, $\\Delta^{\\rm iv}$,\nwhich is presented in Fig. \\ref{fig:avgap} as a function of the inverse size, $L^{-1}$, for\ndifferent values of $\\delta$. The limiting value as $L \\to \\infty$ is monotonously decreasing\nwith $\\delta$ and $\\Delta^{\\rm iv}$ seems to approach zero at a limiting disorder $\\delta_1\\approx 0.45-0.5$.\nSimilar conclusion is obtained from the behavior of the inverse average of the end-to-end\ncorrelation function, $C^{\\rm iv}_1(L)$, which is shown in the inset of Fig. \\ref{fig:avgap}.\n\n\n\\begin{figure}\n\\includegraphics [width=0.7 \\linewidth]{fig1.eps}\n\\caption{The inverse average gap, $\\Delta^{\\rm iv}$, as a function of the\ninverse size of the system for different strengths of\ndisorder, $\\delta=0.1, 0.2, 0.3, 0.4$, and $0.5$ from the top to the bottom, respectively. $\\Delta^{\\rm iv}$\nseems to vanish around $\\delta \\approx 0.5$. Inset: the inverse average end-to-end\ncorrelation function as a function of\ninverse size, with the same values of disorder, as in the main panel. Note that for weak disorder\nthe size dependence of $C_1^{\\rm iv}$ is nonmonotonic, which is due to a finite correlation length\nin the system.\n\\label{fig:avgap}\n}\n\\end{figure}\n\nNote, however, that the average end-to-end correlation function, as shown in\nFig. \\ref{fig:CP} is finite\nat $\\delta_1$.\nThe extrapolated values of $\\Delta^{\\rm iv}$ and $C^{\\rm iv}_1(L)$ are shown\nin Fig. \\ref{fig:extrG}. Close to $\\delta_1$, both curves are compatible with an approximately linear\nvariation with $\\delta_1-\\delta$. At the boundary point, $\\delta_1=\\delta$, the size dependences\nof $\\Delta^{\\rm iv}$ and $C^{\\rm iv}_1(L)$ are shown in the inset of Fig. \\ref{fig:extrG}. Both are\nlinear in $L^{-1}$, in accordance with the criterion that at $\\delta_1$\n the disorder induced dynamical exponent is $z(\\delta_1) = 1$.\n\n\n\\begin{figure}\n\\includegraphics [width=0.7 \\linewidth]{fig2.eps}\n\\caption{Extrapolated values of the inverse average gap and the inverse average\n  end-to-end correlation function as a function of the strength of\n  disorder. At $\\delta=0$ we obtain estimates for the nonrandom\n  model, $\\Delta=0.4105(3)$ and $C_1=0.283(1)$. Inset: Size dependence of the\n  inverse average gap and the inverse average end-to-end correlation\n  function in a log-log plot at the boundary of the gapless Haldane\n  phase. Both lines have an approximate slope, $z=1$, denoted by broken lines.\nThe typical value of the error is indicated; otherwise, the error is smaller than\nthe size of the symbol.\n\\label{fig:extrG}\n}\n\\end{figure}\n\n\\subsubsection{Singular region: $z>1$ }\n\nWe have calculated the average string order parameter and the average\nend-to-end correlation function for different sizes $L$. Also we have\ndetermined the disorder induced dynamical exponent, $z$, which is\ndeduced from the low-energy tail of the gap distribution [see\nEq. (\\ref{eq:scaling-conv})]. The extrapolated values of $O^z$ and\n$C_1$, as well as $1\/z$ are plotted in Fig. \\ref{fig:CP} for different\nstrengths of disorder. All these three quantities tend to zero around the\nsame limiting value of disorder and\nthe border of the Griffiths phase, i.e., the location\nof the critical point can be determined as $\\delta_c=1.0(1)$.\nIn the extrapolation procedure we have made use of the finite-size dependence\nof $O^z(L) \\sim L^{-\\eta_{st}}$ and $C_1(L)\\sim L^{-\\eta_{1}}$ at the critical point,\nwhich is shown in the inset of Fig. \\ref{fig:CP}. For weaker disorder, $\\delta<\\delta_c$,\n$O^z$ tends to a finite limiting value, which is illustrated in Fig. \\ref{fig:GH} using a scale\n$L^{-\\eta_{st}}$. A similar conclusion is obtained for the average end-to-end correlation\nfunction, which is presented in the inset of Fig. \\ref{fig:GH}.\n\n\\begin{figure}\n\\includegraphics [width=0.7 \\linewidth]{fig3.eps}\n\\caption{The average string order parameter, $O^z$, the average end-to-end\n  correlation function, $C_1(L)$ and the disorder induced dynamical\n  exponent, $1\/z$, as a function of the strength of disorder. Inset:\n  finite-size dependence of the string order parameter and the average\n  end-to-end correlation function at the critical point ($\\delta_c=1$,\n  open symbols) and in the RS phase ($\\delta_c=1.5$, full symbols) in\n  a log-log plot. The slope of the broken lines representing the\n  critical exponents are $\\circ, ~0.39 \\pm0.03$[0.509];\n  $\\bullet, ~0.41\\pm0.04[0.382]$; $\\lozenge, ~0.69\\pm0.05$; and\n  $\\blacklozenge, ~0.86\\pm0.06[1.0]$ where in the brackets we presented\n  the theoretical RG results; see Table \\ref{table1}. Typical values\n  of the error are indicated; otherwise, the error is smaller than the\n  size of the symbol.\n\\label{fig:CP}\n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics [width=0.7 \\linewidth]{fig4.eps}\n\\caption{The average string orderparameter as a function of $L^{-\\eta_{st}}$,\nwith $\\eta_{st}=0.39$ as obtained in the inset of Fig. \\ref{fig:CP},\nfor disorder, $\\delta=0.3, \\, 0.5, \\,  0.8, \\, 1.0,$ and $1.5$ from the top to the\n  bottom.  Inset: the average end-to-end correlation function as a function of $L^{-\\eta_1}$,\n$\\eta_1=0.69$ being the critical decay exponent, for the same values of disorder as in the main panel. \nSolid straight lines over the $\\delta=1.0$ points are guide to the eyes.\n\\label{fig:GH}\n}\n\\end{figure}\n\n\\subsection{Critical point and the RS phase}\n\nOur aim with the numerical investigations in this subsection is twofold: first, to\ncheck the properties of the RS phase, thus to present numerical evidences, and second,\nand this is numerically more demanding, to try to discriminate between the properties in\nRS phase and at the critical point. We start to analyze the finite-size dependence\nof the average string order parameter and that of the\naverage end-to-end correlation function, which is shown in the inset of Fig. \\ref{fig:CP}\nat two values of the disorder, $\\delta=1$ and $\\delta=1.5$. The first value should be\nclose to the critical point (see Fig. \\ref{fig:CP}); however, there is certainl;y some uncertainty,\nsee the values of $W_G$ in Table \\ref{table2}. The second value of disorder, $\\delta=1.5$,\nshould be deeply in the RS phase; however, see the RG estimates in Ref. \\onlinecite{comment}.\n\nAt $\\delta=1.5$ the decay of the average\nstring order parameter, as well as that of the end-to-end correlation function is\nalgebraic, and the decay exponents of both quantities are in agreement with the theoretical\nprediction in the RS phase, as given in Table \\ref{table1}. For the average end-to-end correlation\nfunction we have somewhat less accuracy, which could be due to similar crossover effects as\nnoted for the bulk spin-spin correlation function in Ref. \\onlinecite{bergkvist}.\nThe same type of analysis of the results at $\\delta=1$ give somewhat\ndifferent results. The decay of\nthe average end-to-end correlation function is characterized by an exponent,\n$\\eta_1=0.69$, which differs considerably from the value in the RS phase. On the\nother hand, the decay exponent of the average string order parameter, within the error of the\ncalculation, agrees with the value found in the RS phase. We note that\nat the same disorder in the QMC simulation\\cite{bergkvist} also the exponent in\nthe RS phase is observed. One possible explanation is that $\\delta=1$ is already in the RS phase\nand therefore we find the corresponding exponent. Anyway, even at the critical point one expects strong\ncrossover effects due to the vicinity of the RS fixed point, so that probably much larger\nsystems are needed to observe the true asymptotic behavior.