Content-Type: multipart/mixed; boundary="-------------0301140855301" This is a multi-part message in MIME format. ---------------0301140855301 Content-Type: text/plain; name="03-14.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-14.comments" e-mail : vrousse@ujf-grenoble.fr ---------------0301140855301 Content-Type: text/plain; name="03-14.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-14.keywords" Born-Oppenheimer Approximation , Landau-Zener , Eigenvalue Avoided Crossings ---------------0301140855301 Content-Type: application/x-tex; name="LZtACfBO.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="LZtACfBO.tex" \documentclass[a4paper,12pt]{article} \newtheorem{theo}{Theorem} \newtheorem{prop}{Proposition} \newtheorem{defi}{Definition} \newtheorem{lem}{Lemma} \newenvironment{dem}{{\bf Proof }}{$\Box$} \newenvironment{idee}{{\bf Sketch of proof }}{$\Box$} \newenvironment{rmq}{{\bf Remark }}{} \newenvironment{rmqs}{{\bf Remarks}\begin{enumerate} \item}{\end{enumerate}} \newcommand{\h}{\hbar} \newcommand{\eps}{\varepsilon} \newcommand{\sd}{(s,\delta)} \newcommand{\td}{(t,\delta)} \newcommand{\te}{(t,\varepsilon)} \newcommand{\de}{(\delta,\varepsilon)} \newcommand{\sde}{(s,\delta,\varepsilon)} \newcommand{\tde}{(t,\delta,\varepsilon)} \newcommand{\utd}{(\underline{t},\underline{\delta})} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Aa}{\mathcal{A}} \newcommand{\Bb}{\mathcal{B}} \newcommand{\Cc}{\mathcal{C}} \newcommand{\Dd}{\mathcal{D}} \newcommand{\Ff}{\mathcal{F}} \newcommand{\Hh}{\mathcal{H}} \newcommand{\Mm}{\mathcal{M}} \newcommand{\Oo}{\mathcal{O}} \newcommand{\Ss}{\mathcal{S}} \newcommand{\Uu}{\mathcal{U}} \newcommand{\Vv}{\mathcal{V}} \newcommand{\vol}{\mathrm{vol}} \newcommand{\sgn}{\mathrm{sgn}} \newcommand{\Tr}{\mathrm{Tr}} \newcommand{\Com}{\mathrm{Com}} \newcommand{\Sym}{\mathrm{Sym}} \newcommand{\Diag}{\mathrm{Diag}} \newcommand{\Hess}{\mathrm{Hess}} \usepackage{amssymb,epsf,graphics} \addtolength{\hoffset}{-1cm} \addtolength{\textwidth}{2cm} \addtolength{\voffset}{-1.5cm} \addtolength{\textheight}{2cm} \begin{document} \title{\sc Landau-Zener Transitions for Eigenvalue Avoided Crossings in the Adiabatic and Born-Oppenheimer Approximations} \author{Vidian ROUSSE \\ Institut Fourier \\ BP 74, 38402 Saint Martin d'H\`eres Cedex (France) \\ e-mail: vrousse@ujf-grenoble.fr} \date{December, 2002} \maketitle \begin{abstract} In the Born-Oppenheimer approximation context, we study the propagation of Gaussian wave packets through the simplest type of eigenvalue avoided crossings of an electronic Hamiltonian $\Cc^4$ in the nuclear position variable. It yields a two-parameter problem: the mass ratio $\eps^4$ between electrons and nuclei and the minimum gap $\delta$ between the two eigenvalues. We prove that, up to first order, the Landau-Zener formula correctly predicts the transition probability from a level to another when the wave packet propagates through the avoided crossing in the two different regimes: $\delta$ being either asymptotically smaller or greater than $\eps$ when both go to $0$. \end{abstract} \section{Introduction} The Hamiltonian for a molecular system with $K$ nuclei and $N-K$ electrons has the form \begin{equation} \label{hamilreal} H(\eps)=-\sum^K_{j=1}\frac{\eps^4}{2M_j}\Delta_{x_j}-\sum^N_{j=K+1}\frac{1}{2m_j}\Delta_{x_j}+\sum_{i0$ and the $O$ are to be understood in the limit $\|x\|$ and $\delta$ going to $0$. In practice, type I avoided crossings occur for diatomic molecules, where the electron energy levels depend only on the distance between the nuclei because of rotational symmetry. Our main result is the determination of what happens when a standard time-depen\-dent Born-Oppenheimer molecular wave packet propagates through those avoided crossing if the gap size $\delta$ is either asymptotically smaller or greater than $\eps$. Our analysis, together with the results of \cite{HagJoy} where the critical case $\delta=\eps$ is considered, allow to get a complete picture of the dynamics through those avoided crossings for all ranges of $(\delta,\eps)\rightarrow0$ with a regularity of order $\Cc^4$ only on the electronic Hamiltonian $h(x,\delta)$. Using matched asymptotic expansions already used in \cite{HagJoy}, we compute approximate solutions to the molecular Schr\"odinger equation. We observe that, to leading order in $\delta$ and $\eps$, the Landau-Zener formula (see \cite{Joye}) correctly describes the probabilities for the system to remain in the original electronic level or to make a transition to the other electronic level involved in the avoided crossing (Theorem \ref{maintheo} in Section \ref{mainresult}). To apply the Landau-Zener formula in this case, one treats the nuclei as classical point particles to obtain a time-dependent Hamiltonian for the electrons. More precisely, suppose there is a generic type I avoided crossing at nuclear configuration $x=0$. In an appropriate coordinate system, the gap between the electron energy levels is $$2r\sqrt{x_1^2+\delta^2}+O(\Vert x\Vert^2+\delta^2)\ ,$$ with $r>0$. Suppose that a semi-classical nuclear wave packet passes through the avoided crossing with velocity $\mu$, whose first component is $\mu_1\neq0$. Then the Landau-Zener formula predicts that the probability of remaining in the same electronic state is $1-e^{-\pi r\delta^2/(\mu_1\eps^2)}$, and the probability of making a transition to the other electronic level involved in the avoided crossing is $e^{-\pi r\delta^2/(\mu_1\eps^2)}$. This formula yields a transition probability, at leading order, $1$ (respectively $0$) when $\delta/\eps\rightarrow0$ (respectively $+\infty$). Mimicking our analysis in the Born-Oppenheimer context, we also get results for the following simpler problem in the adiabatic context: we want to solve the time-dependent Schr\"odinger equation \begin{equation} \label{eqzero} i\eps^2\frac{\partial}{\partial t}\psi=H\td\psi \end{equation} where $H\td$ is a family of self-adjoint operators with fixed domain $\Dd$ (in any separable Hilbert space $\Hh$) and whose resolvent is strongly $\Cc^4$ in $\td\in]t_0-2T+t_0+2T[\times]-2\delta_0,2\delta_0[$. We assume that $H\td$ displays the simplest case of avoided crossing at $t_0$ (see section \ref{nfadiab} for details) and establish the validity of the Landau-Zener formula to leading order for the two same regimes $\delta$ either asymptotically smaller or greater than $\eps$ (the critical case where $\delta=\eps$ can be found in \cite{Hag} for $H\td$ real symmetric). The result is stated in Theorem \ref{secondtheo} of Section \ref{casadiab}. In a $\Cc^{\infty}$ context, similar microlocal results based on pseudodifferential techniques can be found in \cite{Colin}. The organization of the paper is as follows. Section \ref{wavepacket} gives usual tools for constructing the leading order Born-Oppenheimer approximation with $\delta$ fixed (the eigenvalues do not cross, are simple and isolated). Section \ref{asymptotics} deals with the asymptotics of the classical quantities of the problem. Sections \ref{away} and \ref{near} give the different Ans\"atze used respectively far from and close to the crossing surface $\Gamma$. Section \ref{matching} makes the matching of those different Ans\"atze in an overlapping region, and Section \ref{mainresult} states the main result. Finally, Section \ref{casadiab} deals with avoided crossings in the adiabatic context. \section{Coherent States and Classical Dynamics} \label{wavepacket} We recall the definition of the coherent states $\varphi_l(A,B,\h,a,\eta,x)$ that are described in detail in \cite{Hag2}. A more explicit, but more complicated definition is given in \cite{Hag5}. We adopt the standard multi-index notation. A multi-index $l=(l_1,\ldots,l_d)$ is a $d$-tuple of non-negative integers. We define $|l|=\sum^d_{k=1}l_k$, $x^l=x^{l_1}_1\ldots x^{l_d}_d$, $l!=(l_1!)\ldots(l_d!)$, and $D^l=\frac{\partial^{|l|}}{(\partial x_1)^{l_1}\ldots(\partial x_d)^{l_d}}$. Throughout the paper we assume $a\in\R^d$, $\eta\in\R^d$ and $\h>0$. We also assume that $A$ and $B$ are $d\times d$ complex invertible matrices that satisfy \begin{equation} \label{surfadm} \begin{array}{rcl} A^tB-B^tA & = & 0\ , \\ A^*B+B^*A & = & 2I\ . \end{array} \end{equation} These conditions guarantee that both the real and imaginary parts of $BA^{-1}$ are symmetric. Furthermore, the real part of $BA^{-1}$ is strictly positive definite and has inverse $AA^*$. Our definition of $\varphi_l(A,B,\h,a,\eta,x)$ is based on the following raising operators defined for $j=1,\ldots,d$ by $$\Aa_j(A,B,\h,a,\eta)^*=\frac{1}{\sqrt{2\h}}\left[\sum^d_{k=1}\overline{B_{kj}}(x_k-a_k)-i\sum^d_{k=1}\overline{A_{kj}}\left(-i\h\frac{\partial}{\partial x_j}-\eta_j\right)\right]\ .$$ The corresponding lowering operators $\Aa_j(A,B,\h,a,\eta)$ are their formal adjoints. These operators satisfy the following useful commutation relations : the raising operators $\Aa_j(A,B,\h,a,\eta)^*$ for $j=1,\ldots,d$ commute with one another, the lowering operators $\Aa_j(A,B,\h,a,\eta)$ commute with one another, however, for $j,k=1,\ldots,d$ $$\Aa_j(A,B,\h,a,\eta)\Aa_k(A,B,\h,a,\eta)^*-\Aa_k(A,B,\h,a,\eta)^*\Aa_j(A,B,\h,a,\eta)=\delta_{jk}\ .$$ For the multi-index $l=0$, we define the normalized complex Gaussian wave packet (modulo the sign of a square root) by $$\varphi_0(A,B,\h,a,\eta,x)=(\pi\h)^{-d/4}(\det A)^{-1/2}\exp\left(-\frac{\left\langle x-a,BA^{-1}(x-a)\right\rangle}{2\h}+i\frac{\left\langle\eta,x-a\right\rangle}{\h}\right)\ .$$ Then, for any non-zero multi-index $l$, we define $$\varphi_l(A,B,\h,a,\eta,\cdot)=\frac{1}{\sqrt{l!}}\Aa_1(A,B,\h,a,\eta)^{*l_1}\ldots\Aa_d(A,B,\h,a,\eta)^{*l_d}\varphi_0(A,B,\h,a,\eta,\cdot)\ ,$$ $$\phi_l(A,B,y)=\varphi_l(A,B,1,0,0,y)\ .$$ We have the following properties \begin{enumerate} \item For $A=B=I$, $\h=1$ and $a=\eta=0$, the $\varphi_l(A,B,\h,a,\eta,\cdot)$ are just the standard harmonic oscillator eigenstates with energies $|l|+d/2$. \item For each admissible $A$, $B$, $\h$, $a$ and $\eta$, the set $\left(\varphi_l(A,B,\h,a,\eta,\cdot)\right)_{l\in\N^d}$ is an orthonormal basis for $L^2(\R^d;\C)$. \item In \cite{Hag5}, the state $\varphi_l(A,B,\h,a,\eta,x)$ is defined as a normalization factor times $$H_l(A;\h^{-1/2}|A|^{-1}(x-a))\varphi_0(A,B,\h,a,\eta,x)\ .$$ Here $H_l(A;y)$ is a recursively defined $|l|^{th}$ order polynomial in $y$ that depends on $A$ only through $U_A$, where $A=|A|U_A$ is the polar decomposition of $A$. \item When the dimension $d$ is $1$, the position and momentum uncertainties of the $\varphi_l(A,B,\h,a,\eta,\cdot)$ are $\sqrt{(l+1/2)\h}|A|$ and $\sqrt{(l+1/2)\h}|B|$, respectively. In higher dimensions, they are bounded by $\sqrt{(|l|+d/2)\h}\Vert A\Vert$ and $\sqrt{(|l|+d/2)\h}\Vert B\Vert$, respectively. \item When we approximately solve the Schr\"odinger equation, the choice of the sign of the square root in the definition of $\varphi_0(A,B,\h,a,\eta,\cdot)$ is determined by continuity in time after an arbitrary initial choice. \item The behaviour of $\varphi_l(A,B,\h,a,\eta,\cdot)$ through small perturbations of parameters $A$, $B$ and $a$ is the following \begin{eqnarray*} \lefteqn{\left\Vert\varphi_l(A,B,\h,a,\eta,\cdot)-\varphi_l(A_0,B_0,\h,a_0,\eta,\cdot)\right\Vert_{L^2}} \\ & & \leq C_l(A_0,B_0,a_0)\left[\Vert A-A_0\Vert+\Vert B-B_0\Vert+\frac{\Vert a-a_0\Vert}{\sqrt{\h}}\right]\ , \end{eqnarray*} for $(A,B)$ in a neighbourhood of $(A_0,B_0)$. This estimation is already mentioned in \cite{HagJoy}, but the proof needs some modification (we cannot treat each matrix variable separately) : generalize the one-dimensional formulae of propositions $4$ and $7$ of \cite{HagRob} and give asymptotics when $A_2-A_1$ and $B_2-B_1$ are small. \item If we fix a cutoff function $F\in\Cc^{\infty}(\R_+;[0,1])$ (with $F(x)=1$ for $x\leqslant 1$ and $F(x)=0$ for $x\geqslant 2$), we have the following estimates \begin{eqnarray} \lefteqn{\Vert(1-F)^{(n)}(\gamma^2\Vert y\Vert^2)\phi_l(A,B,y)\Vert_{L^2(\R^d;\C)}} \nonumber \\ & & \leq C_{l,n}\left[1+\left(\Vert A\Vert\gamma\right)^{|l|}\right]\left(\Vert A\Vert\gamma\right)^{d/2}e^{-d\left\Vert A\right\Vert^{-2}\gamma^{-2}}\ , \label{majotroncatureun} \end{eqnarray} \begin{eqnarray} \lefteqn{\Vert(1-F)^{(n)}(\gamma^2\Vert y\Vert^2)y.\nabla_y\phi_l(A,B,y)\Vert} \nonumber \\ & & \leq C'_{l,n}\Vert A\Vert.