\n\nFinally, we compare in Fig. \\ref{fig:SC}\nthe distribution of the gaps at the critical point (a) and in the RS phase (b). For\nboth cases the distribution is broadened with increasing $L$, which is a clear signal of infinite\ndisorder scaling. Indeed, one can obtain a good scaling collapse using the form\nin Eq. \\!(\\ref{eq:scaling-psi}). In the insets we have illustrated this type of behavior\nby using the theoretical predictions, $\\psi=1\/3$ at the\ncritical point and $\\psi=1\/2$ in the RS phase, respectively. The estimated exponents obtained from\nthe optimal scaling collapse are shown in Table \\ref{table1}.\n\n\\begin{figure}\n\\includegraphics [width=0.7 \\linewidth]{fig5.eps}\n\\vspace{5mm}\n\\caption{(Color online) Distribution of the gaps in finite systems at the critical point, $\\delta=1$ (a);\nand in the RS phase, $\\delta=1.4$ (b). In the insets scaling collapse with\nEq. (\\ref{eq:scaling-psi})\nis shown with $\\psi=1\/3$ (a) and $\\psi=1\/2$ (b).\n\\label{fig:SC}\n}\n\\end{figure}\n\n\\section{Conclusion}\n\\label{Sec:4}\n\nThe random antiferromagnetic $S=1$ chain is a paradigm of disorder\ninduced phase transition phenomena (see also Ref. \\onlinecite{Carl01}) for\nwhich detailed strong disorder RG predictions are available. These\npredictions, however, have only been partially verified by numerical\ncalculations and even the numerical results are somewhat conflicting.\nIn this paper we have used extensive DMRG calculations with the aim to\nclarify the low-energy properties of the system with varying strengths\nof disorder. The sizes of the systems we used in the calculation are\ncomparable with those in previous DMRG studies; \\cite{Hida} however, we\nused a power-law distribution of the couplings in\nEq. \\!(\\ref{distribut}), which can be more random, than the box\ndistribution in Eq. \\!(\\ref{box}) used previously. We have also\ncalculated a quantity, the end-to-end spin-spin correlation\nfunction, which can be used to locate the borders of the different\nphases in the system and to obtain an independent estimate of the\ndynamical exponent. Our calculations gave further numerical support of\nthe phase diagram predicted by the strong disorder RG method and our\nresults are basically in agreement with the scenario of disorder\ninduced phase transitions. In the RS phase we made calculations far\nfrom the critical point, which is not possible with boxlike\ndistribution of couplings as given in Eq. (\\ref{box}) and obtained\nestimates for the critical exponents which are compatible with the RG\npredictions. Our results at the critical point are less conclusive,\nwhich is probably due to crossover effects and the inaccurate\nlocation of this point. For the critical exponent of the end-to-end\ncorrelation function, $\\eta_1$, and that of the gap scaling, $\\psi$,\nnumerical estimates at the critical point are clearly different from\nthat in the RS phase, which are in accordance with the RG results. On\nthe other hand, for the average string correlation function our\nnumerical results are in conflict with the RG prediction. We believe\nthat at this point much larger finite systems are necessary to obtain\na precise numerical estimate and thus to be able to test the results\nof RG predictions.\n\nWe close our paper by mentioning that the present day numerical possibilities to\nexplore the properties of the random $S=1$ chain seem to be exhausted, as far as\nDMRG or QMC methods are considered. Some independent and probably more accurate\nresults can be expected, however, by the numerical application of different variants\nof the strong disorder RG method, in particular in the vicinity of the\ncritical point and in the RS phase. Results obtained in this direction will\nbe published in the future.\\cite{Peter}\n\n\\centerline{\\bf ACKNOWLEDGMENT}\nThis work has been supported by Kuwait University Research Grant No. [SP 09\/02].\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nSupersymmetric quantum mechanics (SUSY QM) has constituted an active\nresearch field in theoretical sciences over two decades. Although\nthe original motivation for studying SUSY QM was to unveil the\nmechanisms of its dynamical breaking in quantum field\ntheories~\\cite{Wi81}, it turned out that SUSY QM, as the minimum\nbuilding block of SUSY, contains various relevant concepts\nwhich provide convenient platforms to uncover many useful properties\nof quantum mechanics~\\cite{CKS95,Ju96,Ba00}. In particular, it is\nconsistent with factorization schemes~\\cite{Sc40a} and intertwining\nrelationships~\\cite{Da1882} thereby providing a powerful tool to\nconstruct solvable Schr\\\"{o}dinger equations. Furthermore, interesting\nextensions to higher-order SUSY schemes were carried out by taking\nrecourse to higher-derivative versions of the factorization\noperators~\\cite{AIS93,AST01b,AS03}.\n\nThe original extension in Ref.~\\cite{AIS93} was considered by using\nhigher-order intertwining operators which are expressed as products\nof first-order linear differential operators and then applying\nthe ordinary SUSY results. Later in Refs.~\\cite{AICD95,AIN95a},\nthrough the analysis of the general second-order case, the concept\nof \\emph{reducibility} was introduced. In this regard,\na higher-order intertwining operator is said to be \\emph{reducible}\nif it is factorized into a product of first-order differential\noperators such that with respect to each factor there exists\nan intermediate \\emph{real} Hamiltonian satisfying a (shifted) SUSY\nrelation. Otherwise it is called \\emph{irreducible}. This concept,\nhowever, seems less useful in view of the current status where\nnon-Hermitian quantum theories have been investigated intensively\nsince the discovery of $\\mathcal{P}\\mathcal{T}$ symmetry~\\cite{BB98a}. In fact,\nthe SUSY method turned to be useful also in constructing\na \\emph{complex} potential with real\nspectrum~\\cite{CJT98,BR00,ZCBR00}.\n\nIn addition to the usefulness of the reality constraint, there\narises a natural question about the well-definiteness of the concept\nif we take into account the fact that in general factorization of\nhigher-order linear differential operators is not unique. The latter\nfact indeed has been reported in the context of the factorization\nmethod and ordinary SUSY QM, that is, there exist several\nSchr\\\"{o}dinger operators that admit different\nfactorizations~\\cite{Mi84,Fe84,Zh87,Ku87,AF88,Fi88,MRLB89}, see\nalso Ref.~\\cite{Qu08b} for a recent approach.\nHence, a reducible higher-order intertwining operator may admit\nsimultaneously another factorization for which there are no\nintermediate real Hamiltonians.\n\nOn the other hand, the non-uniqueness in factorizing intertwining\noperators of arbitrary finite orders was, to the best of our\nknowledge, first reported in Ref.~\\cite{AST01a} for the well-known\nquasi-solvable sextic anharmonic oscillator potentials in the framework\nof type A $\\mathcal{N}$-fold supersymmetry (see, Eq.~(47) in the latter\nreference). Later it was shown that the non-uniqueness of\nfactorizations in type A $\\mathcal{N}$-fold SUSY is a consequence of\nthe underlying $GL(2,\\mathbb{C})$ symmetry~\\cite{Ta03a}.\n\nIn a recent communication~\\cite{BQR08}, the following two-parameter\nfamily of second-order supersymmetric (SSUSY) system $(h^{(1)},h^{(2)},\n\\hat{A}\\hat{B})$, characterized by the two parameters $A$ and $B$,\nsatisfying $\\hat{A}\\hat{B}h^{(1)}=h^{(2)}\\hat{A}\\hat{B}$ was constructed:\n\\begin{align}\nh^{(1)}&=\\hat{B}^{\\dagger}\\hat{B}+\\frac{\\bar{c}}{2}\n =-\\frac{\\rmd^{2}}{\\rmd x^{2}}+V_{A,B}(x)-\\tilde{E}+\\frac{\\bar{c}}{2},\n\\label{eq:BQRs}\\\\\nh^{(2)}&=\\hat{A}\\hat{A}^{\\dagger}-\\frac{\\bar{c}}{2}\n =-\\frac{\\rmd^{2}}{\\rmd x^{2}}+V_{A,B,\\text{ext}}(x)-E\n -\\frac{\\bar{c}}{2},\n\\end{align}\nwhere\n\\begin{align}\nV_{A,B}(x)&=[B^{2}+A(A+1)]\\mathrm{csch\\,}^{2}x-B(2A+1)\\mathrm{csch\\,} x\\coth x,\\\\\nV_{A,B,\\textrm{ext}}(x)&=V_{A,B}(x)+\\frac{2(2A+1)}{2B\\cosh x-2A-1}\n -\\frac{2[4B^{2}-(2A+1)^{2}]}{(2B\\cosh x-2A-1)^{2}},\n\\end{align}\nand the constants $E$, $\\tilde{E}$, and $\\bar{c}$ are given by\n\\begin{align}\nE=-\\left(B\\mp\\frac{1}{2}\\right)^{2},\\quad\n \\tilde{E}=-\\left(B\\pm\\frac{1}{2}\\right)^{2},\\quad\\bar{c}=\\mp2B.\n\\end{align}\nThe intertwining operators $\\hat{A}$ and $\\hat{B}$ are respectively\ngiven by\n\\begin{align}\n\\hat{A}&=\\frac{\\rmd}{\\rmd x}\\pm\\left(B\\pm\\frac{1}{2}\\right)\\coth x\n \\mp\\left(A+\\frac{1}{2}\\right)\\mathrm{csch\\,} x-\\frac{2B\\sinh x}{2B\\cosh x-2A-1},\\\\\n\\hat{B}&=\\frac{\\rmd}{\\rmd x}\\mp\\left(B\\pm\\frac{1}{2}\\right)\\coth x\n \\pm\\left(A+\\frac{1}{2}\\right)\\mathrm{csch\\,} x.\n\\end{align}\nIt was further shown in Ref.~\\cite{BQR08} that the system admits the\nintermediate Hamiltonians $h$ given by\n\\begin{align}\nh&=\\hat{A}^{\\dagger}\\hat{A}-\\frac{\\bar{c}}{2}\n =-\\frac{\\rmd^{2}}{\\rmd x^{2}}+V_{A,B\\pm1}(x)\n -E-\\frac{\\bar{c}}{2}\\notag\\\\\n&=\\hat{B}\\hat{B}^{\\dagger}+\\frac{\\bar{c}}{2}\n =-\\frac{\\rmd^{2}}{\\rmd x^{2}}+V_{A,B\\pm1}(x)\n -\\tilde{E}+\\frac{\\bar{c}}{2},\n\\end{align}\nwith\n\\begin{align}\nV_{A,B\\pm1}(x)=[(B\\pm1)^{2}+A(A+1)]\\mathrm{csch\\,}^{2}x\n -(B\\pm1)(2A+1)\\mathrm{csch\\,} x\\coth x,\n\\label{eq:BQRe}\n\\end{align}\nwhich satisfy the ordinary SUSY relations $\\hat{A}h=h^{(2)}\\hat{A}$ and\n$\\hat{B}h^{(1)}=h\\hat{B}$. We shall hereafter call the system\n(\\ref{eq:BQRs})--(\\ref{eq:BQRe}) the \\emph{BQR SSUSY model}.\n\nTaking into account the fact that type A 2-fold SUSY is the necessary\nand sufficient condition for the existence of (at least) two linearly\nindependent analytic (local) solutions to Schr\\\"{o}dinger equation of\none degree of freedom~\\cite{GT06}, we immediately know that the above\nBQR SSUSY model, which is exactly solvable, also belongs to type A\n2-fold SUSY. Therefore, it would be natural that it has (at least)\ntwo different factorizations of the second-order intertwining operator\nin view of the aforementioned $GL(2,\\mathbb{C})$ symmetry and has two\ndifferent intermediate Hamiltonians correspondingly.\n\nRegarding the existence of intermediate Hamiltonians, we note\nthe following fact shown in Ref.~\\cite{AST01a} that\nthe most general form of type A $\\mathcal{N}$-fold SUSY quantum systems\nconstructed directly from the $\\mathcal{N}$th-order intertwining operators\nof type A, namely, type A $\\mathcal{N}$-fold supercharge\n$P_{\\mathcal{N}}^{-}=P_{\\mathcal{N} 1}^{-}\\dots P_{\\mathcal{N}\\cN}^{-}$ (cf.,\nEq.~(\\ref{eq:ANfch})) by solving $P_{\\mathcal{N}}^{-}H^{-}=H^{+}P_{\\mathcal{N}}^{-}$\nis more general than that of the systems constructed from the $\\mathcal{N}$\nrepeated applications of the first-order intertwining operators\n$P_{\\mathcal{N} k}^{-}$ by solving $P_{\\mathcal{N} k}^{-}H^{(k-1)}=H^{(k)}P_{\\mathcal{N} k}^{-}$\n($k=1,\\dots,\\mathcal{N}$) with the identification $H^{-}=H^{(0)}$ and\n$H^{+}=H^{(\\mathcal{N})}$. It is apparent that in the latter construction we\nautomatically obtain a series of the intermediate Hamiltonians\n$H^{(1)},\\dots,H^{(\\mathcal{N}-1)}$ in addition to the $\\mathcal{N}$-fold SUSY pair\nHamiltonians $H^{\\pm}$ at the cost of the generality. In the former\nconstruction, on the other hand, the existence of intermediate\nHamiltonians is not guaranteed in general. Hence, the fact that the\ntype A $\\mathcal{N}$-fold supercharge has the factorized form by definition\ndoes not necessarily imply the existence of intermediate Hamiltonians. \nIn particular, the fact that type A $\\mathcal{N}$-fold supercharge admits\ndifferent factorizations does not automatically mean that the most\ngeneral type A $\\mathcal{N}$-fold SUSY system has different sets of\nintermediate Hamiltonians accordingly.\n\nMotivated by the backgrounds described above, we investigate in this\narticle under what conditions type A $\\mathcal{N}$-fold SUSY systems admit\nintermediate Hamiltonians in the case of $\\mathcal{N}=2$. Furthermore, we\nalso examine under the satisfaction of the conditions how many sets\nof such Hamiltonians are admissible for the type A 2-fold SUSY\nsystems. In addition, we show that any such a system has another\ntype of nonlinear supersymmetries, namely, parasupersymmetry\nof order 2~\\cite{RS88}.\n\nWe organize the article as follows. In the next section, we review\nthe framework of type A $\\mathcal{N}$-fold SUSY by putting emphasis on the\n$GL(2,\\mathbb{C})$ symmetry. In Section~\\ref{sec:inHam}, we investigate\nin details under what conditions a type A 2-fold SUSY quantum system\nhas one or more intermediate Hamiltonians. In particular, we show that\nthe maximum number of different intermediate Hamiltonians in type A\n2-fold SUSY is two. In Section~\\ref{sec:psusy}, we further show that\nwhen a type A 2-fold SUSY system admits (at least) one intermediate\nHamiltonian, the system can have second-order parasupersymmetry.\nA novel generalization of 2-fold superalgebra is discussed briefly.\nAs an application of the results, we construct in\nSection~\\ref{sec:appli} a type A 2-fold SUSY system with two\ndifferent intermediate Hamiltonians of generalized\nP\\\"{o}schl--Teller type which includes the BQR SSUSY model as\na particular case. Then, we close the article with discussion and\nperspectives of further developments in the last section.\n\n\n\\section{Type A $\\mathcal{N}$-fold Supersymmetry and $GL(2,\\mathbb{C})$ Covariance}\n\\label{sec:typeA}\n\nRoughly speaking, type A $\\mathcal{N}$-fold SUSY quantum systems are composed\nof a pair of scalar Hamiltonians $H^{\\pm}$ and an $\\mathcal{N}$th-order\nlinear differential operator $P_{\\mathcal{N}}^{-}$ of the following\nforms:\\footnote{We keep the original notations as far as possible.\nThus, note that the function $E(x)$ is different from the constant\n$E$ in the BQR SSUSY model (\\ref{eq:BQRs})--(\\ref{eq:BQRe}).\nSimilarly, the function $A(z)$ introduced later in (\\ref{eq:gHams})\nis different from the parameter $A$ in the latter model.}\n\\begin{gather}\nH^{\\pm}=-\\frac{1}{2}\\,\\frac{\\rmd^{2}}{\\rmd x^{2}}+\\frac{1}{2}W(x)^{2}\n -\\frac{\\mathcal{N}^{\\,2}-1}{24}\\left(2E'(x)-E(x)^{2}\\right)\\pm\\frac{\\mathcal{N}}{2}\n W'(x)-R,\n\\label{eq:AHams}\\\\\nP_{\\mathcal{N}}^{-}=\\prod_{k=0}^{\\mathcal{N}-1}\\left(\\frac{\\rmd}{\\rmd x}+W(x)+\\frac{\n \\mathcal{N}-1-2k}{2}E(x)\\right),\n\\label{eq:ANfch}\n\\end{gather}\nwhere $R$ is a constant while $E(x)$ and $W(x)$ are analytic functions\nsatisfying\n\\begin{align}\n\\left(\\frac{\\rmd}{\\rmd x}-E(x)\\right)\\frac{\\rmd}{\\rmd x}\\left(\n \\frac{\\rmd}{\\rmd x}+E(x)\\right)W(x)=0\\quad\\text{for}\\quad\\mathcal{N}\\geq2,\n\\label{eq:condW}\\\\\n\\left(\\frac{\\rmd}{\\rmd x}-2E(x)\\right)\\left(\\frac{\\rmd}{\\rmd x}-E(x)\n \\right)\\frac{\\rmd}{\\rmd x}\\left(\\frac{\\rmd}{\\rmd x}+E(x)\\right)\n E(x)=0\\quad\\text{for}\\quad\\mathcal{N}\\geq3.\n\\label{eq:condE}\n\\end{align}\nThe product of operators appeared in (\\ref{eq:ANfch}) is defined by\n\\begin{align}\n\\prod_{k=0}^{n}A_{k}\\equiv A_{n}\\dots A_{1}A_{0}.\n\\end{align}\nThe operators $H^{\\pm}$ and $P_{\\mathcal{N}}^{-}$ satisfy an intertwining\nrelation\n\\begin{align}\nP_{\\mathcal{N}}^{-}H^{-}=H^{+}P_{\\mathcal{N}}^{-}.