\Vert B\Vert\left[1+\left(\Vert A\Vert\gamma\right)^{|l|+2}\right]\left(\Vert A\Vert\gamma\right)^{d/2}e^{-d\left\Vert A\right\Vert^{-2}\gamma^{-2}} \label{majotroncaturedeux} \end{eqnarray} for $n\geqslant 0$ and $l\in\N^d$ when $\gamma$ tends to $0$. \end{enumerate} In the Born-Oppenheimer approximation, the semi-classical dynamics of the nuclei is generated by an effective potential given by a chosen isolated electronic eigenvalue $E(x,\delta_0)$ of the electronic Hamiltonian $h(x,\delta_0)$, $x\in\R^d$ (we keep $\delta$ fixed). For a given effective potential $E(x,\delta_0)$ we describe the semi-classical dynamics of the nuclei by means of the time dependent basis constructed as follows. Associated to $E(x,\delta_0)$, we have the following classical equations of motion \begin{equation} \label{dynclass} \begin{array}{rcl} \dot{a}(t) & = & \eta(t)\ , \\ \dot{\eta}(t) & = & -\nabla_xE(a(t),\delta_0)_ , \\ \dot{A}(t) & = & iB(t)\ , \\ \dot{B}(t) & = & i\Hess_xE(a(t),\delta_0)A(t)\ , \\ \dot{S}(t) & = & \frac{1}{2}\Vert\eta(t)\Vert^2-E(a(t),\delta_0)\ . \end{array} \end{equation} We always assume the initial condition $(A(0),B(0))$ satisfies (\ref{surfadm}). The matrices $A(t)$ and $B(t)$ are related to the linearization of the classical flow through the following identities \begin{eqnarray*} A(t) & = & \frac{\partial a(t)}{\partial a(0)}A(0)+i\frac{\partial a(t)}{\partial\eta(0)}B(0)\ , \\ B(t) & = & \frac{\partial\eta(t)}{\partial\eta(0)}B(0)-i\frac{\partial\eta(t)}{\partial a(0)}A(0)\ . \end{eqnarray*} Furthermore, it is not difficult to prove that conditions (\ref{surfadm}) are preserved by the flow. The usefulness of those wave packets stems from the following important property. If we decompose the potential as $$E(x,\delta_0)=W_a(x,\delta_0)+[E(x,\delta_0)-W_a(x,\delta_0)]$$ where $W_a(x,\delta_0)$ denotes the second order Taylor expansion $$W_a(x,\delta_0)=E(a,\delta_0)+\nabla_xE(a,\delta_0)(x-a)+\langle x-a,\frac{\Hess_xE(a,\delta_0)}{2}(x-a)\rangle$$ then for all multi-indices $l$, \begin{eqnarray*} \lefteqn{i\h\frac{\partial}{\partial t}\left[e^{iS(t)/\h}\varphi_l(A(t),B(t),\h,a(t),\eta(t),x)\right]} \\ & & =\left(-\frac{\h^2}{2}\Delta_x+W_{a(t)}(x,\delta_0)\right)\left[e^{iS(t)/\h}\varphi_l(A(t),B(t),\h,a(t),\eta(t),x)\right] \end{eqnarray*} if $a(t)$, $\eta(t)$, $A(t)$, $B(t)$ and $S(t)$ satisfy (\ref{dynclass}). In other words, those semi-classical wave packets $\varphi_l$ exactly take into account the kinetic energy and quadratic part $W_{a(t)}(x,\delta_0)$ of the potential when propagated by means of the classical flow and its linearization around the classical trajectory selected by the initial conditions. Then, the leading order Born-Oppenheimer approximation for (\ref{equn'}) is \begin{equation} \label{BOlo} \psi(t,x,\delta_0)=e^{\frac{i}{\h}S(t)}\varphi_l(A(t),B(t),\h,a(t),\eta(t),x)\Phi_E(x,\delta_0) \end{equation} where $\Phi_E(x,\delta_0)$ denotes a particular smooth normalized eigenvector associated to the eigenvalue $E(x,\delta_0)$ (see \cite{Hag3}). \section{Asymptotics of Classical Quantities} \label{asymptotics} In our case, we have a supplementary parameter $\delta$ and we deal with two eigenvalues isolated from the rest of the spectrum but that do approach one another. This leads to two different classical dynamics (one for each eigenvalue). Close to the crossing surface, those two dynamics almost reduce to the one corresponding to the mean of those two. For each of those three, we will now give their asymptotics in a neighbourhood of the crossing surface $\Gamma$. We define $$\rho(x,\delta)=\sqrt{b(x,\delta)^2+c(x,\delta)^2+d(x,\delta)^2}\ ,$$ $$E_{\Cc}(x,\delta):=E(x,\delta)+\nu^{\Cc}\rho(x,\delta)$$ where $\nu^{\Aa}=1$, $\nu^{\Bb}=-1$ and we choose $\eta^0,\eta^{0^{\Cc}}\in\Cc^0([-\delta_0,\delta_0];\R^d)$ with $\eta^0(\delta)=\eta^0+O(\delta)$ and $\eta^{0^{\Cc}}(\delta)=\eta^0+O(\delta)$ where the first component of the vector $\eta^0$ satisfies $\eta^0_1>0$. We solve the following systems with the corresponding initial conditions \begin{equation} \label{sysmoy} \left\{ \begin{array}{rcl} \dot{a}(t,\delta) & = & \eta (t,\delta) \\ \dot{\eta }(t,\delta) & = & -\nabla_xE(a(t,\delta),\delta) \\ \dot{A}(t,\delta) & = & iB(t,\delta) \\ \dot{B}(t,\delta) & = & i\Hess_xE(a(t,\delta),\delta) A(t,\delta) \\ \dot{S}(t,\delta) & = & \frac{1}{2}\Vert\eta (t,\delta)\Vert^2-E(a(t,\delta),\delta) \end{array} \right. \qquad,\qquad \left\{ \begin{array}{rcl} a(0,\delta) & = & 0 \\ \eta(0,\delta) & = & \eta^0(\delta) \\ A(0,\delta) & = & A_0 \\ B(0,\delta) & = & B_0 \\ S(0,\delta) & = & 0 \end{array} \right. \quad, \end{equation} \begin{equation} \label{syscomp} \left\{ \begin{array}{rcl} \dot{a}^{\Cc}(t,\delta) & = & \eta^{\Cc}(t,\delta) \\ \dot{\eta }^{\Cc}(t,\delta) & = & -\nabla_xE_{\Cc}(a^{\Cc}(t,\delta),\delta) \\ \dot{A}^{\Cc}(t,\delta) & = & iB^{\Cc}(t,\delta) \\ \dot{B}^{\Cc}(t,\delta) & = & i\Hess_xE_{\Cc}(a^{\Cc}(t,\delta),\delta) A^{\Cc}(t,\delta) \\ \dot{S}^{\Cc}(t,\delta) & = & \frac{1}{2}\Vert\eta^{\Cc}(t,\delta)\Vert^2-E_{\Cc}(a^{\Cc}(t,\delta),\delta) \end{array} \right. \qquad,\qquad \left\{ \begin{array}{rcl} a^{\Cc}(0,\delta) & = & 0 \\ \eta^{\Cc}(0,\delta) & = & \eta^{0^{\Cc}}(\delta) \\ A^{\Cc}(0,\delta) & = & A_0 \\ B^{\Cc}(0,\delta) & = & B_0 \\ S^{\Cc}(0,\delta) & = & 0 \end{array} \right. \quad. \end{equation} We note that the initial momenta can differ by a term of order $O(\delta)$ ; we will explain why in section \ref{consener}. As in \cite{HagJoy}, Picard fixed point theorem techniques yield \begin{prop} \label{prop1} The solutions of differential systems (\ref{sysmoy}) and (\ref{syscomp}) have the following asymptotics when $t$ and $\delta$ tend to $0$ \begin{eqnarray*} a(t,\delta) & = & \eta^0(\delta)t-\nabla_xE(0,\delta)\frac{t^2}{2}+O(t^{3})\ , \\ \eta (t,\delta) & = & \eta^0(\delta)-\nabla_xE(0,\delta)t+O(t^2) \end{eqnarray*} (those two are uniform in $\delta$) ; \begin{eqnarray*} a^{\Cc}(t,\delta) & = & \eta^{0^{\Cc}}(\delta)t-\nabla_xE(0,\delta)\frac{t^2}{2}+O(|t|^{3}+\delta t^2)-\nu^{\Cc}\frac{r}{2\eta^{0^{\Cc}}_1(\delta)}\left(\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array}\right) \\ & & \times\left[t\sqrt{(\eta^{0^{\Cc}}_1(\delta)t)^2+\delta^2}+\frac{\delta^2}{\eta^{0^{\Cc}}_1(\delta)}\ln\left(\frac{\eta^{0^{\Cc}}_1(\delta)t+\sqrt{(\eta^{0^{\Cc}}_1(\delta)t)^2+\delta^2}}{\delta}\right)-2\delta t\right]\ , \\ \eta^{\Cc}(t,\delta) & = & \eta^{0^{\Cc}}(\delta)-\nabla_xE(0,\delta)t+O(t^2+\delta|t|) \\ & & {}-\nu^{\Cc}\frac{r}{\eta^{0^{\Cc}}_1(\delta)}\left(\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array}\right)\left[\sqrt{(\eta^{0^{\Cc}}_1(\delta)t)^2+\delta^2}-\delta\right]\ ; \end{eqnarray*} \begin{eqnarray*} S(t,\delta) & = & \left(\frac{1}{2}\Vert\eta^0(\delta)\Vert^2-E(0,\delta)\right)t-\eta^0(\delta).\nabla_xE(0,\delta)t^2+O(t^{3})\ , \\ S^{\Cc}(t,\delta) & = & \left(\frac{1}{2}\Vert\eta^{0^{\Cc}}(\delta)\Vert^2-E(0,\delta)\right)t-\eta^{0^{\Cc}}(\delta).\nabla_xE(0,\delta)t^2+\nu^{\Cc}r\delta t+O(t^{3}+\delta^2t) \\ & & {}-\nu^{\Cc}r\left[t\sqrt{(\eta^{0^{\Cc}}_1(\delta)t)^2+\delta^2}+\frac{\delta^2}{\eta^{0^{\Cc}}_1(\delta)}\ln\left(\frac{\eta^{0^{\Cc}}_1(\delta)t+\sqrt{(\eta^{0^{\Cc}}_1(\delta)t)^2+\delta^2}}{\delta}\right)\right]\ ; \end{eqnarray*} \begin{eqnarray*} A^{\Cc}(t,\delta) & = & A_0+O(t)\ , \\ B^{\Cc}(t,\delta) & = & B_0+\nu^{\Cc}ir\left(\begin{array}{cccc} 1 & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & & \vdots \\ 0 & 0 & \cdots & 0 \end{array}\right) A_0\frac{t}{\sqrt{(\eta_1^{0^{\Cc}}(\delta)t)^2+\delta^2}}+O(|t|+\delta)\ . \end{eqnarray*} \end{prop} Throughout the rest of this paper, we will drop the $\delta$-dependence of those quantities (in the notation only). \section{Away from the Crossing} \label{away} We fix a cutoff function $F\in\Cc^{\infty}(\R_+;[0,1])$ with $F(x)=1$ for $x\leqslant 1$ and $F(x)=0$ for $x\geqslant 2$. \subsection{Crossing Surface vs Cutoff Zone} \label{csvscoz} We introduce the following sets that are suggested in figure \ref{mafigure}. \begin{figure}[h] \begin{center} \scalebox{.7}{\input{crosscutoff.pstex_t}} \end{center} \caption{Classical propagation of the nuclei through the crossing surface \label{mafigure}} \end{figure} For $\delta\in]-2\delta_0,2\delta_0[$, we define $$Z_-(\delta)=\left\{x\in\R^d/b(x,\delta)<\frac{1}{2}\rho(x,\delta)\right\}\ ,\ Z_+(\delta)=\left\{x\in\R^d/b(x,\delta)>-\frac{1}{2}\rho(x,\delta)\right\}$$ the two overlapping zones where $h_1(x,\delta)$ avoids some diagonal form, $$J(\delta)=\left\{x\in\R^d/\rho(x,\delta)>\frac{r}{2}\sqrt{x_1^2+\delta^2}\right\}$$ the zone where the gap is well bounded from below, $$U^{\Cc}(t,\delta,\eps,\gamma)=\left\{x\in\R^d/\Vert x-a^{\Cc}(t,\delta)\Vert_{\infty}\leqslant\sqrt{2}\frac{\eps}{\gamma}\right\}$$ for $t\in[-T,T]$ the cutoff zone at time $t$, and finally $$W^{\Cc}_-(t,\delta,\eps,\gamma)=\bigcup_{\tau\in [-T,t] }U^{\Cc}(\tau,\delta,\eps,\gamma)\ ,\ W^{\Cc}_+(t,\delta,\eps,\gamma)=\bigcup_{\tau\in [t,T] }U^{\Cc}(\tau,\delta,\eps,\gamma)$$ the two disjoint cutoff zones in the incoming and outgoing time intervals. Anticipating estimates required for the proof of Proposition \ref{lementrant}, we give a precise statement of how close to the crossing surface we can approach if we want to control \begin{itemize} \item the gap between the two eigenvalues $E^{\Aa}(x,\delta)$ and $E^{\Bb}(x,\delta)$, \item the deformation of the eigenvectors $\Phi^*_{\Cc}(x,\delta)$ defined in section \ref{caaose} for $\Cc=\Aa,\Bb$ and $*=+,-$. \end{itemize} \begin{lem} If $\delta_0$ and $T$ are small enough, then for all $\delta$ in $]0,\delta_0]$, $\eps$ and $\gamma$ in $]0,1]$, and $t$ in $]0,T]$ such that $$\frac{\eps}{\gamma}\leqslant\frac{\eta^0_1}{8\sqrt{2}}t\ ,$$ we have $W_-^{\Cc}(-t,\delta,\eps,\gamma)\subseteq(Z_-(\delta)\cap J(\delta))$ and $W_+^{\Cc}(t,\delta,\eps,\gamma)\subseteq(Z_+(\delta)\cap J(\delta))$. \end{lem} \begin{dem} We prove only the first part, the other is analogous. We denote by $C_a$, $C_{\eta}$, $C_b$, $C_c$, $C_d$ and $C_{\rho}$ strictly positive constants such that for every $(t,x,\delta)\in [-T,T]\times\overline{B}(0,\kappa)\times[0,\delta_0]$, we have $$\Vert a^{\Cc}(t,\delta)-\eta^{0^{\Cc}}(\delta) t\Vert_{\infty}\leqslant C_at^2\ ,\ \Vert\eta^{0^{\Cc}}(\delta)-\eta^0\Vert\leqslant C_{\eta }\delta\ ,$$ $$|b(x,\delta)-rx_1|\leqslant C_b(\Vert x\Vert^2_{\infty}+\delta^2)\ ,\ |\rho(x,\delta)^2-r^2(x_1^2+\delta^2)|\leqslant C_{\rho}(\Vert x\Vert^2_{\infty}+\delta^2)^{3/2}\ .$$ If $x\in W_-^{\Cc}(t,\delta,\eps,\gamma)$, we have $$x_1=\eta_1^0t+\left(x_1-a_1^{\Cc}(t,\delta)\right)+\left(a_1^{\Cc}(t,\delta)-\eta_1^{0^{\Cc}}(\delta)t\right)+\left(\eta_1^{0^{\Cc}}(\delta)t-\eta_1^0t\right)\ ,$$ but $$|x_1-a_1^{\Cc}(t,\delta)|\leqslant\sqrt{2}\frac{\eps}{\gamma}\ ,\ |a_1^{\Cc}(t,\delta)-\eta_1^{0^{\Cc}}(\delta)t|\leqslant C_at^2\ ,\ |\eta_1^{0^{\Cc}}(\delta)-\eta_1^0|\leqslant C_{\eta }\delta\ ,$$ hence, if $\eps/\gamma\leqslant\eta_1^0|t|/(8\sqrt{2})$, $|t|\leqslant\eta_1^0/(8C_a)$ and $\delta\leqslant\eta^0_1/(2C_{\eta})$, $$x_1\leqslant\eta^0_1t-\frac{\eta^0_1}{8}t-\frac{\eta^0_1}{8}t-\frac{\eta^0_1}{2}t=\frac{\eta^0_1}{4}t<0\ .$$ Moreover, $$x=\eta^0t+\left(x-a^{\Cc}(t,\delta)\right)+\left(a^{\Cc}(t,\delta)-\eta^{0^{\Cc}}(\delta)t\right)+\left(\eta^{0^{\Cc}}(\delta)t-\eta^0t\right)\ ,$$ hence $$\Vert x\Vert_{\infty}\leqslant\Vert\eta^0\Vert_{\infty}|t|+\sqrt{2}\frac{\eps}{\gamma }+C_at^2+C_{\eta }\delta |t|\leqslant (\Vert\eta^0\Vert_{\infty}+\frac{3\eta_1^0}{4}) |t|=D_{t}|t|\ .$$ Let us show first condition $W_-^{\Cc}(t,\delta,\eps,\gamma)\subseteq J(\delta)$ : from above, we have $$\frac{\rho(x,\delta)^2}{r^2(x_1^2+\delta^2)}\geqslant 1-\frac{C_{\rho}}{r^2}\frac{(\Vert x\Vert^2_{\infty}+\delta^2)^{3/2}}{(x_1^2+\delta^2)}\geqslant 1-\frac{C_{\rho}}{r^2}\frac{(D^2_{t}t^2+\delta^2)^{3/2}}{\frac{\eta_1^{0^2}}{16}t^2+\delta^2}$$ hence, if $t^2+\delta^2\leqslant(r^{4}\min(1,\eta_1^{0^{4}}/256))/(4C_{\rho}^2\max(1,D_{t}^{6}))$, $$\rho(x,\delta)^2\geqslant\frac{r^2}{2}(x_1^2+\delta^2)\ .$$ Let us show now condition $W_-^{\Cc}(t,\delta,\eps,\gamma)\subseteq Z_-(\delta)$ : $$b(x,\delta)=rx_1+(b(x,\delta)-rx_1)\frac{r}{2}\sqrt{\frac{\eta_1^{0^2}}{16}t^2+\delta^2}\geqslant R(|t|+\delta)$ for every $x$ in $U^{\Cc}(t,\delta,\eps,\gamma)$ and that $a^{\Cc}([-T,T],\delta)\subseteq J(\delta)$. \end{rmq} \subsection{Construction and Asymptotics of Selected Eigenvectors} \label{caaose} For $(x,\delta)\in\R^d\times]0,2\delta_0[$, we define $$B(x,\delta)=\frac{b(x,\delta)}{\rho(x,\delta)}\ ,\ C(x,\delta)=\frac{c(x,\delta)}{\rho(x,\delta)}\ ,\ D(x,\delta)=\frac{d(x,\delta)}{\rho(x,\delta)}\ ,$$ $$f^-(x,\delta)=\sqrt{\frac{1-B(x,\delta)}{2}}\ ,\ g^-(x,\delta)=\frac{C(x,\delta)+iD(x,\delta)}{\sqrt{2(1-B(x,\delta))}}$$ when $B(x,\delta)<1$, and $$f^+(x,\delta)=\sqrt{\frac{1+B(x,\delta)}{2}}\ ,\ g^+(x,\delta)=\frac{C(x,\delta)+iD(x,\delta)}{\sqrt{2(1+B(x,\delta))}}$$ when $B(x,\delta)>-1$. We define static eigenvectors by \begin{eqnarray*} \Phi^-_{\Aa}(x,\delta) & = & g^-(x,\delta)\psi_1(x,\delta)+f^-(x,\delta)\psi_2(x,\delta)\ , \\ \Phi^-_{\Bb}(x,\delta) & = & -f^-(x,\delta)\psi_1(x,\delta)+\overline{g^-(x,\delta)}\psi_2(x,\delta) \end{eqnarray*} when $B(x,\delta)<1$, and \begin{eqnarray*} \Phi^+_{\Aa}(x,\delta) & = & f^+(x,\delta)\psi_1(x,\delta)+\overline{g^+(x,\delta)}\psi_2(x,\delta)\ , \\ \Phi^+_{\Bb}(x,\delta) & = & -g^+(x,\delta)\psi_1(x,\delta)+f^+(x,\delta)\psi_2(x,\delta) \end{eqnarray*} when $B(x,\delta)>-1$. We now turn to the asymptotics of those static eigenvectors around $(x,\delta)=(0,0)$. First in the same asymptotic time regime as in \cite{HagJoy}, we have \begin{lem} \label{lemasympfctpropretroit} When $\delta$, $\eps$ and $t$ tend to $0$, we have, uniformly in $\gamma\leqslant1$, $|\eps/(\gamma t)|\leqslant M$ and $|\delta/t|\leqslant M'$, for $t<0$, \begin{eqnarray*} \left\Vert F(\eps^{-2}\gamma^2\Vert x-a^{\Aa}(t)\Vert^2)\left[\Phi_{\Aa}^-(x,\delta)-\psi_2(x,\delta)\right]\right\Vert_{L^{\infty}} & = & O\left(|t|+\left|\frac{\delta}{t}\right|\right)\ , \\ \left\Vert F(\eps^{-2}\gamma^2\Vert x-a^{\Bb}(t)\Vert^2)\left[\Phi_{\Bb}^-(x,\delta)+\psi_1(x,\delta)\right]\right\Vert_{L^{\infty}} & = & O\left(|t|+\left|\frac{\delta}{t}\right|\right)\ , \end{eqnarray*} and for $t>0$, \begin{eqnarray*} \left\Vert F(\eps^{-2}\gamma^2\Vert x-a^{\Aa}(t)\Vert^2)\left[\Phi_{\Aa}^+(x,\delta)-\psi_1(x,\delta)\right]\right\Vert_{L^{\infty}} & = & O\left(|t|+\left|\frac{\delta}{t}\right|\right)\ , \\ \left\Vert F(\eps^{-2}\gamma^2\Vert x-a^{\Bb}(t)\Vert^2)\left[\Phi_{\Bb}^+(x,\delta)-\psi_2(x,\delta)\right]\right\Vert_{L^{\infty}} & = & O\left(|t|+\left|\frac{\delta}{t}\right|\right)\ . \end{eqnarray*} \end{lem} \begin{dem} If $x_1\neq 0$, $$\big|\rho(x,\delta)-r|x_1|\big|\leqslant\frac{r^2\delta^2+C_{\rho}(\Vert x\Vert_{\infty}^2+\delta^2)^{3/2}}{r|x_1|}\ .$$ Thus, for $x\in W_*^{\Cc}(t,\delta,\eps,\gamma)$ with $*=+,-$, \begin{eqnarray*} |B(x,\delta)-\sgn(t)| & = & \left|\frac{\sgn(t)b(x,\delta)-\rho(x,\delta)}{\rho(x,\delta)}\right| \\ & \leqslant & \frac{|b(x,\delta)-rx_1|+\big|\rho(x,\delta)-r|x_1|\big|}{\frac{r}{2}|x_1|} \\ & \leqslant & 2\frac{rC_b|x_1|(\Vert x\Vert_{\infty}^2+\delta^2)+r^2\delta^2+C_{\rho}(\Vert x\Vert_{\infty}^2+\delta^2)^{3/2}}{r^2x^2_1}\ , \end{eqnarray*} hence $$B(x,\delta)-\sgn(t)=O\left(|t|+\frac{\delta^2}{t^2}\right)\ .$$ Moreover, similar calculations yield $$C(x,\delta)+iD(x,\delta)=O\left(|t|+\left|\frac{\delta}{t}\right|\right)$$ which leads to the result. \end{dem} By similar considerations, we get also in the opposite asymptotic case \begin{lem} \label{lemasympfctproprlarge} When $\delta$, $\eps$ and $t$ tend to $0$, we have, uniformly in $\gamma\leqslant1$, $|\eps/(\gamma t)|\leqslant M$ and $|t/\delta|\leqslant M'$, for $t<0$, \begin{eqnarray*} \left\Vert F(\eps^{-2}\gamma^2\Vert x-a^{\Aa}(t)\Vert^2)\left[\Phi_{\Aa}^-(x,\delta)-\frac{\sqrt{2}}{2}(\psi_1(x,\delta)+\psi_2(x,\delta))\right]\right\Vert_{L^{\infty}} & = & O\left(\delta+\left|\frac{t}{\delta}\right|\right)\ , \\ \left\Vert F(\eps^{-2}\gamma^2\Vert x-a^{\Bb}(t)\Vert^2)\left[\Phi_{\Bb}^-(x,\delta)-\frac{\sqrt{2}}{2}(-\psi_1(x,\delta)+\psi_2(x,\delta))\right]\right\Vert_{L^{\infty}} & = & O\left(\delta+\left|\frac{t}{\delta}\right|\right)\ , \end{eqnarray*} and for $t>0$, \begin{eqnarray*} \left\Vert F(\eps^{-2}\gamma^2\Vert x-a^{\Aa}(t)\Vert^2)\left[\Phi_{\Aa}^+(x,\delta)-\frac{\sqrt{2}}{2}(\psi_1(x,\delta)+\psi_2(x,\delta))\right]\right\Vert_{L^{\infty}} & = & O\left(\delta+\left|\frac{t}{\delta}\right|\right)\ , \\ \left\Vert F(\eps^{-2}\gamma^2\Vert x-a^{\Bb}(t)\Vert^2)\left[\Phi_{\Bb}^+(x,\delta)-\frac{\sqrt{2}}{2}(-\psi_1(x,\delta)+\psi_2(x,\delta))\right]\right\Vert_{L^{\infty}} & = & O\left(\delta+\left|\frac{t}{\delta}\right|\right)\ . \end{eqnarray*} \end{lem} We introduce now dynamical eigenvectors $$\Phi^*_{\Cc}(t,x,\delta)=e^{i\omega_{\Cc}^*(t,x,\delta)}\Phi^*_{\Cc}(x,\delta)$$ for $\Cc=\Aa,\Bb$ and $*=+,-$ in order to fulfill the orthogonality condition $$\langle\Phi^*_{\Cc}(t,x,\delta),\left(\frac{\partial}{\partial t}+\eta^{\Cc}(t).\nabla_x\right)\Phi^*_{\Cc}(t,x,\delta)\rangle=0\ .$$ Introducing the new variables $s=t$, $z=x-a^{\Cc}(t)$, we have the sufficient condition \begin{equation} \label{phase} \frac{\partial}{\partial s}\tilde{\omega }^*_{\Cc}(s,z,\delta)=i\langle\Phi^*_{\Cc}(a^{\Cc}(s)+z,\delta),\eta^{\Cc}(s).\nabla_x\Phi^*_{\Cc}(a^{\Cc}(s)+z,\delta)\rangle \end{equation} where $\tilde{\omega }^*_{\Cc}(s,z,\delta)=\omega^*_{\Cc}(s,a^{\Cc}(s)+z,\delta)$. If we suppose $\omega^-_{\Cc}(-T,x,\delta)=\omega^+_{\Cc}(T,x,\delta)=0$, we have the following result \begin{lem} \label{lemasympphase} When $\delta$, $\eps$ and $t$ tend to $0$, we have, for $\Cc=\Aa,\Bb$ and $*=+,-$, $$\left\Vert F(\eps^{-2}\gamma^2\Vert x-a^{\Cc}(t)\Vert^2)\left[\Phi_{\Cc}^*(t,x,\delta)-e^{i\omega_{\Cc}^*(t,a^{\Cc}(t),\delta)}\Phi_{\Cc}^*(x,\delta)\right]\right\Vert_{L^{\infty}(\R^d;\Hh)}=O\left(\frac{\eps}{\gamma}\ln\frac{1}{|t|+\delta}\right)$$ uniformly in $\gamma\leqslant1$ and $|\eps/(\gamma t)|\leqslant M$. \end{lem} \begin{dem} Because of (\ref{phase}), we try to compare with the situation at $z=0$. We treat only the case $(\Cc,*)=(\Bb,-)$, others are analogous. Dropping the parameters $t$, $x$ and $\delta$, we get \begin{eqnarray*} \lefteqn{\langle\Phi^-_{\Bb}(x,\delta),\eta^{\Bb}(t).\nabla_x\Phi^-_{\Bb}(x,\delta)\rangle} \\ & & =\eta^{\Bb}.(f^-\nabla_xf^-+g^-\nabla_x\overline{g^-})+f^{-^2}\lambda_{11}|g^-|^2\lambda_{22}-f^-\overline{g^-}\lambda_{12}-f^-g^-\lambda_{21} \end{eqnarray*} where $\lambda_{ij}(t,x,\delta)=\langle\psi_i(x,\delta),\eta^{\Bb}(t).\nabla_x\psi_j(x,\delta)\rangle$. Short calculations show that we have to estimate the difference between $L(x,\delta)$ and $L(a^{\Bb}(t),\delta)$ for $x\in U^{\Bb}(t,\delta,\eps,\gamma)$ where $L$ is one of the following quantities : $\rho$, $B$, $C$, $D$, $\nabla_xc$, $\nabla_xd$ and $\lambda_{ij}$. Set $[L]^x_t=L(x,\delta)-L(a^{\Bb}(t),\delta)$. Further computations show that, for $x\in U^{\Bb}(t,\delta,\eps,\gamma)$ : \begin{enumerate} \item $[\rho]^{x}_{t}=O\left(\frac{\eps|t|}{\gamma(|t|+\delta)}+\frac{\eps}{\gamma}(|t|+\delta)\right)$ ; \item $[B]^{x}_{t}=O\left(\frac{\eps}{\gamma(|t|+\delta)}\right)$ ; \item $[C]^{x}_{t}=O\left(\frac{\eps}{\gamma}+\frac{\eps|t|}{\gamma(|t|+\delta)^2}\right)$ with same estimate for $D$ ; \item $[L]^{x}_{t}=O(\frac{\eps}{\gamma})$ for $\nabla_xc$, $\nabla_xd$ and $\lambda_{ij}$ ; \item $[f^-\nabla_xf^-+g^-\nabla_x\overline{g^-}]^{x}_{t}=O\left(\frac{\eps}{\gamma(|t|+\delta)}\right)$. \end{enumerate} Finally $$\langle\Phi^-_{\Bb}|\eta^{\Bb}.\nabla_x\Phi^-_{\Bb}\rangle=O\left(\frac{\eps}{\gamma(|t|+\delta)}\right)$$ and the claim is obtained by integration on $[-T,t]$. \end{dem} Now we have constructed those dynamical eigenvectors $\Phi^*_{\Cc}(t,x,\delta)$ and given the classical dynamics of (\ref{syscomp}), we want to use the approximation (\ref{BOlo}) and to estimate how good it is. We just recall the following abstract lemma of \cite{Hag2} \begin{lem} \label{estimation} Suppose $H(\h)$ is a family of self-adjoint operators in any separable Hilbert space $\Hh$ for $\h>0$ and let $\nu$ be a strictly positive real number. Suppose $\psi(r,\h)$ belongs to the domain of $H(\h)$, is continuously differentiable in $r$, and approximately solves the Schr\"odinger equation \begin{equation} \label{eqS} i\h^{\nu}\frac{\partial\psi}{\partial r}=H(\h)\psi\ , \end{equation} in the sense that $$i\h^{\nu}\frac{\partial\psi}{\partial r}(r,\h)=H(\h)\psi(r,\h)+\zeta(r,\h)$$ where $\zeta(r,\h)$ satisfies $$\Vert\zeta(r,\h)\Vert\leqslant\mu(r,\h)\ .$$ If $\Psi(r,\h)$ denotes the solution of the Schr\"odinger equation (\ref{eqS}) with initial condition $\Psi(r_0,\h)=\psi(r_0,\h)$, then $$\Vert\Psi(r,\h)-\psi(r,\h)\Vert\leqslant\h^{-\nu}\left|\int_{r_0}^{r}\mu(\rho,\h)d\rho\right|\ .$$ \end{lem} \subsection{Outer Ansatz} Carefully analyzing the time when the usual Born-Oppenheimer approximation (\ref{BOlo}) actually breaks down and setting for $\Cc=\Aa,\Bb$ and $l\in\N^n$ $$\varphi^{\Cc}_l(t,y,\eps)=\exp\left(i\frac{S^{\Cc}(t)}{\eps^2}+i\frac{\eta^{\Cc}(t).y}{\eps}\right)\phi_l(A^{\Cc}(t),B^{\Cc}(t),y)\ ,$$ we get the following result \begin{prop} \label{lementrant} In the incoming outer region $-T\leqslant t\leqslant-t_o(\delta,\eps)<0$, if \begin{equation} \label{solentrante} \psi_{IO}(t,x,\delta,\eps)=\sum_{\Cc =\Aa,\Bb}\Lambda^-_{\Cc}(\delta,\eps)F(\eps^{-2}\gamma^2\Vert x-a^{\Cc}(t)\Vert^2)\varphi^{\Cc}_l\left(t,\frac{x-a^{\Cc}(t)}{\eps},\eps\right)\Phi^-_{\Cc}(t,x,\delta) \end{equation} where $\Lambda_{\Cc}^-(\delta,\eps)=O(1)$ and if $\psi(\cdot,\delta,\eps)$ denotes the solution of (\ref{equn'}) with initial condition $\psi(-T,\cdot,\delta,\eps)=\psi_{IO}(-T,\cdot,\delta,\eps)$, we have $$\sup_{t\in[-T,-t_o]}\Vert\psi(t,x,\delta,\eps)-\psi_{IO}(t,x,\delta,\eps)\Vert_{L^2(\R^d;\Hh)}=O\left(\eps\ln\frac{1}{t_o+\delta}+\frac{\eps^2}{(t_o+\delta)^2}+\frac{\eps^4}{(t_o+\delta)^3}\right)$$ when $\delta$ and $\eps$ tend to $0$ and where $\gamma(\delta,\eps)$ and $t_o(\delta,\eps)$ are chosen to tend to $0$ with $\eps/(\gamma(\delta,\eps)t_o(\delta,\eps))$ bounded. \end{prop} \begin{rmqs} If we fix $\delta>0$, we recover the usual Born-Oppenheimer approximation with an error of order $O(\eps)$. \item There is a similar result in the outgoing outer region $00$.} We can match the two preceding Ans\"atze with an error of same order as for $t<0$ by choosing \begin{eqnarray*} \Lambda_{\Aa}^+(\delta,\eps) & = & e^{i[\omega_{\Bb}^-(-t_m(\delta,\eps),a^{\Bb}(-t_m(\delta,\eps)),\delta)-\omega_{\Aa}^+(t_m(\delta,\eps),a^{\Aa}(t_m(\delta,\eps)),\delta)]} \\ \Lambda_{\Bb}^+(\delta,\eps) & = & 0. \end{eqnarray*} Then, the error term is of order $o(1)$ if we choose $t=t_m(\delta,\eps)\in[t_o(\delta,\eps),t_i(\delta,\eps)]$ and $\gamma=\gamma(\delta,\eps)$ tending to $0$ with $$\max\left(\delta,\frac{\eps}{\gamma }\right)\ll t\ll\min\left(\eps^{2/3},\sqrt{\eps\gamma},\frac{\eps^2}{\delta}\right)\, \, \mathrm{and}\, \, \max\left(\eps^{1/3},\frac{\delta}{\eps}\right)\ll \gamma \ll 1$$ (which is a non-empty zone). \paragraph{First Order Matching.} By choosing $\eta^{0^{\Cc}}(\delta)=\eta^0(\delta)-\frac{r\nu^{\Cc}}{\eta^0_1}\left(\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array}\right)\delta$, we substitute the error term of order $\frac{\delta}{\eps\gamma}$ in $e_{\varphi}(t,\delta,\eps,\gamma)$ by $\frac{\delta^2}{\eps\gamma|t|}$ (by the way, total energy conservation at $0$, $\left[\frac{\Vert\eta^{0^{\Aa}}(\delta)\Vert^2}{2}-E_{\Aa}(0,\delta)\right]-\left[\frac{\Vert\eta^{0^{\Bb}}(\delta)\Vert^2}{2}-E_{\Bb}(0,\delta)\right]=O(\delta^2) $, is now fulfilled up to first order) ; we go further in solving (\ref{systeme}) by taking \begin{eqnarray} \lefteqn{f_1(s,y,\delta,\eps)=e^{-ir(\eta_1^0\frac{s^2}{2}+sy_1)}} \nonumber \\ & & \times\left[g_1(y,\delta,\eps)+\frac{\delta}{\eps}\left(h_1(y,\delta,\eps)-irg_2(y,\delta,\eps)\int_{-\infty}^{s}e^{ir(\eta^0_1\sigma^2+2\sigma y_1)}d\sigma\right)\right] \label{narrowf1} \\ \lefteqn{f_2(s,y,\delta,\eps)=e^{ir(\eta_1^0\frac{s^2}{2}+sy_1)}} \nonumber \\ & & \times\left[g_2(y,\delta,\eps)+\frac{\delta}{\eps}\left(h_2(y,\delta,\eps)+irg_1(y,\delta,\eps)\int_{-\infty}^{s}e^{-ir(\eta^0_1\sigma^2+2\sigma y_1)}d\sigma\right)\right] \label{narrowf2} \end{eqnarray} with $g_k,h_k\in H^2(\R^d)\cap (1+\Vert y\Vert^2)^{-1}L^2(\R^d)$, thus we substitute the error term of order $\frac{|t|\delta}{\eps^2}$ in $e_I(t,\delta,\eps)$ by $\frac{\delta^2}{\eps}$ ; matching for $t<0$ can be performed by choosing \begin{eqnarray*} g_1(y,\delta,\eps) & = &-\phi_l(A_0,B_0^{\Bb}(-1),y) e^{i\omega_{\Bb}^-(-t_m(\delta,\eps),a^{\Bb}(-t_m(\delta,\eps)),\delta)} \\ g_2(y,\delta,\eps) & = & 0 \\ h_1(y,\delta,\eps) & = & 0 \\ h_2(y,\delta,\eps) & = & 0 \ ; \end{eqnarray*} matching for $t>0$ can be performed too (we use the identity $\phi_l(A_0,B_0^{\Bb}(-1),y)=\exp(-i\frac{r}{\eta_1^0}y_1^2)\phi_l(A_0,B_0^{\Bb}(+1),y)$) by choosing \begin{eqnarray*} \Lambda_{\Aa}^+(\delta,\eps) & = & e^{i[\omega_{\Bb}^-(-t_m(\delta,\eps),a^{\Bb}(-t_m(\delta,\eps)),\delta)-\omega_{\Aa}^+(t_m(\delta,\eps),a^{\Aa}(t_m(\delta,\eps)),\delta)]}\sqrt{1-|\Lambda^+_{\Bb}(\delta,\eps)|^2} \\ \Lambda_{\Bb}^+(\delta,\eps) & = &-\frac{\delta}{\eps}\sqrt{\frac{\pi r}{\eta^0_1}}e^{i\frac{\pi}{4}}e^{i[\omega_{\Bb}^-(-t_m(\delta,\eps),a^{\Bb}(-t_m(\delta,\eps)),\delta)-\omega^+_{\Bb}(t_m(\delta,\eps),a^{\Bb}(t_m(\delta,\eps)),\delta)]} \ ; \end{eqnarray*} thus global error is of order $o(\frac{\delta}{\eps})$ if we can choose $t=t_m(\delta,\eps)$ and $\gamma=\gamma(\delta,\eps)$ tending to $0$ with $$\max\left(\delta,\frac{\eps}{\gamma},\frac{\eps^4}{\delta},\frac{\eps^{3/2}}{\delta^{1/2}},\frac{\eps^{5/3}}{\delta^{1/3}}\right)\ll t\ll\min\left(\delta^{1/3}\eps^{1/3},\sqrt{\delta\gamma},\frac{\delta}{\eps}\right)$$ and $$\max\left(\frac{\eps^{2/3}}{\delta^{1/3}},\frac{\eps^{10/3}}{\delta^{5/3}},\frac{\eps^{3}}{\delta^2},\frac{\eps^8}{\delta^3},\frac{\eps^2}{\delta},\delta\right)\ll\gamma\ll 1$$ which is a non-empty zone with the extra condition $\delta/\eps^{7/5}\rightarrow+\infty$ (a natural condition would be $\delta/\eps^2\rightarrow+\infty$: the predicted first order term is of order $O\left(\frac{\delta}{\eps}\right)$ and the general Born-Oppenheimer error is of order $O(\eps)$ ; the technical condition follows from unknown second order terms of the operator $h_1(x,\delta)$ and from the choice of the phase in (\ref{solinterieure}): with more regularity on $h(x,\delta)$, one can improve this technical condition but the choice of the phase seems to be the limiting factor of improvement). \subsection{Wide Avoided Crossing ($\delta/\eps\rightarrow+\infty$): we use the $t/\delta\rightarrow0$ regime.} Similar calculations lead to estimates \begin{eqnarray*} \lefteqn{\left[S^{\Cc}(t)+\eta^{\Cc}(t).(x-a^{\Cc}(t))\right]-\left[S(t)+\eta(t).(x-a(t))\right] = } \\ & & -r\nu^{\Cc}\left[\delta t+\frac{\eta_1^{0^2}t^{3}}{6\delta}+\frac{\eta^0_1t^2\eps}{2\delta}y_1\right]+O\left(\delta^2|t|+\frac{t^{4}}{\delta^2}+\left(\frac{t^4}{\delta^3}+\delta|t|\right)\eps\Vert y\Vert\right) \\ & & {}+\left(\eta^{0^{\Cc}}(\delta)-\eta^0(\delta)\right).\eps y\ , \end{eqnarray*} we remove the last term which would lead to an error of order $O(\frac{\delta}{\eps\gamma})$ by choosing $\eta^{0^{\Cc}}(\delta)=\eta^0(\delta)$ (an extra choice compared to the narrow avoided crossing case) ; and $$\left\Vert\phi_l\left(A^{\Cc}(t),B^{\Cc}(t),y+\frac{a(t)-a^{\Cc}(t)}{\eps}\right)-\phi_l(A_0,B_0,y)\right\Vert_{L^2}$$ is bounded by a constant times $|t|+\delta+\frac{|t|}{\delta}+\frac{\delta^2|t|}{\eps}+\frac{|t|^{3}}{\delta\eps}$. \paragraph{Incoming Outer Asymptotics.} \begin{eqnarray*} \lefteqn{\psi_{IO}(t,x,\delta,\eps)=e^{\frac{i}{\eps^2}(S(t)+\eta(t).