\n\\label{eq:inter}\n\\end{align}\nOne of the most important features of type A $\\mathcal{N}$-fold SUSY quantum\nsystems is that the gauged Hamiltonians $\\tilde{H}^{-}$ and $\\bar{H}^{+}$\nintroduced by\n\\begin{align}\n\\bar{\\tilde{H}}^{\\pm}=\\rme^{\\mathcal{W}_{\\mathcal{N}}^{\\pm}}H^{\\pm}\\rme^{-\\mathcal{W}_{\\mathcal{N}}^{\\pm}},\n\\qquad\\mathcal{W}_{\\mathcal{N}}^{\\pm}(x)=\\frac{\\mathcal{N}-1}{2}\\int\\rmd x\\,E(x)\\mp\\int\\rmd x\\,\n W(x),\n\\end{align}\npreserve the so-called type A monomial space $\\tilde{\\cV}_{\\mathcal{N}}^{(\\mathrm{A})}$:\n\\begin{align}\n\\bar{\\tilde{H}}^{\\pm}\\tilde{\\cV}_{\\mathcal{N}}^{(\\mathrm{A})}\\subset\\tilde{\\cV}_{\\mathcal{N}}^{(\\mathrm{A})},\\qquad\n \\tilde{\\cV}_{\\mathcal{N}}^{(\\mathrm{A})}=\\bigl\\langle 1,z(x),\\dots,z(x)^{\\mathcal{N}-1}\\bigr\\rangle,\n\\label{eq:Amono}\n\\end{align}\nwhere the new variable $z(x)$ satisfies\n\\begin{align}\nz''(x)=E(x)z'(x).\n\\label{eq:defE}\n\\end{align}\nExplicitly, they are given by\n\\begin{align}\n\\bar{\\tilde{H}}^{\\pm}=&-A(z)\\frac{\\rmd^{2}}{\\rmd z^{2}}+\\left[\\frac{\\mathcal{N}-2}{2}\n A'(z)\\pm Q(z)\\right]\\frac{\\rmd}{\\rmd z}\\notag\\\\\n&-\\left[\\frac{(\\mathcal{N}-1)(\\mathcal{N}-2)}{12}A''(z)\\pm\\frac{\\mathcal{N}-1}{2}Q'(z)+R\\right],\n\\label{eq:gHams}\n\\end{align}\nwhere the new functions $A(z)$ and $Q(z)$ are defined by\n\\begin{align}\n2A(z)=z'(x)^{2},\\qquad Q(z)=-z'(x)W(x).\n\\label{eq:defAQ}\n\\end{align}\nThe conditions (\\ref{eq:condW}) and (\\ref{eq:condE}) for type A $\\mathcal{N}$-fold\nSUSY are reduced to the following simple forms in terms of $z$:\n\\begin{align}\n\\frac{\\rmd^{3}}{\\rmd z^{3}}Q(z)=0\\quad\\text{for}\\quad\\mathcal{N}\\geq2,\n\\label{eq:condQ}\\\\\n\\frac{\\rmd^{5}}{\\rmd z^{5}}A(z)=0\\quad\\text{for}\\quad\\mathcal{N}\\geq3.\n\\label{eq:condA}\n\\end{align}\nIn particular, the condition (\\ref{eq:condQ}) indicates that $Q(z)$ is\na polynomial of at most second-degree in $z$ for all $\\mathcal{N}\\geq2$:\n\\begin{align}\nQ(z)=b_{2}z^{2}+b_{1}z+b_{0}.\n\\label{eq:Qofz}\n\\end{align}\nIn terms of $z(x)$, the potential terms $V^{\\pm}(x)$ of type A $\\mathcal{N}$-fold\nSUSY Hamiltonians in (\\ref{eq:AHams}) are expressed as\n\\begin{multline}\nV^{\\pm}(x)=-\\frac{1}{12A(z)}\\biggl[(\\mathcal{N}^{\\,2}-1)\\left(A(z)A''(z)\n -\\frac{3}{4}A'(z)^{2}\\right)-3Q(z)^{2}\\\\\n\\mp 3\\mathcal{N}\\bigl(A'(z)Q(z)-2A(z)Q'(z)\\bigr)\\biggr]-R\\biggr|_{z=z(x)}.\n\\label{eq:VAQ}\n\\end{multline}\nThe type A $\\mathcal{N}$-fold SUSY systems (\\ref{eq:AHams}) and (\\ref{eq:ANfch})\nhave an underlying symmetry which, as we shall show, plays a central\nrole in investigating the existence of intermediate Hamiltonians. It is\n$GL(2,\\mathbb{C})$ linear fractional transformations on the variable $z$\nintroduced as\n\\begin{align}\nz=\\frac{\\alpha w+\\beta}{\\gamma w+\\delta}\\quad(\\alpha,\\beta,\n \\gamma,\\delta\\in\\mathbb{C},\\ \\Delta\\equiv\\alpha\\delta-\\beta\\gamma\\neq0).\n\\label{eq:GL2}\n\\end{align}\nThen, the type A monomial space is invariant under the $GL(2,\\mathbb{C})$\ntransformations induced by (\\ref{eq:GL2}):\n\\begin{align}\n\\tilde{\\cV}_{\\mathcal{N}}^{(\\mathrm{A})}[z]\\mapsto\\widehat{\\tilde{\\cV}}{}_{\\mathcal{N}}^{(\\mathrm{A})}[w]=\n (\\gamma w+\\delta)^{\\mathcal{N}-1}\\tilde{\\cV}_{\\mathcal{N}}^{(\\mathrm{A})}[z]\\Bigr|_{z=\\frac{\n \\alpha w+\\beta}{\\gamma w+\\delta}}=\\tilde{\\cV}_{\\mathcal{N}}^{(\\mathrm{A})}[w].\n\\end{align}\nThe gauged Hamiltonians $\\tilde{H}^{-}$ and $\\bar{H}^{+}$ are both such linear\ndifferential operators that preserve the type A monomial space, as was\nshown in (\\ref{eq:Amono}). As a consequence, they are covariant under\nthe following $GL(2,\\mathbb{C})$ transformations:\n\\begin{align}\n\\bar{\\tilde{H}}^{\\pm}[z]\\mapsto\\widehat{\\bar{\\tilde{H}}}{}^{\\pm}[w]=(\\gamma w\n +\\delta)^{\\mathcal{N}-1}\\bar{\\tilde{H}}^{\\pm}[z](\\gamma w+\\delta)^{-(\\mathcal{N}-1)}\n \\Bigr|_{z=\\frac{\\alpha w+\\beta}{\\gamma w+\\delta}},\n\\end{align}\nthat is, the transformed operators $\\widehat{\\tilde{H}}{}^{-}$ and\n$\\widehat{\\bar{H}}{}^{+}$ both have the same forms as given in\n(\\ref{eq:gHams}) with $z$ replaced by $w$ and with $A(z)$ and $Q(z)$\nreplaced by the transformed functions $\\widehat{A}(w)$ and $\\widehat{Q}(w)$ given by\n\\begin{align}\nA(z)\\mapsto\\widehat{A}(w)=\\Delta^{-2}(\\gamma w+\\delta)^{4}A(z)\\Bigr|_{z=\\frac{\n \\alpha w+\\beta}{\\gamma w+\\delta}},\\\\\nQ(z)\\mapsto\\widehat{Q}(w)=\\Delta^{-1}(\\gamma w+\\delta)^{2}Q(z)\\Bigr|_{z=\\frac{\n \\alpha w+\\beta}{\\gamma w+\\delta}}.\n\\label{eq:traQ}\n\\end{align}\nIn particular, the explicit form of $\\widehat{Q}(w)$ for arbitrary\n$\\mathcal{N}\\geq2$ is given by\n\\begin{align}\n\\widehat{Q}(w)=\\hat{b}_{2}w^{2}+\\hat{b}_{1}w+\\hat{b}_{0},\n\\end{align}\nwith\n\\begin{align}\n\\left(\\begin{array}{c}\n      \\hat{b}_{2}\\\\ \\hat{b}_{1}\\\\ \\hat{b}_{0}\n      \\end{array}\\right)=\\Delta^{-1}\\left(\n \\begin{array}{rrr}\n \\alpha^{2} & \\alpha\\gamma & \\gamma^{2}\\\\\n 2\\alpha\\beta & \\alpha\\delta+\\beta\\gamma & 2\\gamma\\delta\\\\\n \\beta^{2} & \\beta\\delta & \\delta^{2}\n \\end{array}\\right)\\left(\n \\begin{array}{c}\n b_{2}\\\\ b_{1}\\\\ b_{0}\n \\end{array}\\right).\n\\label{eq:trabi}\n\\end{align}\nUtilizing the transformation (\\ref{eq:traQ}) and the formulas\n\\begin{align}\nw(x)=-\\frac{\\delta z(x)-\\beta}{\\gamma z(x)-\\alpha},\\qquad\n w'(x)=\\frac{\\Delta z'(x)}{(\\gamma z(x)-\\alpha)^{2}}=\\Delta^{-1}\n (\\gamma w(x)+\\delta)^{2}z'(x),\n\\end{align}\nwe obtain the transformations of $W(x)$ and $E(x)$ as\n\\begin{align}\nW(x)&\\mapsto\\widehat{W}(x)=-\\frac{\\widehat{Q}(w)}{w'(x)}\n =-\\frac{Q(z)}{z'(x)}=W(x),\n\\label{eq:traW}\\\\\nE(x)&\\mapsto\\widehat{E}(x)=\\frac{w''(x)}{w'(x)}=\\frac{z''(x)}{z'(x)}\n -\\frac{2\\gamma z'(x)}{\\gamma z(x)-\\alpha}\n =E(x)-\\frac{2\\gamma z'(x)}{\\gamma z(x)-\\alpha}.\n\\label{eq:traE}\n\\end{align}\nThe invariance of the pair of type A $\\mathcal{N}$-fold SUSY Hamiltonians\n$H^{\\pm}$ under the $GL(2,\\mathbb{C})$ transformations also follows from\na direct application of (\\ref{eq:traW}) and (\\ref{eq:traE}):\n\\begin{align}\nH^{\\pm}[W,E]=H^{\\pm}[\\widehat{W},\\widehat{E}],\n\\end{align}\nsince $\\widehat{W}(x)=W(x)$ and from (\\ref{eq:defE}) and (\\ref{eq:traE}) we have\n\\begin{align}\n2\\widehat{E}'(x)-\\widehat{E}(x)^{2}&=2E'(x)-E(x)^{2}.\n\\label{eq:traE2}\n\\end{align}\nOn the other hand, the invariance of the type A $\\mathcal{N}$-fold supercharge\n$P_{\\mathcal{N}}^{-}$ is not manifest in the factorized form (\\ref{eq:ANfch})\nand due to the fact that $\\widehat{E}(x)\\neq E(x)$ the factorized form is\nin appearance not invariant:\n\\begin{align}\nP_{\\mathcal{N}}^{-}[\\widehat{W},\\widehat{E}]&=\\prod_{k=0}^{\\mathcal{N}-1}\\left(\\frac{\\rmd}{\\rmd x}\n +\\widehat{W}(x)+\\frac{\\mathcal{N}-1-2k}{2}\\widehat{E}(x)\\right)\\notag\\\\\n&=\\prod_{k=0}^{\\mathcal{N}-1}\\left(\\frac{\\rmd}{\\rmd x}+W(x)+\\frac{\\mathcal{N}-1-2k}{2}\n \\left(E(x)-\\frac{2\\gamma z'(x)}{\\gamma z(x)-\\alpha}\\right)\\right).\n\\label{eq:traPN}\n\\end{align}\nThe fact that $P_{\\mathcal{N}}^{-}$ is also invariant under the $GL(2,\\mathbb{C})$\ntransformations\n\\begin{align}\nP_{\\mathcal{N}}^{-}[\\widehat{W},\\widehat{E}]=P_{\\mathcal{N}}^{-}[W,E],\n\\end{align}\nproved in Ref.~\\cite{Ta03a}, despite the non-invariance in appearance\nfor $\\gamma\\neq0$, indicates that the type A $\\mathcal{N}$-fold supercharge\nadmits a one-parameter family of different factorizations characterized\nby the parameter $\\alpha\/\\gamma$.\n\n\n\\section{Intermediate Hamiltonians for $\\mathcal{N}=2$}\n\\label{sec:inHam}\n\n{}From now on, we shall restrict ourselves to the case of $\\mathcal{N}=2$.