(x-a(t)))}e^{ir\frac{\delta}{\eps}\frac{t}{\eps}}e^{i\omega^-_{\Bb}(t,a^{\Bb}(t),\delta)}} \\ & & \times\phi_l\left(A_0,B_0,\frac{x-a(t)}{\eps}\right)\frac{\sqrt{2}}{2}[-\psi_1(x,\delta)+\psi_2(x,\delta)] \\ & & \times\left[1+O\left(e_{\varphi}(t,\delta,\eps,\gamma)+e_{\phi}(t,\eps,\gamma)+e_{\Phi}(t,\delta)+e_O(t,\eps)+e_{\omega}(t,\delta,\eps,\gamma)\right)\right] \end{eqnarray*} where $$e_{\phi}(t,\delta,\eps)=|t|+\delta+\frac{|t|}{\delta}+\frac{\delta^2|t|}{\eps}+\frac{|t|^{3}}{\delta\eps}\ ,\ e_{\omega}(t,\delta,\eps,\gamma)=\frac{\delta\eps}{\gamma|t|}+\frac{\eps}{\gamma}\ln\frac{1}{\delta}\ ,$$ $$e_{\varphi}(t,\delta,\eps,\gamma)=\frac{\delta^2|t|}{\eps^2}+\frac{|t|^{3}}{\delta \eps^2}+\frac{t^2}{\delta \eps \gamma }+\frac{\delta|t|}{\eps\gamma}\ ,\ e_{\Phi}(t,\delta)=\delta+\frac{|t|}{\delta}\ ,$$ $$e_O(t,\eps)=\eps\ln\frac{1}{\delta}+\frac{\eps^2}{\delta^2}$$ are error terms analogous to the narrow avoided crossing case. \paragraph{Inner Asymptotics.} \begin{eqnarray*} \lefteqn{\hat{\psi}_I(s,y,\delta,\eps)=e^{\frac{i}{\eps^2}S(\eps s)+\frac{i}{\eps}\eta(\eps s).y}} \\ & & \times\left[g_1(y,\delta,\eps)e^{-ir\frac{\delta}{\eps}s}(\psi_1+\psi_2)+g_2(y,\delta,\eps)e^{ir\frac{\delta}{\eps}s}(-\psi_1+\psi_2)\right](a(\eps s)+\eps y,\delta) \\ & & \times\left[1+O\left(e_I(\eps s,\delta,\eps)\right)\right] \end{eqnarray*} where $e_I(t,\delta,\eps)=\frac{\delta^2|t|}{\eps^2}+\frac{|t|}{\delta}+\frac{t^2}{\delta \eps}+\frac{|t|^{3}}{\delta\eps^2}$ is the error term given by proposition \ref{leminterieur}. \paragraph{Matching for $t<0$.} We can match those two Ans\"atze with an error of order $$O\left(e_{\varphi}(t,\delta,\eps,\gamma)+e_{\phi}(t,\delta,\eps)+e_{\Phi}(t,\delta)+e_E(t,\eps)+e_{\omega}(t,\delta,\eps,\gamma)+e_I(t,\delta,\eps)\right)$$ by choosing \begin{eqnarray*} g_1(y,\delta,\eps) & = & 0 \\ g_2(y,\delta,\eps) & = & \frac{\sqrt{2}}{2}\phi_l(A_0,B_0,y) e^{i\omega_{\Bb}^-(-t_m(\delta,\eps),a^{\Bb}(-t_m(\delta,\eps)),\delta)}\ . \end{eqnarray*} \paragraph{Outgoing Outer Asymptotics.} \begin{eqnarray*} \lefteqn{\psi_{OO}(t,x,\delta,\eps)=e^{\frac{i}{\eps^2}(S(t)+\eta(t).(x-a(t)))}\phi_l\left(A_0,B_0,\frac{x-a(t)}{\eps}\right)} \\ & & \times\sum_{(\Cc,k)=(\Aa,1),(\Bb,2)}\Lambda_{\Cc}^+(\delta,\eps) e^{(-1)^{k}ir\frac{\delta}{\eps}\frac{t}{\eps}}e^{i\omega^+_{\Cc}(t,a^{\Cc}(t),\delta)}\frac{\sqrt{2}}{2}\left((-1)^{k-1}\psi_1(x,\delta)+\psi_2(x,\delta)\right) \\ & & \times\left[1+O\left(e_{\varphi}(t,\delta,\eps,\gamma)+e_{\phi}(t,\delta,\eps)+e_{\Phi}(t,\delta)+e_O(t,\eps)+e_{\omega }(t,\delta,\eps,\gamma)\right)\right]\ . \end{eqnarray*} \paragraph{Matching for $t>0$.} We can match the two preceding Ans\"atze with an error of the same order as for $t<0$ by choosing \begin{eqnarray*} \Lambda_{\Aa}^+(\delta,\eps) & = & 0 \\ \Lambda_{\Bb}^+(\delta,\eps) & = & e^{i\left[\omega_{\Bb}^-(-t_m(\delta,\eps),a^{\Bb}(-t_m(\delta,\eps)),\delta)-\omega_{\Bb}^+(t_m(\delta,\eps),a^{\Bb}(t_m(\delta,\eps)),\delta)\right]}\ . \end{eqnarray*} Then, the error term is of order $o(1)$ if we choose $t=t_m(\delta,\eps)\in[t_o(\delta,\eps),t_i(\delta,\eps)]$ and $\gamma=\gamma(\delta,\eps)$ tending to $0$ with $$\frac{\eps}{\gamma}\ll t\ll\min\left(\delta,\sqrt{\delta\eps\gamma},\frac{\eps\gamma}{\delta},\delta^{1/3}\eps^{2/3},\frac{\eps}{\sqrt{\delta}}\right)\,\,\mathrm{and}\,\,\max\left(\frac{\eps^{1/3}}{\delta^{1/3}},\sqrt{\delta}\right)\ll\gamma\ll1$$ (which is a non-empty zone). Note that a first order result in this regime can not be expected with this method, again because of the choice of the phase in (\ref{solinterieure}). \section{Main Result} \label{mainresult} With the preceding notations, we have \begin{theo} \label{maintheo} Let $h(x,\delta)$ be a Hamiltonian that satisfies the hypothesis above, and let $\psi(t,x,\delta,\eps)$ denote the solution of (\ref{equn'}) with initial condition at $t=-T$ $$\sum_{\Cc=\Aa,\Bb}\Lambda^-_{\Cc}(\delta,\eps)F\left(\gamma^2\frac{\Vert x-a^{\Cc}(-T)\Vert^2}{\eps^2}\right)\varphi^{\Cc}_l\left(-T,\frac{x-a^{\Cc }(-T)}{\eps},\eps\right)\Phi^-_{\Cc}(-T,x,\delta)$$ with $|\Lambda^-_{\Aa}(\delta,\eps)|^2+|\Lambda^-_{\Bb}(\delta,\eps)|^2=1$, then we have, in the limit $\delta$ and $\eps$ tending to $0$, \begin{equation} \label{erreursortie} \left\Vert\psi(T,x,\delta,\eps)-\sum_{\Cc=\Aa,\Bb}\Lambda^+_{\Cc}(\delta,\eps)\varphi^{\Cc}_l\left(T,\frac{x-a^{\Cc}(T)}{\eps},\eps\right)\Phi^+_{\Cc}(T,x,\delta)\right\Vert_{L^2(\R^d;\Hh)}=o(1) \end{equation} where $$\left(\begin{array}{c} \Lambda^+_{\Aa}(\delta,\eps) \\ \Lambda^+_{\Bb}(\delta,\eps) \end{array}\right)=S(\delta,\eps)\left(\begin{array}{c} \Lambda^-_{\Aa}(\delta,\eps) \\ \Lambda^-_{\Bb}(\delta,\eps) \end{array}\right)\ ,$$ with, \begin{itemize} \item if $\delta/\eps\rightarrow0$, $$S(\delta,\eps)=\left(\begin{array}{cc} 0 & e^{i\omega_{\Aa\Bb}(\delta,\eps)} \\ e^{i\omega_{\Bb\Aa}(\delta,\eps)} & 0 \end{array}\right)\ ,$$ \item if $\delta/\eps\rightarrow+\infty$, $$S(\delta,\eps)=\left(\begin{array}{cc} e^{i\omega_{\Aa}(\delta,\eps)} & 0 \\ 0 & e^{i\omega_{\Bb}(\delta,\eps)} \end{array}\right)$$ \end{itemize} where each phase only depends on the choice of an initial phase for dynamic eigenvectors $\Phi^*_{\Cc}(t,x,\delta)$ (the matrix $S(\delta,\eps)$ is unitary). Moreover, in the case $\delta/\eps\rightarrow0$, with the extra condition $\delta/\eps^{7/5}\rightarrow+\infty$, (\ref{erreursortie}) holds with $o\left(\frac{\delta}{\eps}\right)$ on the right-hand side and $$S(\delta,\eps)=\left(\begin{array}{cc} \frac{\delta}{\eps}\sqrt{\frac{\pi r}{\eta^0_1}}e^{i\omega_{\Aa}(\delta,\eps)} & \sqrt{1-\frac{\pi r\delta^2}{\eta^0_1\eps^2}}e^{i\omega_{\Aa\Bb}(\delta,\eps)} \\ \sqrt{1-\frac{\pi r\delta^2}{\eta^0_1\eps^2}}e^{i\omega_{\Bb\Aa}(\delta,\eps)} & \frac{\delta}{\eps}\sqrt{\frac{\pi r}{\eta^0_1}}e^{i\omega_{\Bb}(\delta,\eps)} \end{array}\right)\ .$$ \end{theo} \section{Landau-Zener Transitions for Eigenvalue Avoided Crossings in an Adiabatic Limit} \label{casadiab} In the Born-Oppenheimer approximation, we saw that the $x$-variable was relevant only around the semi-classical position $a\td$ so that the molecular Hamiltonian essentially behaved like $$\frac{1}{2}\left\Vert\eta\td\right\Vert^2+h(a\td,\delta)$$ and equation (\ref{equn'}) essentially turned to equation (\ref{eqzero}) with this time-dependent Hamiltonian. This time-dependent reduced situation leads to a purely adiabatic problem (we have dropped the semi-classical approximation for the nuclei by saying that they exactly follow their classical trajectory) with an avoided crossing for the two eigenvalues $$\frac{1}{2}\left\Vert\eta\td\right\Vert^2+E(a\td,\delta)\pm\rho(a\td,\delta)$$ at $t=0$. Let us now treat a case of a general purely adiabatic problem for equation (\ref{eqzero}) with an avoided crossing of variable width $\delta$. \subsection{Avoided Crossings and Normal Form for the Generic Case} \label{nfadiab} \begin{defi} Suppose $H\td$ is a family of self-adjoint operators with fixed domain $\Dd$ (in any separable Hilbert space $\Hh$) for $]t_0-2T,t_0+2T[\times]-2\delta_0,2\delta_0[$. Assume that for every $\delta>0$, $H\td$ has two distinct eigenvalues $E_{\Aa}\td$ and $E_{\Bb}\td$ uniformly isolated from the rest of the spectrum with $E_{\Aa}\td0$. In what follows, we forget the underlined notation for those variables. Finally, we set $\rho\td=\sqrt{b\td^2+c\td^2+d\td^2}$ and $E_{\Cc}\td=E\td+\nu^{\Cc}\rho\td$ where $\nu^{\Aa}=1$ and $\nu^{\Bb}=-1$. \subsection{Away from the Crossing} Let $$Z_-(\delta)=\{t\in I/b\td<\frac{1}{2}\rho\td\}\ ,\ Z_+(\delta)=\{t\in I/b\td>-\frac{1}{2}\rho\td\}\ ,$$ $$J(\delta)=\{t\in I/\rho\td>\frac{r}{2}\sqrt{t^2+\delta^2}\}\ .$$ Mimicking section \ref{csvscoz}, we get that if $T$ and $\delta_0$ are chosen small enough, for every $\delta\in[0,\delta_0]$, $[-T,0]\subseteq(Z_-(\delta)\cap J(\delta))$ and $[0,T]\subseteq(Z_+(\delta)\cap J(\delta))$. The definition of normalized eigenvectors is similar to section \ref{caaose} substituting $t$ for $x$. Then we have the same asymptotics as in lemmas \ref{lemasympfctpropretroit} and \ref{lemasympfctproprlarge} dropping the cutoff function and substituting $t$ for $x$. We still transform the eigenvectors with $\underline{\Phi}^*_{\Cc}\td:=e^{i\omega^*_{\Cc}\td}\Phi^*_\Cc\td$ for $\Cc=\Aa,\Bb$ and $*=+,-$ in order to fulfill the orthogonality condition $<\underline{\Phi}^*_{\Cc}\td|\frac{\partial}{\partial t}\underline{\Phi}^*_{\Cc}\td>=0$ and we choose initial conditions $\omega^-_{\Cc}(-T,\delta)=0$ and $\omega^+_{\Cc}(T,\delta)=0$. By analogy with proposition \ref{lementrant}, we can prove \begin{prop} \label{estimationhorscroisement} In the incoming outer region $-T\leqslant t\leqslant-t_o(\delta,\eps)<0$, if $$\psi_{IO}\tde=\sum_{\Cc=\Aa,\Bb}{\Lambda^-_{\Cc}(\delta,\eps)\exp\Big(-\frac{i}{\eps^2}\int^t_0{E_{\Cc}(\tau,\delta)d\tau}\Big)\underline{\Phi}^-_{\Cc}\td}$$ where $\Lambda^-_{\Cc}(\delta,\eps)$ are chosen with order $O(1)$ and if $\psi\tde$ denotes the solution of (\ref{eqzero}) with initial condition $\psi(-T,\delta,\eps)=\psi_{IO}(-T,\delta,\eps)$, then there exists a strictly positive constant $C$, such that $$\sup_{t\in[-T,-t_o]}\Vert\psi\tde-\psi_{IO}\tde\Vert\leqslant C\frac{\eps^2}{t_o^2+\delta^2}\ .$$ \end{prop} \begin{rmqs} If we fix $\delta>0$, we recover the classical adiabatic theorem with an error of order $O(\eps^2)$. \item There is a similar result in the outgoing outer region $019 D39 D<387FFFFCA3B5FCA21605799521>45 D<120FEA3FC0127FA212FFA31380EA7F00123C0A 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5308 y Fv(3)p Fp(=)p Fv(2)p 2577 5388 591 4 v 2678 5470 a Fp(\021)2715 5447 y Fi(0)2745 5433 y(2)2713 5491 y(1)p 2678 5499 107 4 v 2697 5556 a Fv(16)2795 5522 y Fr(t)2830 5493 y Fv(2)2892 5522 y Fs(+)g Fr(\016)3037 5493 y Fv(2)1828 5755 y Fs(11)p eop %%Page: 12 12 12 11 bop 21 219 a Fs(hence,)34 b(if)e Fr(t)444 182 y Fv(2)505 219 y Fs(+)22 b Fr(\016)650 182 y Fv(2)718 219 y Ff(6)28 b Fs(\()p Fr(r)908 182 y Fv(4)963 219 y Fs(min)o(\(1)p Fr(;)17 b(\021)1309 182 y Fv(0)1344 159 y Fi(4)1305 243 y Fv(1)1382 219 y Fr(=)p Fs(256\)\))p Fr(=)p Fs(\(4)p Fr(C)1867 182 y Fv(2)1860 243 y Fp(\032)1921 219 y Fs(max\(1)p Fr(;)g(D)2318 182 y Fv(6)2315 243 y Fp(t)2357 219 y Fs(\)\),)1383 486 y Fr(\032)p Fs(\()p Fr(x;)g(\016)t Fs(\))1655 445 y Fv(2)1722 486 y Ff(>)1837 419 y Fr(r)1884 383 y Fv(2)p 1837 464 87 4 v 1856 555 a Fs(2)1933 486 y(\()p Fr(x)2026 445 y Fv(2)2026 511 y(1)2088 486 y Fs(+)22 b Fr(\016)2233 445 y Fv(2)2272 486 y Fs(\))33 b Fr(:)167 720 y Fs(Let)g(us)g(sho)m(w)h (no)m(w)f(condition)e Fr(W)1446 684 y Fl(C)1432 745 y(\000)1492 720 y Fs(\()p Fr(t;)17 b(\016)n(;)g(";)g(\015)5 b Fs(\))27 b Fq(\022)h Fr(Z)2077 735 y Fl(\000)2136 720 y Fs(\()p Fr(\016)t Fs(\))k(:)210 982 y Fr(b)p Fs(\()p Fr(x;)17 b(\016)t Fs(\))28 b(=)g Fr(r)s(x)707 997 y Fv(1)769 982 y Fs(+)22 b(\()p Fr(b)p Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b Fq(\000)h Fr(r)s(x)1392 997 y Fv(1)1431 982 y Fs(\))28 b Fr(<)g(C)1671 997 y Fp(b)1705 982 y Fs(\()p Fq(k)p Fr(x)p Fq(k)1898 941 y Fv(2)1898 1007 y Fl(1)1995 982 y Fs(+)22 b Fr(\016)2140 941 y Fv(2)2179 982 y Fs(\))28 b Ff(6)g Fr(C)2420 997 y Fp(b)2470 982 y Fs(max)2669 842 y Fo(\022)2742 982 y Fs(1)p Fr(;)2844 915 y Fs(16)p Fr(D)3026 879 y Fv(2)3023 939 y Fp(t)p 2844 959 221 4 v 2892 1057 a Fr(\021)2944 1022 y Fv(0)2979 1003 y Fi(2)2940 1081 y Fv(1)3075 842 y Fo(\023)3165 982 y Fs(\()p Fr(x)3258 941 y Fv(2)3258 1007 y(1)3320 982 y Fs(+)22 b Fr(\016)3465 941 y Fv(2)3504 982 y Fs(\))982 1370 y Ff(6)29 b Fr(\032)p Fs(\()p Fr(x;)17 b(\016)t Fs(\))1370 1303 y(2)p Fr(C)1489 1318 y Fp(b)p 1370 1347 153 4 v 1423 1438 a Fr(r)1549 1370 y Fs(max)1747 1230 y Fo(\022)1821 1370 y Fs(1)p Fr(;)1923 1303 y Fs(16)p Fr(D)2105 1266 y Fv(2)2102 1327 y Fp(t)p 1923 1347 221 4 v 1971 1444 a Fr(\021)2023 1410 y Fv(0)2058 1391 y Fi(2)2019 1469 y Fv(1)2154 1230 y Fo(\023)2244 1173 y(s)p 2343 1173 427 4 v 2353 1303 a Fr(\021)2405 1268 y Fv(0)2440 1249 y Fi(2)2401 1327 y Fv(1)p 2353 1347 126 4 v 2367 1438 a Fs(16)2489 1370 y Fr(t)2524 1341 y Fv(2)2586 1370 y Fs(+)22 b Fr(\016)2731 1341 y Fv(2)21 1649 y Fs(hence,)34 b(if)419 1597 y Fp(\021)456 1574 y Fi(0)486 1553 y(2)454 1618 y(1)p 419 1626 107 4 v 437 1683 a Fv(16)536 1649 y Fr(t)571 1613 y Fv(2)632 1649 y Fs(+)22 b Fr(\016)777 1613 y Fv(2)844 1649 y Ff(6)29 b Fr(r)s(=)p Fs(\(4)p Fr(C)1203 1664 y Fp(b)1252 1649 y Fs(max\(1)p Fr(;)17 b Fs(16)p Fr(D)1747 1613 y Fv(2)1744 1673 y Fp(t)1785 1649 y Fr(=\021)1886 1613 y Fv(0)1921 1589 y Fi(2)1882 1673 y Fv(1)1960 1649 y Fs(\)\),)32 b Fr(b)p Fs(\()p Fr(x;)17 b(\016)t Fs(\))28 b Fr(<)2500 1610 y Fv(1)p 2500 1626 36 4 v 2500 1683 a(2)2545 1649 y Fr(\032)p Fs(\()p Fr(x;)17 b(\016)t Fs(\).)167 1782 y(Finally)25 b(b)m(y)k(diminishing)24 b Fr(T)41 b Fs(and)27 b Fr(\016)1471 1797 y Fv(0)1511 1782 y Fs(,)h(the)g(only)f(remaining)e (constrain)m(t)i(is)g Fr("=\015)32 b Ff(6)c Fr(\021)3269 1746 y Fv(0)3265 1806 y(1)3309 1782 y Fq(j)p Fr(t)p Fq(j)p Fr(=)p Fs(\(8)3536 1699 y Fq(p)p 3618 1699 49 4 v 3618 1782 a Fs(2)o(\),)21 1902 y(as)33 b(exp)s(ected.)45 b Ff(\003)167 2072 y Fk(Remark)69 b Fs(By)33 b(the)g(w)m(a)m(y)-8 b(,)33 b(w)m(e)g(note)g(that)f Fr(\032)p Fs(\()p Fr(x;)17 b(\016)t Fs(\))28 b Fr(>)2146 2032 y Fp(r)p 2145 2049 36 4 v 2145 2106 a Fv(2)2190 1938 y Fo(q)p 2290 1938 409 4 v 2300 2019 a Fp(\021)2337 1997 y Fi(0)2367 1983 y(2)2335 2041 y(1)p 2300 2049 107 4 v 2318 2106 a Fv(16)2417 2072 y Fr(t)2452 2043 y Fv(2)2514 2072 y Fs(+)22 b Fr(\016)2659 2043 y Fv(2)2726 2072 y Ff(>)28 b Fr(R)q Fs(\()p Fq(j)p Fr(t)p Fq(j)21 b Fs(+)g Fr(\016)t Fs(\))32 b(for)g(ev)m(ery)i Fr(x)21 2203 y Fs(in)e Fr(U)211 2167 y Fl(C)257 2203 y Fs(\()p Fr(t;)17 