\nThe type A $\\mathcal{N}$-fold SUSY systems (\\ref{eq:AHams}) and (\\ref{eq:ANfch})\nfor $\\mathcal{N}=2$ read\n\\begin{align}\n2H^{\\pm}&=-\\frac{\\rmd^{2}}{\\rmd x^{2}}+W(x)^{2}\n -\\frac{E'(x)}{2}+\\frac{E(x)^{2}}{4}-2R\\pm 2W'(x),\n\\label{eq:A2pm}\\\\\nP_{2}^{-}=P_{21}^{-}P_{22}^{-}&=\\frac{\\rmd^{2}}{\\rmd x^{2}}+2W(x)\n \\frac{\\rmd}{\\rmd x}+W'(x)+W(x)^{2}+\\frac{E'(x)}{2}-\\frac{E(x)^{2}}{4},\n\\label{eq:A2fch}\n\\end{align}\nwhere\n\\begin{align}\nP_{21}^{-}=\\frac{\\rmd}{\\rmd x}+W(x)-\\frac{E(x)}{2},\\qquad\nP_{22}^{-}=\\frac{\\rmd}{\\rmd x}+W(x)+\\frac{E(x)}{2}.\n\\label{eq:defQ-}\n\\end{align}\nThe superHamiltonian $\\boldsymbol{H}_{\\!2}$ and the type A 2-fold supercharges\n$\\boldsymbol{Q}_{2}^{\\pm}$ introduced with the ordinary fermionic variables\n$\\psi^{\\pm}$ as\n\\begin{align}\n\\boldsymbol{H}_{\\!2}=H^{-}\\psi^{-}\\psi^{+}+H^{+}\\psi^{+}\\psi^{-},\\qquad\n \\boldsymbol{Q}_{2}^{\\pm}=P_{2}^{\\mp}\\psi^{\\pm},\n\\end{align}\nsatisfy the type A 2-fold superalgebra~\\cite{Ta03a}:\n\\begin{align}\n\\bigl[\\boldsymbol{Q}_{2}^{\\pm},\\boldsymbol{H}_{\\!2}\\bigr]=\\bigl\\{\\boldsymbol{Q}_{2}^{\\pm},\n \\boldsymbol{Q}_{2}^{\\pm}\\bigr\\}=0,\\qquad\\bigl\\{\\boldsymbol{Q}_{2}^{-},\\boldsymbol{Q}_{2}^{+}\n \\bigr\\}=4(\\boldsymbol{H}_{\\!2}+R)^{2}+4b_{0}b_{2}-b_{1}^{\\,2}.\n\\label{eq:A2alg}\n\\end{align}\nIn the expanded form of the type A 2-fold supercharge components\n(\\ref{eq:A2fch}), its invariance under the $GL(2,\\mathbb{C})$ transformations\nis now manifest by applying (\\ref{eq:traW}) and (\\ref{eq:traE2}):\n\\begin{align}\nP_{2}^{-}[\\widehat{W},\\widehat{E}]=P_{2}^{-}[W,E].\n\\label{eq:invP2}\n\\end{align}\nHowever, each factor of the type A $\\mathcal{N}$-fold supercharge in the factorized\nform is not invariant since $\\widehat{E}(x)\\neq E(x)$ as shown in\n(\\ref{eq:traPN}), and thus we generally have\n\\begin{align}\n\\widehat{P}_{21}^{-}\\equiv P_{21}^{-}[\\widehat{W},\\widehat{E}]\\neq P_{21}^{-}[W,E],\\qquad\n \\widehat{P}_{22}^{-}\\equiv P_{22}^{-}[\\widehat{W},\\widehat{E}]\\neq P_{22}^{-}[W,E].\n\\label{eq:ninv}\n\\end{align}\nNext, we introduce another Hamiltonian $H^{\\rmi1}$, which we shall call\nan \\emph{intermediate} Hamiltonian, as\n\\begin{align}\nP_{22}^{-}H^{-}=H^{\\rmi1}P_{22}^{-},\\qquad\n P_{21}^{-}H^{\\rmi1}=H^{+}P_{21}^{-},\n\\label{eq:defH0}\n\\end{align}\nwhich are compatible with (\\ref{eq:inter}). It is evident that\n$H^{\\rmi1}$ is in general not invariant under the $GL(2,\\mathbb{C})$\ntransformation in contrast with $H^{\\pm}$ due to the fact that\nboth of $P_{21}^{-}$ and $P_{22}^{-}$ which intertwine $H^{\\rmi1}$\nwith $H^{\\pm}$ have no invariance (\\ref{eq:ninv}). Hence, we can\nexpect a family of intermediate Hamiltonians for each given type A\n2-fold SUSY system.\nNeedless to say, however, we do not always have such an intermediate\nHamiltonian for a given system. The necessary and sufficient conditions\nfor its existence are that there exist two constants $C_{22}$ and\n$C_{21}$ such that $H^{\\pm}$ and $H^{\\rmi1}$ are expressed as\n(see, e.g., Refs.~\\cite{CKS95,Ju96,Ba00})\n\\begin{align}\n\\begin{split}\n2H^{-}&=P_{22}^{+}P_{22}^{-}+2C_{22},\\quad 2H^{+}=P_{21}^{-}\n P_{21}^{+}+2C_{21},\\\\\n2H^{\\rmi1}&=P_{22}^{-}P_{22}^{+}+2C_{22}=P_{21}^{+}P_{21}^{-}+2C_{21},\n\\label{eq:cond0}\n\\end{split}\n\\end{align}\nwhere $P_{ij}^{+}$ are the transpositions of $P_{ij}^{-}$, that\nis,\\footnote{Note that we do not assume the reality of the functions\n$W(x)$ and $E(x)$.}\n\\begin{align}\nP_{21}^{+}=-\\frac{\\rmd}{\\rmd x}+W(x)-\\frac{E(x)}{2},\\qquad\nP_{22}^{+}=-\\frac{\\rmd}{\\rmd x}+W(x)+\\frac{E(x)}{2}.\n\\label{eq:defQ+}\n\\end{align}\nMore explicitly, the conditions (\\ref{eq:cond0}) read\n\\begin{subequations}\n\\label{eqs:cond1}\n\\begin{align}\n2H^{-}&=-\\frac{\\rmd^{2}}{\\rmd x^{2}}+W(x)^{2}+E(x)W(x)\n +\\frac{E(x)^{2}}{4}-\\frac{E'(x)}{2}-W'(x)+2C_{22},\\\\\n2H^{\\rmi1}&=-\\frac{\\rmd^{2}}{\\rmd x^{2}}+W(x)^{2}+E(x)W(x)\n +\\frac{E(x)^{2}}{4}+\\frac{E'(x)}{2}+W'(x)+2C_{22}\\notag\\\\\n&=-\\frac{\\rmd^{2}}{\\rmd x^{2}}+W(x)^{2}-E(x)W(x)+\\frac{E(x)^{2}}{4}\n +\\frac{E'(x)}{2}-W'(x)+2C_{21},\n\\label{eq:H0}\\\\\n2H^{+}&=-\\frac{\\rmd^{2}}{\\rmd x^{2}}+W(x)^{2}-E(x)W(x)\n +\\frac{E(x)^{2}}{4}-\\frac{E'(x)}{2}+W'(x)+2C_{21}.\n\\end{align}\n\\end{subequations}\n{}From Eqs.~(\\ref{eq:A2pm}) and (\\ref{eqs:cond1}), the necessary and\nsufficient conditions reduce to\n\\begin{align}\nW'(x)+E(x)W(x)=-2R-2C_{22}=C_{21}-C_{22}=2C_{21}+2R.\n\\label{eq:cond2}\n\\end{align}\nNoting the relation\n\\begin{align}\nW'(x)+E(x)W(x)=-Q'(z),\n\\end{align}\nwhich easily follows from (\\ref{eq:defE}) and (\\ref{eq:defAQ}), we\nfind that the latter conditions (\\ref{eq:cond2}) are equivalent to\n\\begin{align}\nQ(z)=(C_{22}-C_{21})z+b_{0},\\qquad -2R=C_{22}+C_{21},\n\\end{align}\nwith $b_{0}$ being another constant. We recall that for the most general\ntype A $\\mathcal{N}$-fold SUSY systems for all $\\mathcal{N}\\geq2$, $Q(z)$ is given by\na polynomial of at most second-degree (\\ref{eq:Qofz}).\nHence, a given type A 2-fold SUSY system (\\ref{eq:A2pm}) admits an\nintermediate Hamiltonian $H^{\\rmi1}$ satisfying Eq.~(\\ref{eq:defH0}) if\nand only if\n\\begin{align}\nb_{2}=0,\\quad b_{1}=C_{22}-C_{21},\\quad -2R=C_{22}+C_{21}.\n\\label{eq:cond3}\n\\end{align}\nThe last two conditions in (\\ref{eq:cond3}) just determine these\nconstants for the given values of $b_{1}$ and $R$. Hence, only\nthe first condition in (\\ref{eq:cond3}) is essential for the existence\nof an intermediate Hamiltonian.\n\nAs was discussed previously, the type A 2-fold supercharge $P_{2}^{-}$\nis invariant under the $GL(2,\\mathbb{C})$ transformation (\\ref{eq:invP2})\nwhile its factors $P_{22}^{-}$ and $P_{21}^{-}$ are not (\\ref{eq:ninv})\nfor $\\gamma\\neq0$. As a consequence, the necessary and sufficient\nconditions (\\ref{eq:cond3}) for the existence of another intermediate\nHamiltonian $H^{\\rmi2}$ after a $GL(2,\\mathbb{C})$ transformation are\naccordingly changed as\n\\begin{align}\n\\hat{b}_{2}=0,\\quad\\hat{b}_{1}=\\hat{C}_{22}-\\hat{C}_{21},\\quad\n -2R=\\hat{C}_{22}+\\hat{C}_{21},\n\\label{eq:cond3'}\n\\end{align}\nwhere $\\hat{C}_{22}$ and $\\hat{C}_{21}$ are another set of constants.\nAgain, only the first condition in (\\ref{eq:cond3'}) is essential for\nthe existence of another intermediate Hamiltonian $H^{\\rmi2}$ after\nthe transformation. Under the fulfillment of the conditions\n(\\ref{eq:cond3'}), the original type A 2-fold SUSY Hamiltonians\n$H^{\\pm}$ and the new intermediate Hamiltonian $H^{\\rmi2}$ are\nexpressed in terms of the transformed supercharges as\n\\begin{align}\n\\begin{split}\n2H^{-}&=\\widehat{P}_{22}^{+}\\widehat{P}_{22}^{-}+2\\hat{C}_{22},\\quad\n 2H^{+}=\\widehat{P}_{21}^{-}\\widehat{P}_{21}^{+}+2\\hat{C}_{21},\\\\\n2H^{\\rmi2}&=\\widehat{P}_{22}^{-}\\widehat{P}_{22}^{+}+2\\hat{C}_{22}=\\widehat{P}_{21}^{+}\n \\widehat{P}_{21}^{-}+2\\hat{C}_{21},\n\\label{eq:Hi2}\n\\end{split}\n\\end{align}\nwhere $\\widehat{P}_{ij}^{+}$ are the transpositions of $\\widehat{P}_{ij}^{-}$ which\nwere defined in (\\ref{eq:ninv}), that is,\n\\begin{align}\n\\widehat{P}_{21}^{\\pm}=\\mp\\frac{\\rmd}{\\rmd x}+\\widehat{W}(x)-\\frac{\\widehat{E}(x)}{2},\\qquad\n\\widehat{P}_{22}^{\\pm}=\\mp\\frac{\\rmd}{\\rmd x}+\\widehat{W}(x)+\\frac{\\widehat{E}(x)}{2}.\n\\label{eq:hQpm}\n\\end{align}\nIt is now clear that a type A 2-fold SUSY system which satisfies the\nconditions (\\ref{eq:cond3}) and thus admits an intermediate Hamiltonian\n$H^{\\rmi1}$ also admits another different intermediate Hamiltonian\n$H^{\\rmi2}$ after a $GL(2,\\mathbb{C})$ transformation with $\\gamma\\neq0$ if\nand only if the conditions (\\ref{eq:cond3'}) are simultaneously fulfilled\nin addition to (\\ref{eq:cond3}). It essentially means the satisfaction\nof $b_{2}=\\hat{b}_{2}=0$. From the transformation formula\n(\\ref{eq:trabi}) we immediately see that it is only possible for the\n$GL(2,\\mathbb{C})$ transformation with $\\gamma\\neq0$ which satisfies\n\\begin{align}\n\\alpha b_{1}+\\gamma b_{0}=0.\n\\label{eq:cond4}\n\\end{align}\nLet us first consider the case when $b_{1}=0$. In this case we can assume\nthat $b_{0}\\neq0$; otherwise $Q(z)=0$ since $b_{2}=0$ is already assumed\nin order to meet the condition (\\ref{eq:cond3}), and from (\\ref{eq:VAQ})\n$V^{-}(x)=V^{+}(x)$ which means that the system is trivial as 2-fold\nSUSY. But for $b_{1}=0$ and $b_{0}\\neq0$ there is no solution to the\nequation (\\ref{eq:cond4}) except for $\\gamma=0$. But for any $GL(2,\\mathbb{C})$\ntransformation with $\\gamma=0$, the function $E(x)$ is invariant by\nEq.~(\\ref{eq:traE}) and so is the intermediate Hamiltonian $H^{\\rmi1}$.\nHence in the case of $b_{1}=0$ the factorization of type A 2-fold\nsupercharge admitting intermediate Hamiltonians is unique. On the other\nhand, in the case of $b_{1}\\neq0$ the solution to the equation\n(\\ref{eq:cond4}) is given by\n\\begin{align}\n\\alpha\/\\gamma=-b_{0}\/b_{1}.\n\\label{eq:cond5}\n\\end{align}\nAs was shown previously, any type A 2-fold supercharge admits\none-parameter family of factorizations $P_{2}^{-}=P_{21}^{-}P_{22}^{-}$\ncharacterized by the parameter $\\alpha\/\\gamma$. In addition, any type A\n2-fold SUSY system with $b_{2}=0$ has at least one intermediate\nHamiltonian $H^{\\rmi1}$.\nThen, the result (\\ref{eq:cond5}) tells us that if the system further\nsatisfies $b_{1}\\neq0$, it can admit a one and only one additional and\ndifferent intermediate Hamiltonian $H^{\\rmi2}$ at the one point\n(\\ref{eq:cond5}) in the parameter space of $\\alpha\/\\gamma\\in\\mathbb{C}$.\nIn Table~\\ref{tb:numb}, we summarize the results.\n\\begin{table}\n\\begin{center}\n\\tabcolsep=10pt\n\\begin{tabular}{cc}\n\\hline\nConditions & Number of $H^{\\rmi}$\\\\\n\\hline\n$b_{2}\\neq0$ & 0\\\\\n$b_{2}=b_{1}=0$ & 1\\\\\n$b_{2}=0$, $b_{1}\\neq0$ & 2\\\\\n\\hline\n\\end{tabular}\n\\caption{The admissible numbers of different intermediate Hamiltonians\nin type A 2-fold supersymmetry.}\n\\label{tb:numb}\n\\end{center}\n\\end{table}\n\n\n\\section{Second-Order Parasupersymmetry}\n\\label{sec:psusy}\n\nIn the previous section, we have just verified that a type A 2-fold SUSY\nsystem $(H^{\\pm}, P_{2}^{-}=P_{21}^{-}P_{22}^{-})$ admits (at least) one\nintermediate Hamiltonian $H^{\\rmi1}$ if and only if the condition\n$b_{2}=0$ holds. In this section, we shall further show that any such\na system can possess an additional symmetry, namely, parasupersymmetry\nof order 2 introduced in Ref.~\\cite{RS88}.\nIndeed, for a given such type A 2-fold SUSY system we can define\na triple of operators $(\\boldsymbol{H}_{\\!\\mathrm{P}},\\boldsymbol{Q}_{\\mathrm{P}}^{\\pm})$ by\n\\begin{subequations}\n\\label{eqs:pss2}\n\\begin{align}\n\\boldsymbol{H}_{\\!\\mathrm{P}}&=H^{-}(\\psi_{\\mathrm{P}}^{-})^{2}(\\psi_{\\mathrm{P}}^{+})^{2}+H^{\\rmi1}\n (\\psi_{\\mathrm{P}}^{+}\\psi_{\\mathrm{P}}^{-}-(\\psi_{\\mathrm{P}}^{+})^{2}(\\psi_{\\mathrm{P}}^{-}\n )^{2})+H^{+}(\\psi_{\\mathrm{P}}^{+})^{2}(\\psi_{\\mathrm{P}}^{-})^{2},\\\\\n\\boldsymbol{Q}_{\\mathrm{P}}^{-}&=P_{22}^{+}(\\psi_{\\mathrm{P}}^{-})^{2}\\psi_{\\mathrm{P}}^{+}\n +P_{21}^{+}\\psi_{\\mathrm{P}}^{+}(\\psi_{\\mathrm{P}}^{-})^{2},\\quad\\boldsymbol{Q}_{\\mathrm{P}}^{+}\n =P_{22}^{-}\\psi_{\\mathrm{P}}^{-}(\\psi_{\\mathrm{P}}^{+})^{2}+P_{21}^{-}\n (\\psi_{\\mathrm{P}}^{+})^{2}\\psi_{\\mathrm{P}}^{-},\n\\end{align}\n\\end{subequations}\nwhere $\\psi_{\\mathrm{P}}^{\\pm}$ are parafermions of order 2\nsatisfying~\\cite{Ta07a}\n\\begin{align}\n(\\psi_{\\mathrm{P}}^{\\pm})^{2}\\neq0,\\quad(\\psi_{\\mathrm{P}}^{\\pm})^{3}=0,\\quad\n \\bigl\\{\\psi_{\\mathrm{P}}^{-},\\psi_{\\mathrm{P}}^{+}\\bigr\\}+\\bigl\\{(\\psi_{\\mathrm{P}}^{-}\n )^{2},(\\psi_{\\mathrm{P}}^{+})^{2}\\bigr\\}=2I.\n\\end{align}\nThen, using Eq.~(\\ref{eq:cond0}) and the parafermionic algebra of order\n2 in Ref.~\\cite{Ta07a} we can show that the triple $(\\boldsymbol{H}_{\\!\\mathrm{P}},\n\\boldsymbol{Q}_{\\mathrm{P}}^{\\pm})$ defined as (\\ref{eqs:pss2}) satisfies the second-order\nparaSUSY relations in Ref.~\\cite{RS88}:\n\\begin{align}\n(\\boldsymbol{Q}_{\\mathrm{P}}^{\\pm})^{2}\\neq0,\\quad(\\boldsymbol{Q}_{\\mathrm{P}}^{\\pm})^{3}=0,\\quad\n \\bigl[\\boldsymbol{Q}_{\\mathrm{P}}^{\\pm},\\boldsymbol{H}_{\\!\\mathrm{P}}\\bigr]=0,\n\\label{eq:para1}\\\\\n(\\boldsymbol{Q}_{\\mathrm{P}}^{\\pm})^{2}\\boldsymbol{Q}_{\\mathrm{P}}^{\\mp}+\\boldsymbol{Q}_{\\mathrm{P}}^{\\pm}\\boldsymbol{Q}_{\\mathrm{P}}^{\\mp}\n \\boldsymbol{Q}_{\\mathrm{P}}^{\\pm}+\\boldsymbol{Q}_{\\mathrm{P}}^{\\mp}(\\boldsymbol{Q}_{\\mathrm{P}}^{\\pm})^{2}\n =4\\boldsymbol{Q}_{\\mathrm{P}}^{\\pm}\\boldsymbol{H}_{\\!\\mathrm{P}},\n\\label{eq:para2}\n\\end{align}\nif and only if the constants $C_{ij}$ in (\\ref{eq:cond0}) satisfy\n\\begin{align}\nC_{22}=-C_{21}=b_{1}\/2,\n\\label{eq:cpara}\n\\end{align}\nand thus in particular $R=0$ by (\\ref{eq:cond3}). Hence, we conclude\nthat any type A 2-fold SUSY quantum system with (at least) one\nintermediate Hamiltonian also has paraSUSY of order 2 when $R=0$.\nThe additional restriction $R=0$ arises since one of the paraSUSY\nconditions (\\ref{eq:para2}) is not invariant under any constant shift\nof $\\boldsymbol{H}_{\\!\\mathrm{P}}$.\nFurthermore, as was shown in Ref.~\\cite{Ta07a} this type of realization\nof second-order paraSUSY systems admits an additional novel nonlinear\nrelation as the following (cf., Eq.~(6.66) in the latter reference):\n\\begin{align}\n(\\boldsymbol{Q}_{\\mathrm{P}}^{-})^{2}(\\boldsymbol{Q}_{\\mathrm{P}}^{+})^{2}+\\boldsymbol{Q}_{\\mathrm{P}}^{\\pm}\n (\\boldsymbol{Q}_{\\mathrm{P}}^{\\mp})^{2}\\boldsymbol{Q}_{\\mathrm{P}}^{\\pm}+(\\boldsymbol{Q}_{\\mathrm{P}}^{+})^{2}\n (\\boldsymbol{Q}_{\\mathrm{P}}^{-})^{2}=4(\\boldsymbol{H}_{\\!\\mathrm{P}})^{2}-b_{1}^{\\,2},\n\\label{eq:nlrel}\n\\end{align}\nwhich can be regarded as a generalized (type A) 2-fold superalgebra.\nIn fact, on the one hand we immediately have from the paraSUSY relations\nin (\\ref{eq:para1})\n\\begin{align}\n\\bigl\\{(\\boldsymbol{Q}_{\\mathrm{P}}^{\\pm})^{2},(\\boldsymbol{Q}_{\\mathrm{P}}^{\\pm})^{2}\\bigr\\}=0,\\qquad\n \\bigl[(\\boldsymbol{Q}_{\\mathrm{P}}^{\\pm})^{2},\\boldsymbol{H}_{\\!\\mathrm{P}}\\bigr]=0,\n\\label{eq:salg1}\n\\end{align}\nwhile on the other hand the nonlinear relation (\\ref{eq:nlrel})\nreduces, in the subsector with the parafermion number zero and two, to\n\\begin{align}\n\\bigl\\{(\\boldsymbol{Q}_{\\mathrm{P}}^{-})^{2},(\\boldsymbol{Q}_{\\mathrm{P}}^{+})^{2}\\bigr\\}\n =4(\\boldsymbol{H}_{\\!\\mathrm{P}})^{2}-b_{1}^{\\,2}.\n\\label{eq:salg2}\n\\end{align}\nThen, the commutation and anti-commutation relations (\\ref{eq:salg1})\nand (\\ref{eq:salg2}) are, under the assumed condition $b_{2}=0$\nand $R=0$, entirely identical with the type A 2-fold superalgebra\n(\\ref{eq:A2alg}) with the trivial identification of the type A 2-fold\nsupercharges $\\boldsymbol{Q}_{2}^{\\pm}$ with $(\\boldsymbol{Q}_{\\mathrm{P}}^{\\pm})^{2}$ and with\nthe observation that in the subsector $\\boldsymbol{H}_{\\!\\mathrm{P}}$ is essentially\nidentical with $\\boldsymbol{H}_{\\!2}$. The relation between type A 2-fold SUSY\nand second-order paraSUSY was briefly referred to in Ref.~\\cite{Ta07a}.\nHere we have firstly shown the necessary and sufficient conditions\nfor a type A 2-fold SUSY system to admit simultaneously second-order\nparaSUSY, namely, Eqs.