b(\016)n(;)g(";)g(\015)5 b Fs(\))32 b(and)g(that)h Fr(a)1127 2167 y Fl(C)1172 2203 y Fs(\([)p Fq(\000)p Fr(T)8 b(;)17 b(T)d Fs(])p Fr(;)j(\016)t Fs(\))28 b Fq(\022)g Fr(J)9 b Fs(\()p Fr(\016)t Fs(\).)21 2488 y Fd(4.2)135 b(Construction)46 b(and)e(Asymptotics)i(of)f(Selected)h (Eigen)l(v)l(ectors)21 2673 y Fs(F)-8 b(or)32 b(\()p Fr(x;)17 b(\016)t Fs(\))28 b Fq(2)g Fm(R)606 2637 y Fp(d)652 2673 y Fq(\002)p Fs(]0)p Fr(;)17 b Fs(2)p Fr(\016)941 2688 y Fv(0)981 2673 y Fs([,)32 b(w)m(e)i(de\014ne)650 2924 y Fr(B)5 b Fs(\()p Fr(x;)17 b(\016)t Fs(\))27 b(=)1096 2857 y Fr(b)p Fs(\()p Fr(x;)17 b(\016)t Fs(\))p 1092 2901 273 4 v 1092 2992 a Fr(\032)p Fs(\()p Fr(x;)g(\016)t Fs(\))1407 2924 y Fr(;)49 b(C)7 b Fs(\()p Fr(x;)17 b(\016)t Fs(\))27 b(=)1927 2857 y Fr(c)p Fs(\()p Fr(x;)17 b(\016)t Fs(\))p 1923 2901 V 1923 2992 a Fr(\032)p Fs(\()p Fr(x;)g(\016)t Fs(\))2238 2924 y Fr(;)49 b(D)s Fs(\()p Fr(x;)17 b(\016)t Fs(\))27 b(=)2761 2857 y Fr(d)p Fs(\()p Fr(x;)17 b(\016)t Fs(\))p 2761 2901 V 2761 2992 a Fr(\032)p Fs(\()p Fr(x;)g(\016)t Fs(\))3076 2924 y Fr(;)671 3271 y(f)730 3230 y Fl(\000)788 3271 y Fs(\()p Fr(x;)g(\016)t Fs(\))28 b(=)1142 3109 y Fo(r)p 1241 3109 492 4 v 1251 3204 a Fs(1)22 b Fq(\000)h Fr(B)5 b Fs(\()p Fr(x;)17 b(\016)t Fs(\))p 1251 3248 472 4 v 1463 3340 a(2)1765 3271 y Fr(;)50 b(g)1893 3230 y Fl(\000)1951 3271 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))27 b(=)2314 3204 y Fr(C)7 b Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b(+)g Fr(iD)s Fs(\()p Fr(x;)17 b(\016)t Fs(\))p 2314 3248 758 4 v 2345 3268 a Fo(p)p 2445 3268 597 4 v 85 x Fs(2\(1)k Fq(\000)i Fr(B)5 b Fs(\()p Fr(x;)17 b(\016)t Fs(\)\))21 3511 y(when)34 b Fr(B)5 b Fs(\()p Fr(x;)17 b(\016)t Fs(\))28 b Fr(<)f Fs(1,)33 b(and)671 3789 y Fr(f)730 3748 y Fv(+)789 3789 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))28 b(=)1142 3627 y Fo(r)p 1242 3627 490 4 v 1252 3722 a Fs(1)22 b(+)g Fr(B)5 b Fs(\()p Fr(x;)17 b(\016)t Fs(\))p 1252 3766 471 4 v 1463 3858 a(2)1764 3789 y Fr(;)50 b(g)1892 3748 y Fv(+)1950 3789 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))28 b(=)2313 3722 y Fr(C)7 b Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b(+)g Fr(iD)s Fs(\()p Fr(x;)17 b(\016)t Fs(\))p 2313 3766 758 4 v 2345 3786 a Fo(p)p 2445 3786 595 4 v 85 x Fs(2\(1)k(+)h Fr(B)5 b Fs(\()p Fr(x;)17 b(\016)t Fs(\)\))21 4061 y(when)34 b Fr(B)5 b Fs(\()p Fr(x;)17 b(\016)t Fs(\))28 b Fr(>)f Fq(\000)p Fs(1.)167 4181 y(W)-8 b(e)33 b(de\014ne)h(static)e(eigen)m(v)m(ectors)j(b)m(y)820 4377 y(\010)890 4336 y Fl(\000)890 4404 y(A)951 4377 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))83 b(=)g Fr(g)1466 4336 y Fl(\000)1524 4377 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))p Fr( )1809 4392 y Fv(1)1849 4377 y Fs(\()p Fr(x;)g(\016)t Fs(\))22 b(+)g Fr(f)2250 4336 y Fl(\000)2309 4377 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))p Fr( )2594 4392 y Fv(2)2633 4377 y Fs(\()p Fr(x;)g(\016)t Fs(\))33 b Fr(;)821 4532 y Fs(\010)891 4491 y Fl(\000)891 4559 y(B)951 4532 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))83 b(=)g Fq(\000)p Fr(f)1551 4491 y Fl(\000)1610 4532 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))p Fr( )1895 4547 y Fv(1)1935 4532 y Fs(\()p Fr(x;)g(\016)t Fs(\))22 b(+)p 2277 4445 332 4 v 22 w Fr(g)2328 4503 y Fl(\000)2386 4532 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))p Fr( )2671 4547 y Fv(2)2711 4532 y Fs(\()p Fr(x;)g(\016)t Fs(\))21 4728 y(when)34 b Fr(B)5 b Fs(\()p Fr(x;)17 b(\016)t Fs(\))28 b Fr(<)f Fs(1,)33 b(and)820 4924 y(\010)890 4883 y Fv(+)890 4951 y Fl(A)951 4924 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))83 b(=)g Fr(f)1474 4883 y Fv(+)1532 4924 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))p Fr( )1817 4939 y Fv(1)1857 4924 y Fs(\()p Fr(x;)g(\016)t Fs(\))22 b(+)p 2199 4837 V 22 w Fr(g)2250 4895 y Fv(+)2309 4924 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))p Fr( )2594 4939 y Fv(2)2633 4924 y Fs(\()p Fr(x;)g(\016)t Fs(\))33 b Fr(;)821 5069 y Fs(\010)891 5028 y Fv(+)891 5096 y Fl(B)951 5069 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))83 b(=)g Fq(\000)p Fr(g)1543 5028 y Fv(+)1601 5069 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))p Fr( )1886 5084 y Fv(1)1926 5069 y Fs(\()p Fr(x;)g(\016)t Fs(\))22 b(+)g Fr(f)2327 5028 y Fv(+)2386 5069 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))p Fr( )2671 5084 y Fv(2)2711 5069 y Fs(\()p Fr(x;)g(\016)t Fs(\))21 5265 y(when)34 b Fr(B)5 b Fs(\()p Fr(x;)17 b(\016)t Fs(\))28 b Fr(>)f Fq(\000)p Fs(1.)167 5386 y(W)-8 b(e)36 b(no)m(w)h(turn)e(to)h(the)g (asymptotics)f(of)g(those)h(static)f(eigen)m(v)m(ectors)i(around)f(\()p Fr(x;)17 b(\016)t Fs(\))33 b(=)f(\(0)p Fr(;)17 b Fs(0\).)21 5506 y(First)32 b(in)g(the)h(same)f(asymptotic)g(time)f(regime)g(as)i (in)f([12],)g(w)m(e)i(ha)m(v)m(e)1828 5755 y(12)p eop %%Page: 13 13 13 12 bop 21 219 a Fk(Lemma)37 b(2)49 b Fj(When)36 b Fr(\016)t Fj(,)f Fr(")h Fj(and)f Fr(t)h Fj(tend)f(to)h Fs(0)p Fj(,)g(we)f(have,)g(uniformly)h(in)f Fr(\015)f Ff(6)c Fs(1)p Fj(,)36 b Fq(j)p Fr("=)p Fs(\()p Fr(\015)5 b(t)p Fs(\))p Fq(j)28 b Ff(6)i Fr(M)46 b Fj(and)21 339 y Fq(j)p Fr(\016)t(=t)p Fq(j)27 b Ff(6)h Fr(M)444 303 y Fl(0)468 339 y Fj(,)35 b(for)g Fr(t)28 b(<)f Fs(0)p Fj(,)388 511 y Fo(\015)388 571 y(\015)444 595 y Fr(F)14 b Fs(\()p Fr(")605 554 y Fl(\000)p Fv(2)698 595 y Fr(\015)754 554 y Fv(2)794 595 y Fq(k)p Fr(x)22 b Fq(\000)h Fr(a)1072 554 y Fl(A)1133 595 y Fs(\()p Fr(t)p Fs(\))p Fq(k)1294 554 y Fv(2)1333 595 y Fs(\))1388 515 y Fo(\002)1429 595 y Fs(\010)1499 554 y Fl(\000)1499 622 y(A)1560 595 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b Fq(\000)h Fr( )1967 610 y Fv(2)2007 595 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))2229 515 y Fo(\003)2270 511 y(\015)2270 571 y(\015)2325 634 y Fp(L)2373 616 y Fh(1)2526 595 y Fs(=)82 b Fr(O)2778 455 y Fo(\022)2852 595 y Fq(j)p Fr(t)p Fq(j)21 b Fs(+)3062 451 y Fo(\014)3062 511 y(\014)3062 571 y(\014)3062 630 y(\014)3106 528 y Fr(\016)p 3106 573 47 4 v 3112 664 a(t)3162 451 y Fo(\014)3162 511 y(\014)3162 571 y(\014)3162 630 y(\014)3196 455 y(\023)3321 595 y Fr(;)400 783 y Fo(\015)400 843 y(\015)455 868 y Fr(F)14 b Fs(\()p Fr(")616 827 y Fl(\000)p Fv(2)710 868 y Fr(\015)766 827 y Fv(2)805 868 y Fq(k)p Fr(x)23 b Fq(\000)f Fr(a)1083 827 y Fl(B)1136 868 y Fs(\()p Fr(t)p Fs(\))p Fq(k)1297 827 y Fv(2)1336 868 y Fs(\))1391 787 y Fo(\002)1432 868 y Fs(\010)1502 827 y Fl(\000)1502 895 y(B)1562 868 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b(+)g Fr( )1967 883 y Fv(1)2007 868 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))2229 787 y Fo(\003)2270 783 y(\015)2270 843 y(\015)2325 907 y Fp(L)2373 888 y Fh(1)2526 868 y Fs(=)82 b Fr(O)2778 727 y Fo(\022)2852 868 y Fq(j)p Fr(t)p Fq(j)21 b Fs(+)3062 723 y Fo(\014)3062 783 y(\014)3062 843 y(\014)3062 903 y(\014)3106 800 y Fr(\016)p 3106 845 V 3112 936 a(t)3162 723 y Fo(\014)3162 783 y(\014)3162 843 y(\014)3162 903 y(\014)3196 727 y(\023)3321 868 y Fr(;)21 1119 y Fj(and)34 b(for)h Fr(t)28 b(>)g Fs(0)p Fj(,)388 1285 y Fo(\015)388 1345 y(\015)444 1370 y Fr(F)14 b Fs(\()p Fr(")605 1329 y Fl(\000)p Fv(2)698 1370 y Fr(\015)754 1329 y Fv(2)794 1370 y Fq(k)p Fr(x)22 b Fq(\000)h Fr(a)1072 1329 y Fl(A)1133 1370 y Fs(\()p Fr(t)p Fs(\))p Fq(k)1294 1329 y Fv(2)1333 1370 y Fs(\))1388 1289 y Fo(\002)1429 1370 y Fs(\010)1499 1329 y Fv(+)1499 1397 y Fl(A)1560 1370 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b Fq(\000)h Fr( )1967 1385 y Fv(1)2007 1370 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))2229 1289 y Fo(\003)2270 1285 y(\015)2270 1345 y(\015)2325 1409 y Fp(L)2373 1390 y Fh(1)2526 1370 y Fs(=)82 b Fr(O)2778 1229 y Fo(\022)2852 1370 y Fq(j)p Fr(t)p Fq(j)21 b Fs(+)3062 1225 y Fo(\014)3062 1285 y(\014)3062 1345 y(\014)3062 1405 y(\014)3106 1302 y Fr(\016)p 3106 1347 V 3112 1438 a(t)3162 1225 y Fo(\014)3162 1285 y(\014)3162 1345 y(\014)3162 1405 y(\014)3196 1229 y(\023)3321 1370 y Fr(;)398 1557 y Fo(\015)398 1617 y(\015)454 1642 y Fr(F)14 b Fs(\()p Fr(")615 1601 y Fl(\000)p Fv(2)708 1642 y Fr(\015)764 1601 y Fv(2)804 1642 y Fq(k)p Fr(x)22 b Fq(\000)h Fr(a)1082 1601 y Fl(B)1134 1642 y Fs(\()p Fr(t)p Fs(\))p Fq(k)1295 1601 y Fv(2)1334 1642 y Fs(\))1389 1561 y Fo(\002)1431 1642 y Fs(\010)1501 1601 y Fv(+)1501 1669 y Fl(B)1560 1642 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b Fq(\000)h Fr( )1967 1657 y Fv(2)2007 1642 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))2229 1561 y Fo(\003)2270 1557 y(\015)2270 1617 y(\015)2325 1681 y Fp(L)2373 1662 y Fh(1)2526 1642 y Fs(=)82 b Fr(O)2778 1502 y Fo(\022)2852 1642 y Fq(j)p Fr(t)p Fq(j)21 b Fs(+)3062 1498 y Fo(\014)3062 1557 y(\014)3062 1617 y(\014)3062 1677 y(\014)3106 1575 y Fr(\016)p 3106 1619 V 3112 1710 a(t)3162 1498 y Fo(\014)3162 1557 y(\014)3162 1617 y(\014)3162 1677 y(\014)3196 1502 y(\023)3321 1642 y Fr(:)167 1899 y Fk(Pro)s(of)70 b Fs(If)33 b Fr(x)662 1914 y Fv(1)729 1899 y Fq(6)p Fs(=)28 b(0,)935 2158 y Fq(j)o Fr(\032)p Fs(\()p Fr(x;)17 b(\016)t Fs(\))23 b Fq(\000)f Fr(r)s Fq(j)p Fr(x)1486 2173 y Fv(1)1525 2158 y Fq(jj)27 b Ff(6)1724 2091 y Fr(r)1771 2055 y Fv(2)1810 2091 y Fr(\016)1857 2055 y Fv(2)1918 2091 y Fs(+)22 b Fr(C)2086 2106 y Fp(\032)2126 2091 y Fs(\()p Fq(k)p Fr(x)p Fq(k)2319 2055 y Fv(2)2319 2116 y Fl(1)2416 2091 y Fs(+)g Fr(\016)2561 2055 y Fv(2)2601 2091 y Fs(\))2639 2055 y Fv(3)p Fp(=)p Fv(2)p 1724 2136 1025 4 v 2138 2227 a Fr(r)s Fq(j)p Fr(x)2268 2242 y Fv(1)2307 2227 y Fq(j)2791 2158 y Fr(:)21 2420 y Fs(Th)m(us,)34 b(for)e Fr(x)d Fq(2)f Fr(W)728 2384 y Fl(C)714 2445 y(\003)772 2420 y Fs(\()p Fr(t;)17 b(\016)n(;)g(";)g(\015)5 b Fs(\))32 b(with)g Fq(\003)c Fs(=)f(+)p Fr(;)17 b Fq(\000)p Fs(,)346 2677 y Fq(j)p Fr(B)5 b Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b Fq(\000)g Fs(sgn)q(\()p Fr(t)p Fs(\))p Fq(j)83 b Fs(=)1320 2533 y Fo(\014)1320 2592 y(\014)1320 2652 y(\014)1320 2712 y(\014)1363 2610 y Fs(sgn)q(\()p Fr(t)p Fs(\))p Fr(b)p Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b Fq(\000)h Fr(\032)p Fs(\()p Fr(x;)17 b(\016)t Fs(\))p 1363 2654 910 4 v 1682 2745 a Fr(\032)p Fs(\()p Fr(x;)g(\016)t Fs(\))2283 2533 y Fo(\014)2283 2592 y(\014)2283 2652 y(\014)2283 2712 y(\014)1160 2947 y Ff(6)1330 2880 y Fq(j)p Fr(b)p Fs(\()p Fr(x;)g(\016)t Fs(\))22 b Fq(\000)h Fr(r)s(x)1845 2895 y Fv(1)1885 2880 y Fq(j)e Fs(+)h Fq(j)p Fr(\032)p Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b Fq(\000)h Fr(r)s Fq(j)p Fr(x)2584 2895 y Fv(1)2623 2880 y Fq(jj)p 1330 2924 1349 4 v 1912 2976 a Fp(r)p 1912 2992 36 4 v 1912 3050 a Fv(2)1957 3015 y Fq(j)p Fr(x)2040 3030 y Fv(1)2079 3015 y Fq(j)1160 3236 y Ff(6)83 b Fs(2)1379 3169 y Fr(r)s(C)1496 3184 y Fp(b)1530 3169 y Fq(j)p Fr(x)1613 3184 y Fv(1)1652 3169 y Fq(j)p Fs(\()p Fq(k)p Fr(x)p Fq(k)1873 3133 y Fv(2)1873 3193 y Fl(1)1970 3169 y Fs(+)22 b Fr(\016)2115 3133 y Fv(2)2154 3169 y Fs(\))g(+)g Fr(r)2359 3133 y Fv(2)2398 3169 y Fr(\016)2445 3133 y Fv(2)2507 3169 y Fs(+)g Fr(C)2675 3184 y Fp(\032)2715 3169 y Fs(\()p Fq(k)p Fr(x)p Fq(k)2908 3133 y Fv(2)2908 3193 y Fl(1)3005 3169 y Fs(+)g Fr(\016)3150 3133 y Fv(2)3189 3169 y Fs(\))3227 3133 y Fv(3)p Fp(=)p Fv(2)p 1379 3213 1959 4 v 2268 3305 a Fr(r)2315 3276 y Fv(2)2354 3305 y Fr(x)2409 3270 y Fv(2)2409 3329 y(1)3380 3236 y Fr(;)21 3485 y Fs(hence)1156 3640 y Fr(B)5 b Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b Fq(\000)h Fs(sgn\()p Fr(t)p Fs(\))28 b(=)g Fr(O)2057 3500 y Fo(\022)2130 3640 y Fq(j)p Fr(t)p Fq(j)22 b Fs(+)2351 3573 y Fr(\016)2398 3536 y Fv(2)p 2351 3617 87 4 v 2357 3708 a Fr(t)2392 3680 y Fv(2)2447 3500 y Fo(\023)2569 3640 y Fr(:)21 3857 y Fs(Moreo)m(v)m(er,)34 b(similar)c(calculations)g (yield)1139 4108 y Fr(C)7 b Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b(+)g Fr(iD)s Fs(\()p Fr(x;)17 b(\016)t Fs(\))28 b(=)g Fr(O)2122 3968 y Fo(\022)2196 4108 y Fq(j)p Fr(t)p Fq(j)22 b Fs(+)2407 3964 y Fo(\014)2407 4024 y(\014)2407 4083 y(\014)2407 4143 y(\014)2450 4041 y Fr(\016)p 2450 4085 47 4 v 2456 4177 a(t)2507 3964 y Fo(\014)2507 4024 y(\014)2507 4083 y(\014)2507 4143 y(\014)2540 3968 y(\023)21 4359 y Fs(whic)m(h)33 b(leads)g(to)f(the)h(result.)43 b Ff(\003)167 4480 y Fs(By)34 b(similar)29 b(considerations,)j(w)m(e)i (get)e(also)g(in)g(the)h(opp)s(osite)f(asymptotic)g(case)21 4684 y Fk(Lemma)37 b(3)49 b Fj(When)36 b Fr(\016)t Fj(,)f Fr(")h Fj(and)f Fr(t)h Fj(tend)f(to)h Fs(0)p Fj(,)g(we)f(have,)g (uniformly)h(in)f Fr(\015)f Ff(6)c Fs(1)p Fj(,)36 b Fq(j)p Fr("=)p Fs(\()p Fr(\015)5 b(t)p Fs(\))p Fq(j)28 b Ff(6)i Fr(M)46 b Fj(and)21 4805 y Fq(j)p Fr(t=\016)t Fq(j)27 b Ff(6)h Fr(M)444 4769 y Fl(0)468 4805 y Fj(,)35 b(for)g Fr(t)28 b(<)f Fs(0)p Fj(,)91 4917 y Fo(\015)91 4977 y(\015)91 5036 y(\015)91 5096 y(\015)91 5156 y(\015)146 5091 y Fr(F)14 b Fs(\()p Fr(")307 5050 y Fl(\000)p Fv(2)401 5091 y Fr(\015)457 5050 y Fv(2)496 5091 y Fq(k)p Fr(x)23 b Fq(\000)f Fr(a)774 5050 y Fl(A)835 5091 y Fs(\()p Fr(t)p Fs(\))p Fq(k)996 5050 y Fv(2)1036 5091 y Fs(\))1091 4921 y Fo(")1148 5091 y Fs(\010)1218 5050 y Fl(\000)1218 5118 y(A)1279 5091 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b Fq(\000)1633 4941 y(p)p 1716 4941 49 4 v 83 x Fs(2)p 1633 5068 132 4 v 1674 5159 a(2)1775 5091 y(\()p Fr( )1876 5106 y Fv(1)1916 5091 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b(+)g Fr( )2321 5106 y Fv(2)2360 5091 y Fs(\()p Fr(x;)17 b(\016)t Fs(\)\))2620 4921 y Fo(#)2678 4917 y(\015)2678 4977 y(\015)2678 5036 y(\015)2678 5096 y(\015)2678 5156 y(\015)2734 5220 y Fp(L)2782 5201 y Fh(1)2934 5091 y Fs(=)83 b Fr(O)3187 4951 y Fo(\022)3260 5091 y Fr(\016)26 b Fs(+)3427 