~(\\ref{eq:cond3}) and (\\ref{eq:cpara}). Finally,\nit is evident that we can construct two sets of second-order paraSUSY\nsystems whenever a type A 2-fold SUSY system has two different\nintermediate Hamiltonians.\n\n\n\\section{An Application to the generalized P\\\"{o}schl--Teller potential}\n\\label{sec:appli}\n\nAs an application of the general framework discussed in the previous\ntwo sections, we shall reconstruct the BQR SSUSY model (\\ref{eq:BQRs})%\n--(\\ref{eq:BQRe}) and its generalization which preserves all the SUSY\nand SSUSY structure therein. To this end, let us first choose\nthe change of variable $z=z(x)$ which determines the relation\nbetween physical Hamiltonians and gauged ones as\n\\begin{align}\nz(x)&=-(\\sinh x)^{-2B}\\left(\\tanh\\frac{x}{2}\\right)^{2A+1}\n \\notag\\\\\n&=-2^{-2B}\\left(\\sinh\\frac{x}{2}\\right)^{2A-2B+1}\n \\left(\\cosh\\frac{x}{2}\\right)^{-(2A+2B+1)},\n\\end{align}\nwhere $A$ and $B$ are both constants. The function $A(z)=z'(x)^{2}\/2$\ndefined in (\\ref{eq:defAQ}) and its derivatives with respect to $z$\nare in general transcendental functions of $z$. Explicitly, they read\n\\begin{align}\nz'(x)&=-z(x)\\frac{2B\\cosh x-2A-1}{\\sinh x},\n\\label{eq:z'x}\\\\\nA(z)&=\\frac{z(x)^{2}(2B\\cosh x-2A-1)^{2}}{2\\sinh^{2}x},\n\\label{eq:Aofz}\\\\\nA'(z)&=z(x)\\left(4B^{2}-\\frac{\\alpha_{1}^{+}\\cosh x-\\alpha_{2}^{+}}{\n \\sinh^{2}x}\\right),\\\\\nA''(z)&=4B^{2}-\\frac{\\beta_{1}\\cosh x-\\beta_{2}}{\\sinh^{2}x}\n -\\frac{2A+1}{2B\\cosh x-2A-1},\n\\end{align}\nwhere $\\alpha_{i}^{+}$ and $\\beta_{i}$ are all constants given by\n\\begin{alignat}{2}\n\\alpha_{1}^{\\pm}&=(2A+1)(4B\\pm1),&\\qquad\n \\alpha_{2}^{\\pm}&=(2A+1)^{2}+2B(2B\\pm1),\n\\label{eq:alpha}\\\\\n\\beta_{1}&=(2A+1)(4B+3),&\\beta_{2}&=(2A+1)^{2}+2(B+1)(2B+1).\n\\end{alignat}\nSubstituting them into the most general form of a pair of type A 2-fold\nSUSY potentials, Eq.~(\\ref{eq:VAQ}) with $\\mathcal{N}=2$, we obtain\n\\begin{align}\nV^{\\pm}(x)=&\\,\\frac{Q(z(x))^{2}\\sinh^{2}x}{2z(x)^{2}(2B\\cosh x-2A-1)^{2}}\n+\\frac{4(2A+1)B\\cosh x+(2A+1)^{2}-12B^{2}}{8(2B\\cosh x-2A-1)^{2}}\\notag\\\\\n&-\\frac{(2A+1)B\\cosh x-A(A+1)-B^{2}}{2\\sinh^{2}x}+\\frac{B^{2}}{2}\n -R\\notag\\\\\n&\\pm\\frac{Q(z(x))(4B^{2}\\sinh^{2}x-\\alpha_{1}^{+}\\cosh x+\\alpha_{2}^{+}\n )}{z(x)(2B\\cosh x-2A-1)^{2}}\\mp Q'(z(x)),\n\\label{eq:Vpm}\n\\end{align}\nwhere $Q(z)$ is a polynomial of at most second-degree given as in\n(\\ref{eq:Qofz}). The function $W(x)$ characterizing the type A system\nin this case reads\n\\begin{align}\nW(x)=-\\frac{Q(z(x))}{z'(x)}=\\frac{Q(z(x))\\sinh x}{z(x)(2B\\cosh x-2A-1)}.\n\\label{eq:Wx2}\n\\end{align}\nThe other function $E(x)$ defined through the relation (\\ref{eq:defE}),\nwhich also characterizes the type A system, is calculated as\n\\begin{align}\nE(x)=\\frac{z''(x)}{z'(x)}=-\\frac{(2B+1)\\cosh x-2A-1}{\\sinh x}\n +\\frac{2B\\sinh x}{2B\\cosh x-2A-1}.\n\\label{eq:Ex}\n\\end{align}\nNext, let us consider the case when the type A 2-fold SUSY system admits\nan intermediate Hamiltonian, namely, $b_{2}=0$. From (\\ref{eq:Wx2}) and\n(\\ref{eq:Ex}) we have\n\\begin{align}\nW'(x)=&\\,\\frac{Q(z(x))}{z(x)}-Q'(z(x))\n -\\frac{(2A+1)\\cosh x-2B}{z(x)(2B\\cosh x-2A-1)^{2}}Q(z(x)),\\\\\nE'(x)=&-\\frac{2(2A+1)B\\cosh x-4B^{2}}{(2B\\cosh x-2A-1)^{2}}\n -\\frac{(2A+1)\\cosh x-2B-1}{\\sinh^{2}x},\\\\\nE(x)^{2}=&\\,4B^{2}+\\frac{(2A+1)^{2}-4B^{2}}{(2B\\cosh x-2A-1)^{2}}\\notag\\\\\n &-\\frac{2(2A+1)(2B+1)\\cosh x-(2A+1)^{2}-(2B+1)^{2}}{\\sinh^{2}x}.\n\\label{eq:Ex^2}\n\\end{align}\nSubstituting (\\ref{eq:Wx2})--(\\ref{eq:Ex^2}) into (\\ref{eq:H0}), using\nthe relations (\\ref{eq:cond3}) among the constants, and noting that\n$Q'(z)=b_{1}$ when $b_{2}=0$, we obtain the intermediate potential\n$V^{\\rmi1}(x)$ as\n\\begin{align}\nV^{\\rmi1}(x)=&\\,\\frac{Q(z(x))^{2}\\sinh^{2}x}{2z(x)^{2}(2B\\cosh x\n -2A-1)^{2}}-\\frac{4(2A+1)B\\cosh x-(2A+1)^{2}-4B^{2}}{8(2B\\cosh x\n -2A-1)^{2}}\\notag\\\\\n&-\\frac{(2A+1)(B+1)\\cosh x-A(A+1)-(B+1)^{2}}{2\\sinh^{2}x}\n +\\frac{B^{2}}{2}-R.\n\\label{eq:Vi1}\n\\end{align}\nNext, we shall consider the case when the system admits another different\nintermediate Hamiltonian $H^{\\rmi2}$, namely, $b_{1}\\neq0$. The\n$GL(2,\\mathbb{C})$ transformation which takes the type A 2-fold supercharge to\nanother factorization for which $H^{\\rmi2}$ exists must satisfy\nthe condition (\\ref{eq:cond5}). The parameter $\\delta$ does not play an\nimportant role in our context, so we set $\\delta=0$ without any loss of\ngenerality. But in this case $\\beta$ cannot be $0$ otherwise $\\Delta=0$.\nThus, we fix the parameters as\n\\begin{align}\n\\alpha\/\\gamma=-b_{0}\/b_{1}\\equiv -z_{0},\\qquad\\beta\/\\gamma=-1,\n \\qquad\\delta=0.\n\\label{eq:parav}\n\\end{align}\nIn other words, we choose the following $GL(2,\\mathbb{C})$ transformation on\nthe variable $z(x)$:\n\\begin{align}\nw(x)=-\\frac{1}{z(x)+z_{0}}=\\frac{1}{(\\sinh x)^{-2B}(\\tanh x\/2\n )^{2A+1}-z_{0}}.\n\\end{align}\nThe function $W(x)$ is invariant under the transformation (see,\nEq.~(\\ref{eq:traW})) while $E(x)$ is transformed according to\n(\\ref{eq:traE}) as\n\\begin{align}\n\\widehat{E}(x)=&\\,E(x)-\\frac{2z'(x)}{z(x)+z_{0}}=E(x)+\\frac{2z(x)}{z(x)+z_{0}}\n \\frac{2B\\cosh x-2A-1}{\\sinh x}\\notag\\\\\n=&\\,\\frac{[(2B-1)z(x)-(2B+1)z_{0}]\\cosh x-(2A+1)(z(x)-z_{0})}{\n (z(x)+z_{0})\\sinh x}\\notag\\\\\n&+\\frac{2B\\sinh x}{2B\\cosh x-2A-1}.\n\\label{eq:hatE}\n\\end{align}\nNoting the relation $Q(z)=b_{1}(z+z_{0})$ when $b_{2}=0$, we have\n\\begin{align}\n\\widehat{E}(x)\\widehat{W}(x)=E(x)W(x)+2b_{1}.\n\\end{align}\n{}From the second expression of $\\widehat{E}(x)$ in (\\ref{eq:hatE}), we obtain\nthe following formulas:\n\\begin{align}\n\\widehat{E}'(x)=&\\,E'(x)+\\frac{2z(x)}{z(x)+z_{0}}\n \\frac{(2A+1)\\cosh x-2B}{\\sinh^{2}x}\\notag\\\\\n&-\\frac{2z_{0}z(x)}{(z(x)+z_{0})^{2}}\n \\frac{(2B\\cosh x-2A-1)^{2}}{\\sinh^{2}x},\\\\\n\\widehat{E}(x)^{2}=&\\,E(x)^{2}+\\frac{8Bz(x)}{z(x)+z_{0}}-\\frac{4z(x)\\cosh x\n (2B\\cosh x-2A-1)}{(z(x)+z_{0})\\sinh^{2}x}\\notag\\\\\n&-\\frac{4z_{0}z(x)(2B\\cosh x-2A-1)^{2}}{(z(x)+z_{0})^{2}\\sinh^{2}x}.\n\\label{eq:hE^2}\n\\end{align}\nSubstituting (\\ref{eq:hatE})--(\\ref{eq:hE^2}) into Eq.~(\\ref{eq:Hi2})\nand noting that $\\hat{b}_{1}=-b_{1}$ by the transformation formula\n(\\ref{eq:trabi}) in our choice of the parameters (\\ref{eq:parav}), we\nobtain for $H^{\\pm}$ the same potentials as the ones in (\\ref{eq:Vpm}),\nas they should be, while for the other intermediate Hamiltonian\n$H^{\\rmi2}$ the following form of the potential:\n\\begin{align}\nV^{\\rmi2}(x)=&\\,V^{\\rmi1}(x)+\\frac{z(x)^{2}[(2A+1)\\cosh x-2B]}{\n (z(x)+z_{0})^{2}\\sinh^{2}x}\\notag\\\\\n&+\\frac{z_{0}z(x)(\\alpha_{1}^{+}\\cosh x-\\alpha_{2}^{+})}{(z(x)+z_{0})^{2}\n \\sinh^{2}x}-\\frac{4B^{2}z_{0}z(x)}{(z+z_{0})^{2}}.\n\\end{align}\nIf we substitute (\\ref{eq:Vi1}) for $V^{\\rmi1}(x)$ into the above, we\nfinally obtain the full expression of $V^{\\rmi2}(x)$ as\n\\begin{align}\nV^{\\rmi2}(x)=&\\,\\frac{Q(z(x))^{2}\\sinh^{2}x}{2z(x)^{2}(2B\\cosh x-\n 2A-1)^{2}}-\\frac{4(2A+1)B\\cosh x-(2A+1)^{2}-4B^{2}}{8(2B\\cosh x\n -2A-1)^{2}}\\notag\\\\\n&-\\frac{(2A+1)(B-1)\\cosh x-A(A+1)-(B-1)^{2}}{2\\sinh^{2}x}\n +\\frac{B^{2}}{2}-R\\notag\\\\\n&+\\frac{z_{0}z(x)(\\alpha_{1}^{-}\\cosh x-\\alpha_{2}^{-})}{\n (z(x)+z_{0})^{2}\\sinh^{2}x}\n-\\frac{z_{0}^{\\,2}[(2A+1)\\cosh x-2B]}{(z(x)+z_{0})^{2}}\n -\\frac{4B^{2}z_{0}z(x)}{(z(x)+z_{0})^{2}},\n\\label{eq:Vi2}\n\\end{align}\nwhere $\\alpha_{i}^{-}$ are defined in (\\ref{eq:alpha}). We are now\nin a position to show that the type A 2-fold system with the two\nintermediate Hamiltonians (\\ref{eq:Vpm}), (\\ref{eq:Vi1}), and\n(\\ref{eq:Vi2}) contains as a special case the BQR SSUSY model\n(\\ref{eq:BQRs})--(\\ref{eq:BQRe}). For the purpose, let put $b_{0}=0$\nand $b_{1}=bB$. In this case, $Q(z)=bBz$ and $z_{0}=b_{0}\/b_{1}=0$.