4947 y Fo(\014)3427 5006 y(\014)3427 5066 y(\014)3427 5126 y(\014)3476 5024 y Fr(t)p 3470 5068 47 4 v 3470 5159 a(\016)3527 4947 y Fo(\014)3527 5006 y(\014)3527 5066 y(\014)3527 5126 y(\014)3560 4951 y(\023)3685 5091 y Fr(;)24 5253 y Fo(\015)24 5313 y(\015)24 5373 y(\015)24 5432 y(\015)24 5492 y(\015)79 5427 y Fr(F)14 b Fs(\()p Fr(")240 5386 y Fl(\000)p Fv(2)334 5427 y Fr(\015)390 5386 y Fv(2)429 5427 y Fq(k)p Fr(x)22 b Fq(\000)h Fr(a)707 5386 y Fl(B)759 5427 y Fs(\()p Fr(t)p Fs(\))p Fq(k)920 5386 y Fv(2)960 5427 y Fs(\))1015 5257 y Fo(")1072 5427 y Fs(\010)1142 5386 y Fl(\000)1142 5454 y(B)1202 5427 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b Fq(\000)1555 5278 y(p)p 1639 5278 49 4 v 1639 5360 a Fs(2)p 1555 5404 132 4 v 1597 5496 a(2)1697 5427 y(\()p Fq(\000)p Fr( )1875 5442 y Fv(1)1916 5427 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b(+)g Fr( )2321 5442 y Fv(2)2360 5427 y Fs(\()p Fr(x;)17 b(\016)t Fs(\)\))2620 5257 y Fo(#)2678 5253 y(\015)2678 5313 y(\015)2678 5373 y(\015)2678 5432 y(\015)2678 5492 y(\015)2734 5556 y Fp(L)2782 5537 y Fh(1)2934 5427 y Fs(=)83 b Fr(O)3187 5287 y Fo(\022)3260 5427 y Fr(\016)26 b Fs(+)3427 5283 y Fo(\014)3427 5343 y(\014)3427 5402 y(\014)3427 5462 y(\014)3476 5360 y Fr(t)p 3470 5404 47 4 v 3470 5496 a(\016)3527 5283 y Fo(\014)3527 5343 y(\014)3527 5402 y(\014)3527 5462 y(\014)3560 5287 y(\023)3685 5427 y Fr(;)1828 5755 y Fs(13)p eop %%Page: 14 14 14 13 bop 21 219 a Fj(and)34 b(for)h Fr(t)28 b(>)g Fs(0)p Fj(,)91 336 y Fo(\015)91 396 y(\015)91 456 y(\015)91 516 y(\015)91 575 y(\015)146 511 y Fr(F)14 b Fs(\()p Fr(")307 469 y Fl(\000)p Fv(2)401 511 y Fr(\015)457 469 y Fv(2)496 511 y Fq(k)p Fr(x)23 b Fq(\000)f Fr(a)774 469 y Fl(A)835 511 y Fs(\()p Fr(t)p Fs(\))p Fq(k)996 469 y Fv(2)1036 511 y Fs(\))1091 340 y Fo(")1148 511 y Fs(\010)1218 469 y Fv(+)1218 538 y Fl(A)1279 511 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b Fq(\000)1633 361 y(p)p 1716 361 49 4 v 82 x Fs(2)p 1633 488 132 4 v 1674 579 a(2)1775 511 y(\()p Fr( )1876 526 y Fv(1)1916 511 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b(+)g Fr( )2321 526 y Fv(2)2360 511 y Fs(\()p Fr(x;)17 b(\016)t Fs(\)\))2620 340 y Fo(#)2678 336 y(\015)2678 396 y(\015)2678 456 y(\015)2678 516 y(\015)2678 575 y(\015)2734 639 y Fp(L)2782 620 y Fh(1)2934 511 y Fs(=)83 b Fr(O)3187 370 y Fo(\022)3260 511 y Fr(\016)26 b Fs(+)3427 366 y Fo(\014)3427 426 y(\014)3427 486 y(\014)3427 545 y(\014)3476 443 y Fr(t)p 3470 488 47 4 v 3470 579 a(\016)3527 366 y Fo(\014)3527 426 y(\014)3527 486 y(\014)3527 545 y(\014)3560 370 y(\023)3685 511 y Fr(;)24 672 y Fo(\015)24 732 y(\015)24 792 y(\015)24 852 y(\015)24 912 y(\015)79 847 y Fr(F)14 b Fs(\()p Fr(")240 806 y Fl(\000)p Fv(2)334 847 y Fr(\015)390 806 y Fv(2)429 847 y Fq(k)p Fr(x)22 b Fq(\000)h Fr(a)707 806 y Fl(B)759 847 y Fs(\()p Fr(t)p Fs(\))p Fq(k)920 806 y Fv(2)960 847 y Fs(\))1015 676 y Fo(")1072 847 y Fs(\010)1142 806 y Fv(+)1142 874 y Fl(B)1202 847 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b Fq(\000)1555 697 y(p)p 1639 697 49 4 v 1639 779 a Fs(2)p 1555 824 132 4 v 1597 915 a(2)1697 847 y(\()p Fq(\000)p Fr( )1875 862 y Fv(1)1916 847 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b(+)g Fr( )2321 862 y Fv(2)2360 847 y Fs(\()p Fr(x;)17 b(\016)t Fs(\)\))2620 676 y Fo(#)2678 672 y(\015)2678 732 y(\015)2678 792 y(\015)2678 852 y(\015)2678 912 y(\015)2734 976 y Fp(L)2782 957 y Fh(1)2934 847 y Fs(=)83 b Fr(O)3187 706 y Fo(\022)3260 847 y Fr(\016)26 b Fs(+)3427 702 y Fo(\014)3427 762 y(\014)3427 822 y(\014)3427 882 y(\014)3476 779 y Fr(t)p 3470 824 47 4 v 3470 915 a(\016)3527 702 y Fo(\014)3527 762 y(\014)3527 822 y(\014)3527 882 y(\014)3560 706 y(\023)3685 847 y Fr(:)167 1150 y Fs(W)-8 b(e)33 b(in)m(tro)s(duce)g(no)m(w)g(dynamical)e(eigen)m(v)m (ectors)1259 1360 y(\010)1329 1319 y Fl(\003)1329 1385 y(C)1374 1360 y Fs(\()p Fr(t;)17 b(x;)g(\016)t Fs(\))28 b(=)f Fr(e)1851 1319 y Fp(i!)1921 1296 y Fh(\003)1919 1342 y(C)1959 1319 y Fv(\()p Fp(t;x;\016)r Fv(\))2157 1360 y Fs(\010)2227 1319 y Fl(\003)2227 1385 y(C)2272 1360 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))21 1571 y(for)32 b Fq(C)i Fs(=)28 b Fq(A)p Fr(;)17 b Fq(B)35 b Fs(and)e Fq(\003)27 b Fs(=)h(+)p Fr(;)17 b Fq(\000)32 b Fs(in)g(order)h(to)f (ful\014ll)e(the)j(orthogonalit)m(y)e(condition)901 1833 y Fq(h)p Fs(\010)1010 1792 y Fl(\003)1010 1858 y(C)1055 1833 y Fs(\()p Fr(t;)17 b(x;)g(\016)t Fs(\))p Fr(;)1400 1693 y Fo(\022)1501 1766 y Fr(@)p 1483 1810 93 4 v 1483 1901 a(@)5 b(t)1607 1833 y Fs(+)22 b Fr(\021)1757 1792 y Fl(C)1802 1833 y Fs(\()p Fr(t)p Fs(\))p Fr(:)p Fq(r)2023 1848 y Fp(x)2067 1693 y Fo(\023)2157 1833 y Fs(\010)2227 1792 y Fl(\003)2227 1858 y(C)2273 1833 y Fs(\()p Fr(t;)17 b(x;)g(\016)t Fs(\))p Fq(i)27 b Fs(=)g(0)33 b Fr(:)167 2108 y Fs(In)m(tro)s(ducing)g(the)g(new)g(v)-5 b(ariables)32 b Fr(s)27 b Fs(=)h Fr(t)p Fs(,)33 b Fr(z)f Fs(=)c Fr(x)22 b Fq(\000)h Fr(a)2150 2071 y Fl(C)2195 2108 y Fs(\()p Fr(t)p Fs(\),)33 b(w)m(e)g(ha)m(v)m(e)h(the)f(su\016cien)m(t)h (condition)643 2300 y Fr(@)p 620 2345 103 4 v 620 2436 a(@)5 b(s)740 2367 y Fs(~)-56 b Fr(!)798 2326 y Fl(\003)794 2392 y(C)838 2367 y Fs(\()p Fr(s;)17 b(z)t(;)g(\016)t Fs(\))28 b(=)g Fr(i)p Fq(h)p Fs(\010)1418 2326 y Fl(\003)1418 2392 y(C)1463 2367 y Fs(\()p Fr(a)1552 2326 y Fl(C)1597 2367 y Fs(\()p Fr(s)p Fs(\))22 b(+)g Fr(z)t(;)17 b(\016)t Fs(\))p Fr(;)g(\021)2113 2326 y Fl(C)2158 2367 y Fs(\()p Fr(s)p Fs(\))p Fr(:)p Fq(r)2390 2382 y Fp(x)2434 2367 y Fs(\010)2504 2326 y Fl(\003)2504 2392 y(C)2549 2367 y Fs(\()p Fr(a)2638 2326 y Fl(C)2684 2367 y Fs(\()p Fr(s)p Fs(\))22 b(+)g Fr(z)t(;)17 b(\016)t Fs(\))p Fq(i)415 b Fs(\(15\))21 2616 y(where)41 b(~)-57 b Fr(!)367 2580 y Fl(\003)363 2641 y(C)408 2616 y Fs(\()p Fr(s;)17 b(z)t(;)g(\016)t Fs(\))27 b(=)h Fr(!)910 2580 y Fl(\003)906 2641 y(C)951 2616 y Fs(\()p Fr(s;)17 b(a)1130 2580 y Fl(C)1174 2616 y Fs(\()p Fr(s)p Fs(\))j(+)g Fr(z)t(;)d(\016)t Fs(\).)43 b(If)31 b(w)m(e)i(supp)s(ose)f Fr(!)2329 2574 y Fl(\000)2325 2643 y(C)2388 2616 y Fs(\()p Fq(\000)p Fr(T)8 b(;)17 b(x;)g(\016)t Fs(\))28 b(=)f Fr(!)2992 2574 y Fv(+)2988 2643 y Fl(C)3051 2616 y Fs(\()p Fr(T)8 b(;)17 b(x;)g(\016)t Fs(\))27 b(=)h(0,)j(w)m(e)21 2736 y(ha)m(v)m(e)j(the)f(follo)m(wing)d (result)21 2953 y Fk(Lemma)37 b(4)49 b Fj(When)35 b Fr(\016)t Fj(,)f Fr(")h Fj(and)f Fr(t)h Fj(tend)g(to)g Fs(0)p Fj(,)g(we)f(have,)g (for)h Fq(C)f Fs(=)27 b Fq(A)p Fr(;)17 b Fq(B)38 b Fj(and)c Fq(\003)28 b Fs(=)f(+)p Fr(;)17 b Fq(\000)p Fj(,)50 3101 y Fo(\015)50 3161 y(\015)50 3221 y(\015)105 3216 y Fr(F)d Fs(\()p Fr(")266 3174 y Fl(\000)p Fv(2)360 3216 y Fr(\015)416 3174 y Fv(2)455 3216 y Fq(k)p Fr(x)22 b Fq(\000)h Fr(a)733 3174 y Fl(C)778 3216 y Fs(\()p Fr(t)p Fs(\))p Fq(k)939 3174 y Fv(2)979 3216 y Fs(\))1034 3105 y Fo(h)1080 3216 y Fs(\010)1150 3174 y Fl(\003)1150 3240 y(C)1196 3216 y Fs(\()p Fr(t;)17 b(x;)g(\016)t Fs(\))22 b Fq(\000)g Fr(e)1663 3174 y Fp(i!)1733 3151 y Fh(\003)1731 3197 y(C)1772 3174 y Fv(\()p Fp(t;a)1881 3151 y Fh(C)1922 3174 y Fv(\()p Fp(t)p Fv(\))p Fp(;\016)r Fv(\))2087 3216 y Fs(\010)2157 3174 y Fl(\003)2157 3240 y(C)2203 3216 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))2425 3105 y Fo(i)2472 3101 y(\015)2472 3161 y(\015)2472 3221 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737 5468 249 4 v 737 5526 a(\015)t Fv(\()p Fl(j)p Fp(t)p Fl(j)p Fv(+)p Fp(\016)r Fv(\))995 5381 y Fo(\021)1087 5491 y Fs(;)1828 5755 y(14)p eop %%Page: 15 15 15 14 bop 140 233 a Fs(3.)49 b([)p Fr(C)7 b Fs(])396 197 y Fp(x)396 258 y(t)468 233 y Fs(=)27 b Fr(O)665 123 y Fo(\020)738 194 y Fp(")p 735 211 41 4 v 735 268 a(\015)807 233 y Fs(+)1008 186 y Fp(")p Fl(j)p Fp(t)p Fl(j)p 915 211 284 4 v 915 268 a Fp(\015)t Fv(\()p Fl(j)p Fp(t)p Fl(j)p Fv(+)p Fp(\016)r Fv(\))1162 249 y Fi(2)1208 123 y Fo(\021)1300 233 y Fs(with)32 b(same)h(estimate)e(for)h Fr(D)k Fs(;)140 464 y(4.)49 b([)p Fr(L)p Fs(])385 428 y Fp(x)385 489 y(t)457 464 y Fs(=)28 b Fr(O)s Fs(\()691 425 y Fp(")p 687 441 41 4 v 687 499 a(\015)736 464 y Fs(\))33 b(for)f Fq(r)1039 479 y Fp(x)1082 464 y Fr(c)p Fs(,)h Fq(r)1267 479 y Fp(x)1311 464 y Fr(d)f Fs(and)h Fr(\025)1641 479 y Fp(ij)1734 464 y Fs(;)140 717 y(5.)49 b([)p Fr(f)351 681 y Fl(\000)410 717 y Fq(r)493 732 y Fp(x)537 717 y Fr(f)596 681 y Fl(\000)677 717 y Fs(+)22 b Fr(g)826 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b(\016)t Fs(\))48 b(is)f(the)h(restriction)f(to)21 5265 y Fr(P)98 5229 y Fl(?)157 5265 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))p Fq(H)25 b Fs(of)f(the)h(resolv)m(en)m(t)h(of)e Fr(h)p Fs(\()p Fr(x;)17 b(\016)t Fs(\))24 b(tak)m(en)i(in)d Fr(E)1993 5280 y Fl(B)2046 5265 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))24 b(and)g(w)m(e)i(ha)m(v)m(e)g(dropp)s(ed)f(the)g(v)-5 b(ariables)21 5386 y(in)35 b(the)h(dynamical)e(eigen)m(v)m(ectors)k (\010)1400 5345 y Fl(\000)1400 5413 y(C)1459 5386 y Fs(\()p Fr(t;)17 b(x;)g(\016)t Fs(\))36 b(and)g(in)f(the)h(eigen)m(v)-5 b(alues)36 b Fr(E)2857 5401 y Fl(C)2902 5386 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))36 b(for)f Fq(C)40 b Fs(=)33 b Fq(A)p Fr(;)17 b Fq(B)s Fs(.)21 5506 y(T)-8 b(erms)32 b(that)g(in)m(v)m(olv)m(e)g(deriv)-5 b(ativ)m(es)31 b(of)g(the)i (cuto\013)e(function)g(turn)h(out)g(to)f(b)s(e)h(exp)s(onen)m(tially)f (small)1828 5755 y(16)p eop %%Page: 17 17 17 16 bop 21 219 a Fs(and)43 b(w)m(e)h(can)f(neglect)g(them)g (comparing)e(with)h(others.)75 b(Th)m(us)45 b(w)m(e)e(only)g(treat)f (the)i(remaining)21 339 y(terms)33 b(follo)m(wing)d(the)j(same)f(metho) s(d)g(as)h(on)f(pages)h(108-110)e(of)h([9].)167 459 y(The)k(quan)m(tit) m(y)f(exp)928 379 y Fo(\000)974 459 y Fq(\000)1083 420 y Fp(i)p 1061 436 68 4 v 1061 494 a(")1094 475 y Fi(2)1138 459 y Fr(S)1204 423 y Fl(B)1256 459 y Fs(\()p Fr(t)p Fs(\))22 b Fq(\000)1503 420 y Fp(i)p 1499 436 33 4 v 1499 494 a(")1541 459 y Fr(\021)1593 423 y Fl(B)1645 459 y Fs(\()p Fr(t)p Fs(\))p Fr(:y)1835 379 y Fo(\001)16 b(\002)1938 459 y Fr(i")2017 423 y Fv(2)2079 420 y Fp(@)p 2067 436 67 4 v 2067 494 a(@)t(t)2165 459 y Fq(\000)23 b Fr(H)8 b Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))2640 379 y Fo(\003)2697 459 y Fs(\011\()p Fr(t;)g(x;)g(y)t(;)g(\016)n(;)g(") p Fs(\))33 b(is)h(the)h(sum)21 580 y(of)23 b(35)h(pro)s(duct)g(terms.) 40 b(P)m(erforming)23 b(brute)h(force)g(estimates)g(on)f(eac)m(h)i(pro) s(duct)f(with)g(the)g Fr(L)3437 544 y Fv(2)3477 580 y Fs(-norm)21 700 y(for)35 b Fr(y)t Fs(-dep)s(enden)m(t)g(factors)h(and)f (the)g Fr(L)1479 664 y Fl(1)1554 700 y Fs(-norm)f(for)h Fr(x)p Fs(-dep)s(enden)m(t)i(ones,)f(w)m(e)g(need)h(the)e(follo)m(wing) 21 820 y(estimates)d(of)g(the)h(singular)f(terms)g(on)h(the)g(supp)s (ort)f(of)h Fr(F)46 b Fs(:)166 1049 y Fq(\017)j Fs(successiv)m(e)35 b(deriv)-5 b(ativ)m(es)33 b(of)f(the)h(gap)f(b)s(et)m(w)m(een)j(eigen)m (v)-5 b(alues)822 1253 y(1)p 435 1298 823 4 v 435 1389 a Fr(E)507 1404 y Fl(A)567 1389 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))22 b Fq(\000)h Fr(E)983 1404 y Fl(B)1035 1389 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))1295 1321 y(=)28 b Fr(O)1492 1180 y Fo(\022)1680 1253 y Fs(1)p 1576 1298 258 4 v 1576 1389 a Fq(j)p Fr(t)p Fq(j)21 b Fs(+)h Fr(\016)1843 1180 y Fo(\023)1966 1321 y Fr(;)49 b Fq(r)2125 1336 y Fp(x)2186 1180 y Fo(\022)2434 1253 y Fs(1)p 2269 1298 379 4 v 2269 1389 a Fr(E)2341 1404 y Fl(A)2424 1389 y Fq(\000)22 b Fr(E)2595 1404 y Fl(B)2658 1180 y Fo(\023)2759 1321 y Fs(=)27 b Fr(O)2956 1180 y Fo(\022)3201 1253 y Fs(1)p 3039 1298 373 4 v 3039 1389 a(\()p Fq(j)p Fr(t)p Fq(j)22 b Fs(+)g Fr(\016)t Fs(\))3373 1360 y Fv(2)3422 1180 y Fo(\023)3545 1321 y Fr(;)1234 1668 y Fs(\001)1315 1683 y Fp(x)1376 1527 y Fo(\022)1624 1600 y Fs(1)p 1459 1645 379 4 v 1459 1736 a Fr(E)1531 1751 y Fl(A)1614 1736 y Fq(\000)h Fr(E)1786 1751 y Fl(B)1848 1527 y Fo(\023)1949 1668 y Fs(=)28 b Fr(O)2146 1527 y Fo(\022)2392 1600 y Fs(1)p 2230 1645 373 4 v 2230 1736 a(\()p Fq(j)p Fr(t)p Fq(j)21 b Fs(+)h Fr(\016)t Fs(\))2563 1707 y Fv(3)2613 1527 y Fo(\023)2735 1668 y Fr(;)166 1935 y Fq(\017)49 b Fs(successiv)m(e)35 b(deriv)-5 b(ativ)m(es)33 b(of)f(the)h(dynamic)f (eigen)m(v)m(ectors)577 2207 y Fq(r)660 2222 y Fp(x)704 2207 y Fs(\010)774 2166 y Fl(\000)774 2234 y(C)834 2207 y Fs(\()p Fr(t;)17 b(x;)g(\016)t Fs(\))27 b(=)h Fr(O)1360 2066 y Fo(\022)1547 2140 y Fs(1)p 1443 2184 258 4 v 1443 2275 a Fq(j)p Fr(t)p Fq(j)22 b Fs(+)g Fr(\016)1711 2066 y Fo(\023)1833 2207 y Fr(;)1909 2066 y Fo(\022)2010 2140 y Fr(@)p 1993 2184 93 4 v 1993 2275 a(@)5 b(t)2117 2207 y Fs(+)22 b Fr(\021)2267 2166 y Fl(B)2319 2207 y Fr(:)p Fq(r)2429 2222 y Fp(x)2473 2066 y Fo(\023)2563 2207 y Fs(\010)2633 2166 y Fl(\000)2633 2234 y(A)2721 2207 y Fs(=)28 b Fr(O)2919 2066 y Fo(\022)3106 2140 y Fs(1)p 3002 2184 258 4 v 3002 2275 a Fq(j)p Fr(t)p Fq(j)22 b Fs(+)g Fr(\016)3270 2066 y Fo(\023)3392 2207 y Fr(;)675 2413 y Fo(\022)776 2487 y Fr(@)p 759 2531 93 4 v 759 2622 a(@)5 b(t)883 2554 y Fs(+)22 b Fr(\021)1033 2513 y Fl(C)1078 2554 y Fr(:)p Fq(r)1188 2569 y Fp(x)1232 2413 y Fo(\023)1322 2554 y Fs(\010)1392 2513 y Fl(\000)1392 2581 y(C)1479 2554 y Fs(=)27 b Fr(O)1676 2413 y Fo(\022)1864 2487 y Fs(1)p 1759 2531 258 4 v 1759 2622 a Fq(j)p Fr(t)p Fq(j)22 b Fs(+)g Fr(\016)2027 2413 y Fo(\023)2149 2554 y Fr(;)50 b Fs(\001)2307 2569 y Fp(x)2351 2554 y Fs(\010)2421 2513 y Fl(\000)2421 2581 y(C)2508 2554 y Fs(=)28 b Fr(O)2705 2413 y Fo(\022)2951 2487 y Fs(1)p 2789 2531 373 4 v 2789 2622 a(\()p Fq(j)p Fr(t)p Fq(j)21 b Fs(+)h Fr(\016)t Fs(\))3122 2593 y Fv(2)3172 2413 y Fo(\023)3294 2554 y Fr(;)266 2731 y Fo(\022)366 2804 y Fr(@)p 349 2849 93 4 v 349 2940 a(@)5 b(t)473 2872 y Fs(+)22 b Fr(\021)623 2831 y Fl(C)668 2872 y Fr(:)p Fq(r)778 2887 y Fp(x)822 2731 y Fo(\023)895 2754 y Fv(2)951 2872 y Fs(\010)1021 2831 y Fl(\000)1021 2899 y(C)1108 2872 y Fs(=)28 b Fr(O)1306 2731 y Fo(\022)1551 2804 y Fs(1)p 1389 2849 373 4 v 1389 2940 a(\()p Fq(j)p Fr(t)p Fq(j)22 b Fs(+)g Fr(\016)t Fs(\))1723 2911 y Fv(2)1772 2731 y Fo(\023)1895 2872 y Fr(;)49 b Fq(r)2054 2887 y Fp(x)2114 2731 y Fo(\022)2215 2804 y Fr(@)p 2198 2849 93 4 v 2198 2940 a(@)5 b(t)2322 2872 y Fs(+)22 b Fr(\021)2472 2831 y Fl(C)2517 2872 y Fr(:)p Fq(r)2627 2887 y Fp(x)2671 2731 y Fo(\023)2761 2872 y Fs(\010)2831 2831 y Fl(\000)2831 2899 y(C)2918 2872 y Fs(=)27 b Fr(O)3115 2731 y Fo(\022)3360 2804 y Fs(1)p 3198 2849 373 4 v 3198 2940 a(\()p Fq(j)p Fr(t)p Fq(j)22 b Fs(+)g Fr(\016)t Fs(\))3532 2911 y Fv(2)3581 2731 y Fo(\023)3704 2872 y Fr(;)1119 3173 y Fs(\001)1200 3188 y Fp(x)1261 3033 y Fo(\022)1362 3106 y Fr(@)p 1344 3150 93 4 v 1344 3242 a(@)5 b(t)1468 3173 y Fs(+)22 b Fr(\021)1618 3132 y Fl(C)1663 3173 y Fr(:)p Fq(r)1773 3188 y Fp(x)1817 3033 y Fo(\023)1907 3173 y Fs(\010)1977 3132 y Fl(\000)1977 3200 y(C)2064 3173 y Fs(=)28 b Fr(O)2262 3033 y Fo(\022)2507 3106 y Fs(1)p 2345 3150 373 4 v 2345 3242 a(\()p Fq(j)p Fr(t)p Fq(j)22 b Fs(+)g Fr(\016)t Fs(\))2679 3213 y Fv(3)2728 3033 y Fo(\023)2850 3173 y Fr(;)167 3499 y Fs(Finally)499 3384 y Fo(\015)499 3444 y(\015)499 3504 y(\015)555 3418 y(\002)596 3499 y Fr(i")675 3462 y Fv(2)737 3459 y Fp(@)p 725 3476 67 4 v 725 3533 a(@)t(t)823 3499 y Fq(\000)h Fr(H)8 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))1219 3418 y Fo(\003)1276 3499 y Fs(\011)1369 3388 y Fo(\020)1428 3499 y Fr(t;)g(x;)1616 3451 y Fp(x)p Fl(\000)p Fp(a)1748 3428 y Fh(B)1794 3451 y Fv(\()p Fp(t)p Fv(\))p 1616 3476 259 4 v 1729 3533 a Fp(")1884 3499 y Fr(;)g(\016)n(;)g(")2059 3388 y Fo(\021)2118 3384 y(\015)2118 3444 y(\015)2118 3504 y(\015)2174 3568 y Fp(L)2222 3549 y Fi(2)2256 3568 y Fv(\()p Fp(x)p Fv(\))2388 3499 y Fs(is)32 b(b)s(ounded)h(b)m(y)h(a)e(constan)m(t)h(times)124 3698 y Fo(\022)197 3838 y Fs(1)22 b(+)520 3771 y Fr(")566 3735 y Fv(2)p 376 3816 373 4 v 376 3907 a Fs(\()p Fq(j)p Fr(t)p Fq(j)g Fs(+)g Fr(\016)t Fs(\))710 3878 y Fv(2)759 3698 y Fo(\023)17 b(\024)1055 3771 y Fr(")1101 3735 y Fv(4)p 912 3816 V 912 3907 a Fs(\()p Fq(j)p Fr(t)p Fq(j)k Fs(+)h Fr(\016)t Fs(\))1245 3878 y Fv(3)1294 3838 y Fq(k)p Fr(\036)1402 3853 y Fp(l)1428 3838 y Fq(k)1478 3855 y Fp(L)1526 3836 y Fi(2)1560 3855 y Fv(\()p Fp(y)r Fv(\))1679 3838 y Fs(+)1873 3771 y Fr(")1919 3735 y Fv(3)p 1787 3816 258 4 v 1787 3907 a Fq(j)p Fr(t)p Fq(j)g Fs(+)g Fr(\016)2054 3838 y Fq(kr)2187 3853 y Fp(y)2229 3838 y Fr(\036)2287 3853 y Fp(l)2312 3838 y Fq(k)2362 3855 y Fp(L)2410 3836 y Fi(2)2445 3855 y Fv(\()p Fp(y)r Fv(\))2563 3838 y Fs(+)g Fr(")2707 3797 y Fv(3)2763 3754 y Fo(\015)2763 3814 y(\015)2818 3838 y Fq(k)p Fr(y)t Fq(k)2970 3797 y Fv(3)3009 3838 y Fr(\036)3067 3853 y Fp(l)3092 3754 y Fo(\015)3092 3814 y(\015)3148 3877 y Fp(L)3196 3859 y Fi(2)3230 3877 y Fv(\()p Fp(y)r Fv(\))3326 3698 y Fo(\025)3428 3838 y Fr(:)103 b Fs(\(20\))167 4120 y(T)-8 b(o)29 b(conclude,)i(w)m(e) e(apply)g(lemma)e(5)h(with)h(estimate)f(\(20\))g(and)h(note)g(that)f (the)i Fr(")3150 4084 y Fv(2)3189 4120 y Fs(-term)e(of)g(\(19\))21 4268 y(is)k(of)g(order)h Fr(O)579 4157 y Fo(\020)736 4229 y Fp(")769 4205 y Fi(2)p 648 4245 243 4 v 648 4302 a Fv(\()p Fl(j)p Fp(t)p Fl(j)p Fv(+)p Fp(\016)r Fv(\))855 4283 y Fi(2)901 4157 y Fo(\021)961 4268 y Fs(.)21 4601 y Ft(5)161 b(Near)54 b(the)f(Crossing)21 4820 y Fs(W)-8 b(e)32 b(no)m(w)h(need)f(an)g(Ansatz)h(around)e(the)h(crossing)g(time)e (\(when)j(the)f(semi-classical)d(dynamics)j(of)21 4940 y(the)j(n)m(uclei)f(reac)m(h)i(the)f(crossing)g(surface)g(\000\))f (where)i(the)f(eigen)m(v)m(ectors)i(are)d(not)h(w)m(ell)f(de\014ned,)i (so)21 5060 y(w)m(e)e(mak)m(e)f(this)g(Ansatz)g(essen)m(tially)g(liv)m (e)f(in)g(the)i(t)m(w)m(o-dimensional)c(eigenspace)k Fr(P)14 b Fs(\()p Fr(x;)j(\016)t Fs(\))p Fq(H)34 b Fs(of)e(the)21 5181 y(t)m(w)m(o)h(eigen)m(v)-5 b(alues.)1828 5755 y(17)p eop %%Page: 18 18 18 17 bop 21 219 a Fk(Prop)s(osition)36 b(3)48 b Fj(In)35 b(the)g(inner)f(r)-5 b(e)g(gion)34 b Fq(j)p Fr(t)p Fq(j)27 b Ff(6)h Fr(t)1829 234 y Fp(i)1858 219 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fj(,)34 b(we)h(set)117 496 y Fr( )180 511 y Fp(I)220 496 y Fs(\()p Fr(t;)17 b(x;)g(\016)n(;)g(")p Fs(\))27 b(=)h(exp)902 355 y Fo(\022)975 496 y Fr(i)1018 428 y(S)6 b Fs(\()p Fr(t)p Fs(\))22 b(+)g Fr(\021)t Fs(\()p Fr(t)p Fs(\))p Fr(:)p Fs(\()p Fr(x)h Fq(\000)f Fr(a)p Fs(\()p Fr(t)p Fs(\)\))p 1018 473 903 4 v 1427 564 a Fr(")1473 535 y Fv(2)1930 355 y Fo(\023)2040 401 y(X)2020 613 y Fp(k)r Fv(=1)p Fp(;)p Fv(2)2221 496 y Fr(f)2269 511 y Fp(k)2328 355 y Fo(\022)2416 428 y Fr(t)p 2411 473 46 4 v 2411 564 a(")2467 496 y(;)2521 428 y(x)g Fq(\000)h Fr(a)p Fs(\()p Fr(t)p Fs(\))p 2521 473 340 4 v 2667 564 a Fr(")2870 496 y(;)17 b(\016)n(;)g(")3045 355 y Fo(\023)3135 496 y Fr( )3198 511 y Fp(k)3241 496 y Fs(\()p Fr(x;)g(\016)t Fs(\))95 b(\(21\))21 813 y Fj(with)271 1108 y Fo(\022)385 1188 y Fr(f)433 1203 y Fv(1)473 1188 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))385 1308 y Fr(f)433 1323 y Fv(2)473 1308 y Fs(\()p Fr(s;)g(y)t(;)g(\016)n(;)g(")p Fs(\))906 1108 y Fo(\023)1048 1249 y Fs(=)1152 925 y Fo(8)1152 1015 y(>)1152 1044 y(>)1152 1074 y(>)1152 1104 y(>)1152 1134 y(<)1152 1313 y(>)1152 1343 y(>)1152 1373 y(>)1152 1403 y(>)1152 1433 y(:)1552 929 y( )1672 1038 y Fr(g)1719 1053 y Fv(1)1759 1038 y Fs(\()p Fr(y)t(;)g(\016)n(;)g(")p Fs(\))p Fr(e)2107 1001 y Fl(\000)p Fp(ir)r Fv(\()p Fp(\021)2284 978 y Fi(0)2282 1022 y(1)2328 974 y Fn(s)2357 953 y Fi(2)p 2328 986 64 3 v 2345 1027 a(2)2402 1001 y Fv(+)p Fp(sy)2525 1010 y Fi(1)2559 1001 y Fv(\))1700 1191 y Fr(g)1747 1206 y Fv(2)1786 1191 y Fs(\()p Fr(y)t(;)g(\016)n(;)g(")p Fs(\))p Fr(e)2134 1154 y Fp(ir)r Fv(\()p Fp(\021)2256 1131 y Fi(0)2254 1175 y(1)2300 1127 y Fn(s)2329 1106 y Fi(2)p 2300 1139 V 2317 1181 a(2)2374 1154 y Fv(+)p Fp(sy)2497 1163 y Fi(1)2531 1154 y Fv(\))2632 929 y Fo(!)2762 1099 y Fj(if)35 b Fr(\016)t(=")27 b Fq(!)g Fs(0)1282 1231 y Fo( )1402 1341 y Fr(g)1449 1356 y Fv(1)1488 1341 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))p Fr(e)1836 1304 y Fl(\000)p Fp(ir)1958 1277 y Fn(\016)p 1958 1289 30 3 v 1958 1330 a(")1998 1304 y Fp(s)2057 1341 y Fq(\000)22 b Fr(g)2203 1356 y Fv(2)2242 1341 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))p Fr(e)2590 1304 y Fp(ir)2657 1277 y Fn(\016)p 2657 1289 V 2657 1330 a(")2697 1304 y Fp(s)1403 1475 y Fr(g)1450 1490 y Fv(1)1489 1475 y Fs(\()p Fr(y)t(;)g(\016)n(;)g(")p Fs(\))p Fr(e)1837 1439 y Fl(\000)p Fp(ir)1959 1411 y Fn(\016)p 1959 1423 V 1959 1465 a(")1999 1439 y Fp(s)2057 1475 y Fs(+)22 b Fr(g)2202 1490 y Fv(2)2242 1475 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))p Fr(e)2590 1439 y Fp(ir)2656 1411 y Fn(\016)p 2656 1423 V 2656 1465 a(")2696 1439 y Fp(s)2775 1231 y Fo(!)2906 1402 y Fj(if)34 b Fr(\016)t(=")27 b Fq(!)h Fs(+)p Fq(1)21 1709 y Fj(wher)-5 b(e)34 b Fr(g)343 1724 y Fp(k)386 1709 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))33 b Fj(satisfy)166 1912 y Fq(\017)49 b Fr(g)312 1927 y Fp(k)382 1912 y Fq(2)28 b Fr(H)565 1876 y Fv(2)604 1912 y Fs(\()p Fm(R)708 1876 y Fp(d)754 1912 y Fs(\))23 b Fq(\\)f Fs(\(1)g(+)g Fq(k)p Fr(y)t Fq(k)1262 1876 y Fv(2)1300 1912 y Fs(\))1338 1876 y Fl(\000)p Fv(1)1432 1912 y Fr(L)1498 1876 y Fv(2)1538 1912 y Fs(\()p Fm(R)1642 1876 y Fp(d)1689 1912 y Fs(\))34 b Fj(if)h Fr(\016)t(=")27 b Fq(!)g Fs(0)p Fj(,)166 2116 y Fq(\017)49 b Fr(g)312 2131 y Fp(k)398 2116 y Fq(2)c Fr(H)598 2080 y Fv(2)637 2116 y Fs(\()p Fm(R)740 2080 y Fp(d)787 2116 y Fs(\))29 b Fq(\\)g Fs(\(1)f(+)g Fq(k)p Fr(y)t Fq(k)1320 2080 y Fv(2)1359 2116 y Fs(\))1397 2080 y Fl(\000)p Fv(3)p Fp(=)p Fv(2)1561 2116 y Fr(L)1627 2080 y Fv(2)1667 2116 y Fs(\()p Fm(R)1771 2080 y Fp(d)1818 2116 y Fs(\))43 b Fj(and)g Fq(r)2180 2131 y Fp(y)2222 2116 y Fr(g)2269 2131 y Fp(k)2311 2116 y Fr(;)17 b Fs(\001)2436 2131 y Fp(y)2477 2116 y Fr(g)2524 2131 y Fp(k)2611 2116 y Fq(2)44 b Fs(\(1)28 b(+)h Fq(k)p Fr(y)t Fq(k)3093 2080 y Fv(2)3131 2116 y Fs(\))3169 2080 y Fl(\000)p Fv(1)p Fp(=)p Fv(2)3334 2116 y Fr(L)3400 2080 y Fv(2)3440 2116 y Fs(\()p Fm(R)3544 2080 y Fp(d)3590 2116 y Fs(\))44 b Fj(if)265 2236 y Fr(\016)t(=")27 b Fq(!)g Fs(+)p Fq(1)p Fj(,)21 2439 y(and)35 b Fs(\(1)22 b Fq(\000)h Fr(F)14 b Fs(\()p Fr(\015)591 2403 y Fv(2)630 2439 y Fq(k)p Fr(y)t Fq(k)782 2403 y Fv(2)820 2439 y Fs(\))p Fr(g)905 2454 y Fp(k)948 2439 y Fs(\()p Fr(y)t(;)j(\016)n(;)g (")p Fs(\))34 b Fj(with)h(their)g(sp)-5 b(atial)35 b(derivatives)g(up)g (to)h(se)-5 b(c)g(ond)34 b(or)-5 b(der)35 b(ar)-5 b(e)35 b(exp)-5 b(o-)21 2560 y(nential)5 b(ly)35 b(smal)5 b(l)34 b(in)g Fr(\015)5 b Fj(.)167 2680 y(If)36 b Fr( )t Fs(\()p Fr(t;)17 b Fq(\001)p Fr(;)g(\016)n(;)g(")p Fs(\))35 b Fj(denotes)g(the)h(solution)g(of)f(\(4\))h(with)g(initial)f(c)-5 b(ondition)35 b(at)h Fr(t)30 b Fs(=)g(0)p Fj(,)36 b Fr( )3291 2695 y Fp(I)3331 2680 y Fs(\(0)p Fr(;)17 b Fq(\001)p Fr(;)g(\016)n(;)g(")p Fs(\))p Fj(,)21 2801 y(then)35 b(the)g(quantity)1061 2921 y Fs(sup)990 3007 y Fp(t)p Fl(2)p Fv([)p Fl(\000)p Fp(t)1162 3017 y Fn(i)1188 3007 y Fp(;t)1233 3017 y Fn(i)1260 3007 y Fv(])1296 2921 y Fq(k)p Fr( )t Fs(\()p Fr(t;)17 b(x;)g(\016)n(;)g(")p Fs(\))22 b Fq(\000)h Fr( )1983 2936 y Fp(I)2023 2921 y Fs(\()p Fr(t;)17 b(x;)g(\016)n(;)g(")p Fs(\))p Fq(k)2458 2940 y Fp(L)2506 2921 y Fi(2)2540 2940 y Fv(\()p Fg(R)2615 2921 y Fn(d)2652 2940 y Fv(;)p Fl(H)p Fv(\))21 3165 y Fj(is)35 b(b)-5 b(ounde)g(d)34 b(by)h(a)g(c)-5 b(onstant)34 b(times)166 3383 y Fq(\017)49 b Fr(t)300 3398 y Fp(i)351 3383 y Fs(+)462 3331 y Fp(t)487 3308 y Fi(3)487 3353 y Fn(i)p 459 3361 68 4 v 459 3418 a Fp(")492 3399 y Fi(2)558 3383 y Fs(+)666 3342 y Fp(t)691 3352 y Fn(i)718 3342 y Fp(\016)p 666 3361 86 4 v 675 3418 a(")708 3399 y Fi(2)783 3383 y Fs(+)891 3331 y Fp(t)916 3308 y Fi(2)916 3353 y Fn(i)p 891 3361 61 4 v 905 3418 a Fp(")996 3383 y Fj(if)35 b Fr(\016)t(=")27 b Fq(!)g Fs(0)p Fj(,)166 3617 y Fq(\017)275 3576 y Fp(t)300 3586 y Fn(i)327 3576 y Fp(\016)360 3553 y Fi(2)p 275 3594 121 4 v 301 3651 a Fp(")334 3633 y Fi(2)427 3617 y Fs(+)535 3565 y Fp(t)560 3541 y Fi(2)560 3586 y Fn(i)595 3565 y Fp(\016)p 535 3594 94 4 v 549 3651 a(")582 3633 y Fi(2)661 3617 y Fs(+)769 3576 y Fp(t)794 3586 y Fn(i)p 769 3594 52 4 v 778 3651 a Fp(\016)853 3617 y Fs(+)964 3565 y Fp(t)989 3541 y Fi(2)989 3586 y Fn(i)p 961 3594 67 4 v 961 3651 a Fp(\016)r(")1059 3617 y Fs(+)1188 3565 y Fp(t)1213 3541 y Fi(3)1213 3586 y Fn(i)p 1167 3594 101 4 v 1167 3651 a Fp(\016)r(")1233 3633 y Fi(2)1300 3617 y Fs(+)1429 3565 y Fp(t)1454 3541 y Fi(4)1454 3586 y Fn(i)p 1408 3594 V 1408 3651 a Fp(\016)r(")1474 3633 y Fi(2)1541 3617 y Fs(+)1650 3578 y Fp(")p 1649 3594 34 4 v 1649 3651 a(\016)1728 3617 y Fj(if)34 b Fr(\016)t(=")28 b Fq(!)f Fs(+)p Fq(1)21 3820 y Fj(when)34 b Fr(\016)39 b Fj(and)34 b Fr(")h Fj(tend)g(to)g Fs(0)p Fj(.)167 4049 y Fk(Remarks)140 4252 y Fs(1.)49 b(In)38 b(the)g(expression)h(of)e(the) h(Ansatz)g(\(21\),)h(the)f(cuto\013)f(function)g(do)s(es)h(not)g(app)s (ear)f(as)h(the)265 4373 y(v)m(ectors)f Fr( )662 4388 y Fp(k)705 4373 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))36 b(are)f(de\014ned)i(ev)m(erywhere.)56 b(But)36 b(without)f(extra)h(kno) m(wledge)h(ab)s(out)e(the)265 4493 y(gro)m(wth)26 b(at)g(in\014nit)m(y) f(of)g(some)h(spatial)e(deriv)-5 b(ativ)m(es)26 b(of)f(those)i(v)m (ectors,)i(w)m(e)e(ha)m(v)m(e)g(to)e(in)m(tro)s(duce)265 4613 y(it)43 b(in)g(the)h(pro)s(of)e(or)i(to)f(imp)s(ose)g(some)g (extra)h(conditions)f(on)g(functions)h Fr(g)3165 4628 y Fp(k)3207 4613 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))43 b(that)265 4734 y(balance)32 b(this)h(gro)m(wth.)140 4937 y(2.)49 b(Note)23 b(that)g(in)f(the)h(pro)s(of)f(b)s(elo)m(w)h (\(equations)g(\(24\))f(and)h(\(25\)\))f(and)h(in)f(section)h(6.1)f (\(equations)265 5057 y(\(27\))41 b(and)i(\(28\)\),)g(w)m(e)g(men)m (tion)e(corrections)i(for)e(the)i Fr(f)2389 5072 y Fp(k)2431 5057 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))40 b(that)i(lead)g(to)f(a)h(more)265 5178 y(precise)33 b(result.)1828 5755 y(18)p eop %%Page: 19 19 19 18 bop 167 219 a Fk(Pro)s(of)70 b Fs(After)33 b(rescaling)e(time)h (b)m(y)h Fr(")f Fs(\()p Fr(s)c Fs(=)f Fr(t=")p Fs(\),)33 b(equation)f(\(4\))g(b)s(ecomes)1495 488 y Fr(i")1607 421 y(@)p 1584 465 103 4 v 1584 556 a(@)5 b(s)1717 462 y Fs(^)1697 488 y Fr( )32 b Fs(=)27 b Fr(H)8 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))2210 462 y(^)2191 488 y Fr( )21 760 y Fs(where)323 733 y(^)303 760 y Fr( )t Fs(\()p Fr(s;)g(x;)g(\016)n (;)g(")p Fs(\))27 b(=)h Fr( )t Fs(\()p Fr("s;)17 b(x;)g(\016)n(;)g(")p Fs(\).)42 b(Then,)34 b(w)m(e)g(substitute)f(the)g(Ansatz)329 1011 y(^)309 1037 y Fr( )372 1052 y Fp(I)413 1037 y Fs(\()p Fr(s;)17 b(x;)g(\016)n(;)g(")p Fs(\))82 b(=)h Fr(F)14 b Fs(\()p Fr(")1211 996 y Fl(\000)p Fv(2)1305 1037 y Fr(\015)1361 996 y Fv(2)1400 1037 y Fq(k)p Fr(x)23 b Fq(\000)f Fr(a)p Fs(\()p Fr("s)p Fs(\))p Fq(k)1896 996 y Fv(2)1935 1037 y Fs(\))17 b(exp)2155 897 y Fo(\022)2229 1037 y Fr(i)2272 970 y(S)6 b Fs(\()p Fr("s)p Fs(\))21 b(+)h Fr(\021)t Fs(\()p Fr("s)p Fs(\))p Fr(:)p Fs(\()p Fr(x)g Fq(\000)h Fr(a)p Fs(\()p Fr("s)p Fs(\)\))p 2272 1014 1072 4 v 2765 1105 a Fr(")2811 1077 y Fv(2)3353 897 y Fo(\023)1050 1309 y Fq(\002)1164 1215 y Fo(X)1144 1427 y Fp(k)r Fv(=1)p Fp(;)p Fv(2)1344 1309 y Fr(f)1392 1324 y Fp(k)1452 1169 y Fo(\022)1525 1309 y Fr(s;)1625 