\nThen, the 2-fold SUSY pair of the potentials (\\ref{eq:Vpm}) and the\ntwo intermediate potentials (\\ref{eq:Vi1}) and (\\ref{eq:Vi2})\nreduce to, respectively,\n\\begin{align}\nV^{\\pm}(x)=&\\,\\frac{4(b^{2}+1)(2A+1)B\\cosh x-(b^{2}-1)(2A+1)^{2}\n -4(b^{2}+3)B^{2}}{8(2B\\cosh x-2A-1)^{2}}\\notag\\\\\n&-\\frac{(2A+1)B\\cosh x-A(A+1)-B^{2}}{2\\sinh^{2}x}+\\frac{b^{2}}{8}\n +\\frac{B^{2}}{2}-R\\notag\\\\\n&\\mp bB\\frac{(2A+1)\\cosh x-2B}{(2B\\cosh x-2A-1)^{2}},\n\\label{eq:Vpm'}\n\\end{align}\nand\n\\begin{align}\nV^{\\rmi1}(x)=&\\,(b^{2}-1)\\frac{4(2A+1)B\\cosh x-(2A+1)^{2}-4B^{2}}{\n 8(2B\\cosh x-2A-1)^{2}}\\notag\\\\\n&-\\frac{(2A+1)(B+1)\\cosh x-A(A+1)-(B+1)^{2}}{\\sinh^{2}x}\n +\\frac{b^{2}}{8}+\\frac{B^{2}}{2}-R,\n\\label{eq:Vi1'}\n\\end{align}\nand\n\\begin{align}\nV^{\\rmi2}(x)=&\\,(b^{2}-1)\\frac{4(2A+1)B\\cosh x-(2A+1)^{2}-4B^{2}}{\n 8(2B\\cosh x-2A-1)^{2}}\\notag\\\\\n&-\\frac{(2A+1)(B-1)\\cosh x-A(A+1)-(B-1)^{2}}{\\sinh^{2}x}\n +\\frac{b^{2}}{8}+\\frac{B^{2}}{2}-R.\n\\label{eq:Vi2'}\n\\end{align}\nThe components of supercharges $P_{ij}^{\\pm}$ and $\\widehat{P}_{ij}^{\\pm}$\ngiven by (\\ref{eq:defQ-}), (\\ref{eq:defQ+}), and (\\ref{eq:hQpm}) in\nthis case read\n\\begin{align}\nP_{21}^{\\pm}&=\\mp\\frac{\\rmd}{\\rmd x}+\\frac{(2B+1)\\cosh x-2A-1}{2\\sinh x}\n +\\frac{(b-1)B\\sinh x}{2B\\cosh x-2A-1},\\\\\nP_{22}^{\\pm}&=\\mp\\frac{\\rmd}{\\rmd x}-\\frac{(2B+1)\\cosh x-2A-1}{2\\sinh x}\n +\\frac{(b+1)B\\sinh x}{2B\\cosh x-2A-1},\\\\\n\\widehat{P}_{21}^{\\pm}&=\\mp\\frac{\\rmd}{\\rmd x}-\\frac{(2B-1)\\cosh x-2A-1}{2\\sinh x}\n +\\frac{(b-1)B\\sinh x}{2B\\cosh x-2A-1},\\\\\n\\widehat{P}_{22}^{\\pm}&=\\mp\\frac{\\rmd}{\\rmd x}+\\frac{(2B-1)\\cosh x-2A-1}{2\\sinh x}\n +\\frac{(b+1)B\\sinh x}{2B\\cosh x-2A-1}.\n\\end{align}\nIt is now easy to see that the BQR SSUSY model is realized when $b=-1$.\nIndeed, we have the following correspondences when $b=-1$:\n\\begin{align}\n2V^{-}(x)=V_{A,B}(x)+\\frac{1}{4}+B^{2}-2R,\\quad\n 2V^{+}(x)=V_{A,B,\\textrm{ext}}(x)+\\frac{1}{4}+B^{2}-2R,\\\\\n2V^{\\rmi1}(x)=V_{A,B+1}(x)+\\frac{1}{4}+B^{2}-2R,\\quad\n 2V^{\\rmi2}(x)=V_{A,B-1}(x)+\\frac{1}{4}+B^{2}-2R,\\\\\nP_{21}^{-}\\ \\textrm{or}\\ \\widehat{P}_{21}^{-}=\\hat{A},\\quad\n P_{21}^{+}\\ \\textrm{or}\\ \\widehat{P}_{21}^{+}=\\hat{A}^{\\dagger},\\quad\n P_{22}^{-}\\ \\textrm{or}\\ \\widehat{P}_{22}^{-}=\\hat{B},\\quad\n P_{22}^{+}\\ \\textrm{or}\\ \\widehat{P}_{22}^{+}=\\hat{B}^{\\dagger},\n\\end{align}\nand in particular $P_{2}^{-}=P_{21}^{-}P_{22}^{-}=\\widehat{P}_{21}^{-}\\widehat{P}_{22}^{-}\n=\\hat{A}\\hat{B}$. The relations among the constants are given by\n\\begin{align}\n\\bar{c}=4C_{22}=-4C_{21}=-2B\\quad\\textrm{or}\n \\quad\\bar{c}=4\\hat{C}_{22}=-4\\hat{C}_{21}=2B,\\\\\n-\\tilde{E}+\\frac{\\bar{c}}{2}=-E-\\frac{\\bar{c}}{2}\n =B^{2}+\\frac{1}{4},\\qquad R=0.\n\\end{align}\nThe last equality $R=0$ means from the results in Section~\\ref{sec:psusy}\nthat the BQR SSUSY model also has second-order paraSUSY.\nWe note that the reason why the BQR SSUSY model is realized as the\nparticular case $b_{2}=b_{0}=0$ of the most general type A 2-fold SUSY\nis the same as the one discussed in Ref.~\\cite{GT06}, Section~5.\n\n\n\\section{Discussion and Summary}\n\\label{sec:discus}\n\nIn this article, we have investigated in detail under what conditions\ntype A $\\mathcal{N}$-fold SUSY systems can have intermediate Hamiltonians in\nthe case of $\\mathcal{N}=2$. It turns out that although type A 2-fold\nsupercharge admits a one-parameter family of factorization into\nproduct of two first-order linear differential operators due to\nthe underlying $GL(2,\\mathbb{C})$ symmetry, at most two different\nintermediate Hamiltonians are admissible. As a by product of the\nstudies, we have also obtained the necessary and sufficient conditions\nfor a type A 2-fold SUSY system to possess paraSUSY of order 2 as well.\nWhen it is the case, the type A 2-fold superalgebra together with\nthe second-order parasuperalgebra constitute a generalized 2-fold\nsuperalgebra. As a demonstration of the general arguments, we have\nconstructed the generalized P\\\"{o}schl--Teller potentials which\nare components of type A 2-fold SUSY with two intermediate Hamiltonians\nand reduce to the BQR SSUSY model in a particular case.\n\nAs for the concept like the reducibility in Ref.~\\cite{AICD95},\nthe present investigations indicate that it would be more natural\nand useful to classify higher-order intertwining operators according\nto the existence and the number of intermediate Hamiltonians as has\nbeen done in Table~\\ref{tb:numb}. After employing the latter\nclassification scheme, we can further classify them according to\nthe properties of the intermediate Hamiltonians such as Hermiticity,\n$\\mathcal{P}\\mathcal{T}$ symmetry, and so on.\n\nRegarding the generalized P\\\"{o}schl--Teller potentials constructed in\nSection~\\ref{sec:appli}, it is worth noticing that the framework of\n$\\mathcal{N}$-fold SUSY works well even when the function $A(z)$, which\ncontrols the change of variable $z=z(x)$ from the physical coordinate\n$x$ to the variable $z$ in the gauged space, is a transcendental\nfunction of $z$ without destroying quasi-solvability. For all\n$\\mathcal{N}\\geq3$ cases type A $\\mathcal{N}$-fold SUSY requires the additional\ncondition (\\ref{eq:condA}) so that $A(z)$ is allowed to be at most\na polynomial of fourth-degree in $z$, which results in the admissible\nchange of variable to be at most an elliptic function, see, e.g.,\nRef.~\\cite{Ta03a}. For the $\\mathcal{N}=2$ case, on the other hand, there\nare no such restrictions and, to the best of our knowledge, our\ngeneralized P\\\"{o}schl--Teller potentials are the first quasi-solvable\nexamples where $A(z)$ is given by a transcendental function of $z$\nas Eq.~(\\ref{eq:Aofz}).\n\nThe analyses for $\\mathcal{N}=2$ carried out in this article are easily\ngeneralized to the cases $\\mathcal{N}\\geq3$, but we anticipate that richer\nstructure could emerge for the higher $\\mathcal{N}$ cases. In the case of\n$\\mathcal{N}=3$, for instance, according to the factorization of type A 3-fold\nsupercharge $P_{3}^{-}=P_{31}^{-}P_{32}^{-}P_{33}^{-}$ we can consider\nnot only the case where intermediate Hamiltonians between $P_{31}^{-}$\nand $P_{32}^{-}$ and between $P_{32}^{-}$ and $P_{33}^{-}$ both exist,\nbut also the cases where they exist only between the former place or\nonly between the latter place exclusively. It is also interesting to\nstudy whether or not type A $\\mathcal{N}$-fold SUSY systems for higher $\\mathcal{N}$,\nwhen they have intermediate Hamiltonians, can admit another symmetry.\nThe fact that in the case of $\\mathcal{N}=2$ they have second-order paraSUSY\nindicates that they could have higher-order paraSUSY~\\cite{To92,Kh92}\nfor $\\mathcal{N}\\geq3$. Indeed, it was shown in Ref.~\\cite{Ta07c} that a\ncertain realization of paraSUSY of order 3 also admits a generalized\n3-fold superalgebra. Hence, at least in the case of $\\mathcal{N}=3$ we have\na reasonable basis to expect paraSUSY as an additional symmetry.\nOther candidates might be quasi-paraSUSY introduced in\nRef.~\\cite{Ta07a} and $\\mathcal{N}$-fold paraSUSY in Ref.~\\cite{Ta07b}.\n\n\n\\begin{acknowledgments\n This work (T.Tanaka) was partially supported by the National Cheng Kung\n University under the grant No.\\ OUA:95-3-2-071.\n\\end{acknowledgments\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}