1242 y(x)f Fq(\000)h Fr(a)p Fs(\()p Fr("s)p Fs(\))p 1625 1286 396 4 v 1800 1378 a Fr(")2031 1309 y(;)17 b(\016)n(;)g(")2206 1169 y Fo(\023)2295 1309 y Fr( )2358 1324 y Fp(k)2401 1309 y Fs(\()p Fr(x;)g(\016)t Fs(\))33 b Fr(;)21 1672 y Fs(in)j(this)g(equation.)55 b(The)38 b(error)e(term)g(exp)1631 1562 y Fo(\020)1690 1672 y Fq(\000)p Fr(i)1810 1625 y Fp(S)t Fv(\()p Fp("s)p Fv(\))p 1810 1649 167 4 v 1861 1707 a Fp(")1894 1688 y Fi(2)2010 1672 y Fq(\000)23 b Fr(i)2153 1625 y Fp(\021)r Fv(\()p Fp("s)p Fv(\))p Fp(:y)p 2153 1649 215 4 v 2244 1707 a(")2378 1562 y Fo(\021)2454 1591 y(\002)2495 1672 y Fr(i")2600 1633 y Fp(@)p 2584 1649 74 4 v 2584 1707 a(@)t(s)2690 1672 y Fq(\000)g Fr(H)8 b Fs(\()p Fr(s;)17 b(\016)n(;)g(")p Fs(\))3176 1591 y Fo(\003)3260 1646 y Fs(^)3233 1672 y Fr( )3296 1687 y Fp(I)3336 1672 y Fs(\()p Fr(s;)g(x;)g(\016)n(;)g(")p Fs(\))21 1820 y(is,)31 b(remo)m(ving)f(the)i(con)m(tribution)d(of)i (the)g(cuto\013)g(function)g(and)g(its)f(deriv)-5 b(ativ)m(es)31 b(\(whic)m(h)h(turns)g(out)21 1940 y(to)g(b)s(e)h(exp)s(onen)m(tially)f (small\),)615 2123 y Fo(X)595 2335 y Fp(k)r Fv(=1)p Fp(;)p Fv(2)795 2077 y Fo(\022)869 2218 y Fr(i")958 2150 y(@)5 b(f)1062 2165 y Fp(k)p 958 2195 148 4 v 980 2286 a Fr(@)g(s)1115 2218 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))p Fr( )1571 2233 y Fp(k)1613 2218 y Fs(\()p Fr(x;)g(\016)t Fs(\))22 b Fq(\000)g Fr(f)2004 2233 y Fp(k)2047 2218 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))p Fr(h)2496 2233 y Fv(1)2534 2218 y Fs(\()p Fr(x;)g(\016)t Fs(\))p Fr( )2819 2233 y Fp(k)2862 2218 y Fs(\()p Fr(x;)g(\016)t Fs(\))3084 2077 y Fo(\023)408 2571 y Fq(\000)g Fs([)p Fr(E)6 b Fs(\()p Fr("y)t(;)17 b(\016)t Fs(\))j Fq(\000)j Fr(E)6 b Fs(\()p Fr(a)p Fs(\()p Fr("s)p Fs(\))p Fr(;)17 b(\016)t Fs(\))k Fq(\000)i Fr("y)t(:)p Fq(r)1785 2586 y Fp(x)1828 2571 y Fr(E)6 b Fs(\()p Fr(a)p Fs(\()p Fr("s)p Fs(\))p Fr(;)17 b(\016)t Fs(\)])2355 2476 y Fo(X)2335 2688 y Fp(k)r Fv(=1)p Fp(;)p Fv(2)2535 2571 y Fr(f)2583 2586 y Fp(k)2626 2571 y Fs(\()p Fr(s;)g(y)t(;)g(\016)n(;)g(")p Fs(\))p Fr( )3082 2586 y Fp(k)3123 2571 y Fs(\()p Fr(x;)g(\016)t Fs(\))392 2920 y(+)478 2852 y Fr(")524 2816 y Fv(2)p 478 2897 86 4 v 496 2988 a Fs(2)609 2825 y Fo(X)589 3037 y Fp(k)r Fv(=1)p Fp(;)p Fv(2)790 2920 y Fs(\001)871 2935 y Fp(y)912 2920 y Fr(f)960 2935 y Fp(k)1003 2920 y Fs(\()p Fr(s;)g(y)t(;)g(\016)n(;)g(")p Fs(\))p Fr( )1459 2935 y Fp(k)1500 2920 y Fs(\()p Fr(x;)g(\016)t Fs(\))22 b(+)g Fr(i")1921 2878 y Fv(2)1997 2825 y Fo(X)1977 3037 y Fp(k)r Fv(=1)p Fp(;)p Fv(2)2178 2920 y Fr(f)2226 2935 y Fp(k)2268 2920 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))p Fr(\021)t Fs(\()p Fr("s)p Fs(\))p Fr(:)p Fq(r)2991 2935 y Fp(x)3033 2920 y Fr( )3096 2935 y Fp(k)3139 2920 y Fs(\()p Fr(x;)g(\016)t Fs(\))197 b(\(22\))455 3269 y(+)p Fr(")577 3227 y Fv(3)652 3174 y Fo(X)632 3386 y Fp(k)r Fv(=1)p Fp(;)p Fv(2)832 3269 y Fq(r)915 3284 y Fp(y)957 3269 y Fr(f)1005 3284 y Fp(k)1048 3269 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))p Fr(:)p Fq(r)1551 3284 y Fp(x)1593 3269 y Fr( )1656 3284 y Fp(k)1699 3269 y Fs(\()p Fr(x;)g(\016)t Fs(\))22 b(+)2051 3201 y Fr(")2097 3165 y Fv(4)p 2051 3246 V 2069 3337 a Fs(2)2182 3174 y Fo(X)2163 3386 y Fp(k)r Fv(=1)p Fp(;)p Fv(2)2363 3269 y Fr(f)2411 3284 y Fp(k)2454 3269 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\)\001)2928 3284 y Fp(x)2970 3269 y Fr( )3033 3284 y Fp(k)3076 3269 y Fs(\()p Fr(x;)g(\016)t Fs(\))21 3546 y(where)34 b Fr(h)359 3561 y Fv(1)399 3546 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))32 b(is)g(giv)m(en)h(in)f(\(5\).)167 3666 y(T)-8 b(o)36 b(reduce)h(the)e(error,)h(the)g(\014rst)g(term)f(is) g(already)g(to)g(b)s(e)g(remo)m(v)m(ed)h(b)m(y)h(a)e(suitable)f(c)m (hoice)i(of)21 3787 y(the)d Fr(f)237 3802 y Fp(k)280 3787 y Fs(,)g(i.e.,)f(w)m(e)h(appro)m(ximately)f(solv)m(e)257 4059 y Fr(i")368 3992 y(@)p 346 4036 103 4 v 346 4128 a(@)5 b(s)475 3919 y Fo(\022)590 3998 y Fr(f)638 4013 y Fv(1)590 4119 y Fr(f)638 4134 y Fv(2)719 3919 y Fo(\023)820 4059 y Fs(=)923 3919 y Fo(\022)1179 3998 y Fr(b)p Fs(\()p Fr(a)p Fs(\()p Fr("s)p Fs(\))23 b(+)f Fr("y)t(;)17 b(\016)t Fs(\))222 b(\()p Fr(c)22 b Fs(+)g Fr(id)p Fs(\)\()p Fr(a)p Fs(\()p Fr("s)p Fs(\))g(+)g Fr("y)t(;)17 b(\016)t Fs(\))1038 4119 y(\()p Fr(c)22 b Fq(\000)h Fr(id)p Fs(\)\()p Fr(a)p Fs(\()p Fr("s)p Fs(\))f(+)g Fr("y)t(;)17 b(\016)t Fs(\))182 b Fq(\000)p Fr(b)p Fs(\()p Fr(a)p Fs(\()p Fr("s)p Fs(\))23 b(+)f Fr("y)t(;)17 b(\016)t Fs(\))3013 3919 y Fo(\023)g(\022)3217 3998 y Fr(f)3265 4013 y Fv(1)3217 4119 y Fr(f)3265 4134 y Fv(2)3346 3919 y Fo(\023)3469 4059 y Fr(:)21 4337 y Fs(Ignoring)32 b(the)h(\014v)m(e)g(remaining)e(terms)h(in)g(\(22\))g (leads)g(to)h(an)f(error)h(of)f(order)938 4557 y Fr(O)1031 4477 y Fo(\000)1077 4557 y Fr(")1123 4516 y Fv(2)1162 4557 y Fs(\(1)22 b(+)g Fq(k)p Fr(y)t Fq(k)1521 4516 y Fv(2)1559 4557 y Fs(\))p Fq(j)p Fr(f)1673 4572 y Fp(k)1716 4557 y Fq(j)f Fs(+)h Fr(")1909 4516 y Fv(3)1949 4557 y Fq(jr)2060 4572 y Fp(y)2101 4557 y Fr(f)2149 4572 y Fp(k)2191 4557 y Fq(j)g Fs(+)g Fr(")2385 4516 y Fv(2)2424 4557 y Fq(j)p Fs(\001)2533 4572 y Fp(y)2575 4557 y Fr(f)2623 4572 y Fp(k)2665 4557 y Fq(j)2693 4477 y Fo(\001)2788 4557 y Fr(:)167 4778 y Fs(F)-8 b(rom)32 b(\(6\))g(and)g(asymptotics)h (of)f(prop)s(osition)e(1,)j(w)m(e)g(appro)m(ximate)f(this)g(system)i(b) m(y)638 5056 y Fr(i")750 4988 y(@)p 727 5033 V 727 5124 a(@)5 b(s)856 4915 y Fo(\022)971 4995 y Fr(f)1019 5010 y Fv(1)971 5115 y Fr(f)1019 5130 y Fv(2)1100 4915 y Fo(\023)1201 5056 y Fs(=)27 b Fr(r)1368 4915 y Fo(\022)1483 4995 y Fr(\021)1535 4958 y Fv(0)1531 5019 y(1)1574 4995 y Fr("s)21 b Fs(+)h Fr("y)1879 5010 y Fv(1)2272 4995 y Fr(\016)1677 5115 y(\016)281 b Fq(\000)p Fs(\()p Fr(\021)2168 5079 y Fv(0)2164 5140 y(1)2208 5115 y Fr("s)21 b Fs(+)h Fr("y)2513 5130 y Fv(1)2552 5115 y Fs(\))2632 4915 y Fo(\023)16 b(\022)2836 4995 y Fr(f)2884 5010 y Fv(1)2836 5115 y Fr(f)2884 5130 y Fv(2)2965 4915 y Fo(\023)3088 5056 y Fr(:)443 b Fs(\(23\))21 5338 y(Doing)31 b(so)i(leads)f(to)g(an)h(error) f(term)g(of)g(order)h Fr(O)19 b Fs(\(\()p Fr(")1979 5302 y Fv(2)2018 5338 y Fr(s)2064 5302 y Fv(2)2126 5338 y Fs(+)j Fr(")2270 5302 y Fv(2)2309 5338 y Fq(k)p Fr(y)t Fq(k)2461 5302 y Fv(2)2521 5338 y Fs(+)g Fr(\016)2666 5302 y Fv(2)2705 5338 y Fs(\))p Fq(j)p Fr(f)2819 5353 y Fp(k)2862 5338 y Fq(j)p Fs(\))o(.)167 5458 y(W)-8 b(e)33 b(no)m(w)h(deal)e(with)g(t)m(w)m(o)h(situations)f(:)1828 5755 y(19)p eop %%Page: 20 20 20 19 bop 140 219 a Fs(1.)49 b Fr(\016)t(=")27 b Fq(!)g Fs(0)33 b(and)f(the)h(system)h(is)e(almost)874 462 y Fr(i)940 395 y(@)p 917 439 103 4 v 917 531 a(@)5 b(s)1046 322 y Fo(\022)1161 401 y Fr(f)1209 416 y Fv(1)1161 522 y Fr(f)1209 537 y Fv(2)1290 322 y Fo(\023)1391 462 y Fs(=)28 b Fr(r)1558 322 y Fo(\022)1673 401 y Fr(\021)1725 365 y Fv(0)1721 426 y(1)1764 401 y Fr(s)22 b Fs(+)g Fr(y)1978 416 y Fv(1)2325 401 y Fs(0)1821 522 y(0)230 b Fq(\000)p Fs(\()p Fr(\021)2267 486 y Fv(0)2263 546 y(1)2307 522 y Fr(s)22 b Fs(+)g Fr(y)2521 537 y Fv(1)2560 522 y Fs(\))2639 322 y Fo(\023)17 b(\022)2844 401 y Fr(f)2892 416 y Fv(1)2844 522 y Fr(f)2892 537 y Fv(2)2973 322 y Fo(\023)3096 462 y Fr(;)265 701 y Fs(with)32 b(solution)1179 896 y Fr(f)1227 911 y Fv(1)1267 896 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))81 b(=)i Fr(g)1947 911 y Fv(1)1986 896 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))p Fr(e)2334 854 y Fl(\000)p Fp(ir)r Fv(\()p Fp(\021)2511 831 y Fi(0)2509 875 y(1)2555 827 y Fn(s)2584 806 y Fi(2)p 2555 839 64 3 v 2572 881 a(2)2629 854 y Fv(+)p Fp(sy)2752 863 y Fi(1)2786 854 y Fv(\))1179 1075 y Fr(f)1227 1090 y Fv(2)1267 1075 y Fs(\()p 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y(in)32 b Fr("=\016)991 3151 y(f)1039 3166 y Fp(k)1081 3151 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))26 b(=)i Fr(f)1663 3110 y Fv(0)1652 3176 y Fp(k)1702 3151 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))j(+)2224 3084 y Fr(")p 2223 3128 47 4 v 2223 3219 a(\016)2280 3151 y(f)2339 3110 y Fv(1)2328 3176 y Fp(k)2378 3151 y Fs(\()p Fr(s;)d(y)t(;)g(\016)n(;)g(")p Fs(\))k(+)h Fq(\001)17 b(\001)g(\001)265 3336 y Fs(for)32 b Fr(f)462 3351 y Fp(k)505 3336 y Fs(.)43 b(W)-8 b(e)33 b(solv)m(e)g(successiv)m(ely)1268 3575 y Fr(i)1335 3508 y(@)p 1311 3552 103 4 v 1311 3644 a(@)5 b(s)1441 3435 y Fo(\022)1556 3514 y Fr(f)1615 3478 y Fv(0)1604 3539 y(1)1556 3635 y Fr(f)1615 3599 y Fv(0)1604 3659 y(2)1696 3435 y Fo(\023)1797 3575 y Fs(=)27 b Fr(r)1964 3435 y Fo(\022)2081 3514 y Fs(0)2225 3475 y Fp(\016)p 2225 3491 34 4 v 2226 3549 a(")2088 3597 y(\016)p 2088 3613 V 2089 3670 a(")2217 3636 y Fs(0)2310 3435 y Fo(\023)17 b(\022)2515 3514 y Fr(f)2574 3478 y Fv(0)2563 3539 y(1)2515 3635 y 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Fr(f)3411 3938 y Fv(0)3400 3998 y(1)3352 4094 y Fr(f)3411 4058 y Fv(0)3400 4119 y(2)3491 3894 y Fo(\023)3614 4035 y Fr(:)265 4274 y Fs(Hence)34 b(the)f(solutions)601 4379 y Fo(\002)642 4460 y Fr(f)701 4419 y Fv(0)690 4485 y(1)763 4460 y Fs(+)22 b Fr(f)920 4419 y Fv(0)909 4485 y(2)959 4379 y Fo(\003)1017 4460 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))81 b(=)i Fr(g)1697 4475 y Fv(1)1736 4460 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))p Fr(e)2084 4419 y Fl(\000)p Fp(ir)2206 4392 y Fn(\016)p 2206 4404 30 3 v 2206 4445 a(")2246 4419 y Fp(s)523 4550 y Fo(\002)565 4631 y Fq(\000)p Fr(f)701 4590 y Fv(0)690 4655 y(1)763 4631 y Fs(+)22 b Fr(f)920 4590 y Fv(0)909 4655 y(2)959 4550 y Fo(\003)1017 4631 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))81 b(=)i Fr(g)1697 4646 y Fv(2)1736 4631 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))p Fr(e)2084 4590 y Fp(ir)2151 4563 y Fn(\016)p 2151 4575 V 2151 4616 a(")2191 4590 y Fp(s)601 4721 y Fo(\002)642 4802 y Fr(f)701 4761 y Fv(1)690 4826 y(1)763 4802 y Fs(+)22 b Fr(f)920 4761 y Fv(1)909 4826 y(2)959 4721 y Fo(\003)1017 4802 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))81 b(=)i Fr(h)1706 4817 y Fv(1)1746 4802 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))p Fr(e)2094 4761 y Fl(\000)p Fp(ir)2215 4734 y Fn(\016)p 2215 4746 V 2215 4787 a(")2255 4761 y Fp(s)2314 4802 y Fq(\000)23 b Fr(g)2461 4817 y Fv(2)2500 4802 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))1650 5016 y Fq(\002)1744 4876 y Fo(\024)1807 4949 y Fr(\021)1859 4913 y Fv(0)1855 4973 y(1)p 1807 4993 92 4 v 1828 5085 a Fs(2)1908 5016 y Fr(se)1999 4975 y Fp(ir)2067 4948 y Fn(\016)p 2067 4960 30 3 v 2067 5001 a(")2107 4975 y Fp(s)2166 5016 y Fq(\000)2265 4876 y Fo(\022)2349 4949 y Fr(\021)2401 4913 y Fv(0)2397 4973 y(1)p 2349 4993 92 4 v 2370 5085 a Fs(2)2460 4949 y Fr(")p 2460 4993 47 4 v 2460 5085 a(\016)2539 5016 y Fq(\000)22 b Fr(iy)2719 5031 y Fv(1)2759 4876 y Fo(\023)2848 5016 y Fs(sin)2985 4876 y Fo(\022)3058 5016 y Fr(r)3115 4949 y(\016)p 3115 4993 V 3116 5085 a(")3172 5016 y(s)3218 4876 y Fo(\023\025)3558 5016 y Fs(\(24\))523 5166 y Fo(\002)565 5247 y Fq(\000)p Fr(f)701 5206 y Fv(1)690 5272 y(1)763 5247 y Fs(+)g Fr(f)920 5206 y Fv(1)909 5272 y(2)959 5166 y Fo(\003)1017 5247 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))81 b(=)i Fr(h)1706 5262 y Fv(2)1746 5247 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))p Fr(e)2094 5206 y Fp(ir)2160 5179 y Fn(\016)p 2160 5191 30 3 v 2160 5232 a(")2200 5206 y Fp(s)2259 5247 y Fs(+)22 b Fr(g)2404 5262 y Fv(1)2443 5247 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))1650 5461 y Fq(\002)1744 5321 y Fo(\024)1807 5394 y Fr(\021)1859 5358 y Fv(0)1855 5419 y(1)p 1807 5438 92 4 v 1828 5530 a Fs(2)1908 5461 y Fr(se)1999 5420 y Fl(\000)p Fp(ir)2122 5393 y Fn(\016)p 2122 5405 30 3 v 2122 5446 a(")2162 5420 y Fp(s)2220 5461 y Fq(\000)2320 5321 y Fo(\022)2403 5394 y Fr(\021)2455 5358 y Fv(0)2451 5419 y(1)p 2403 5438 92 4 v 2425 5530 a Fs(2)2515 5394 y Fr(")p 2514 5438 47 4 v 2514 5530 a(\016)2594 5461 y Fs(+)22 b Fr(iy)2773 5476 y Fv(1)2812 5321 y Fo(\023)2902 5461 y Fs(sin)3038 5321 y Fo(\022)3111 5461 y Fr(r)3168 5394 y(\016)p 3168 5438 V 3169 5530 a(")3225 5461 y(s)3271 5321 y Fo(\023\025)3446 5461 y Fr(:)85 b Fs(\(25\))1828 5755 y(20)p eop %%Page: 21 21 21 20 bop 265 219 a Fs(W)-8 b(e)34 b(c)m(ho)s(ose)h Fr(h)801 234 y Fv(1)871 219 y Fs(=)29 b Fr(h)1032 234 y Fv(2)1102 219 y Fs(=)h(0)j(and)i(stop)f(to)f(\014rst)i(order.)47 b(W)-8 b(e)35 b(obtain)e(an)g(extra)i(error)e(term)h(of)265 361 y(order)f Fr(O)614 250 y Fo(\020)683 322 y Fp(")716 298 y Fi(2)p 683 338 68 4 v 700 395 a Fp(\016)760 361 y Fs(\()p Fq(j)p Fr(s)p Fq(j)22 b Fs(+)g Fq(k)p Fr(y)t Fq(k)p Fs(\))p Fq(j)p Fr(f)1297 325 y Fv(1)1286 387 y Fp(k)1334 361 y Fq(j)1362 250 y Fo(\021)1421 361 y Fs(.)167 598 y(F)-8 b(or)32 b(eac)m(h)i(case,)f(w)m(e)h(ha)m(v)m(e)140 791 y(1.)1752 912 y Fr(f)1800 927 y Fp(k)1871 912 y Fs(=)27 b Fr(O)s Fs(\()p Fr(g)2137 927 y Fp(k)2179 912 y Fs(\))p Fr(;)1387 1081 y Fq(r)1470 1096 y Fp(y)1512 1081 y Fr(f)1560 1096 y Fp(k)1630 1081 y Fs(=)h Fr(O)s Fs(\()p Fq(jr)1961 1096 y Fp(y)2001 1081 y Fr(g)2048 1096 y Fp(k)2091 1081 y Fq(j)21 b Fs(+)h Fq(j)p Fr(s)p Fq(j)p Fr(:)p Fq(j)p Fr(g)2442 1096 y Fp(k)2484 1081 y Fq(j)p Fs(\))32 b Fr(;)1152 1251 y Fs(\001)1233 1266 y Fp(y)1275 1251 y Fr(f)1323 1266 y Fp(k)1393 1251 y Fs(=)c Fr(O)s Fs(\()p Fq(j)p Fs(\001)1722 1266 y Fp(y)1762 1251 y Fr(g)1809 1266 y Fp(k)1851 1251 y Fq(j)22 b Fs(+)g Fq(j)p Fr(s)p Fq(j)p Fr(:)p Fq(jr)2239 1266 y Fp(y)2280 1251 y Fr(g)2327 1266 y Fp(k)2369 1251 y Fq(j)g Fs(+)g Fr(s)2563 1209 y Fv(2)2602 1251 y Fq(j)p Fr(g)2677 1266 y Fp(k)2719 1251 y Fq(j)p Fs(\))33 b(;)140 1460 y(2.)1559 1580 y Fr(f)1618 1539 y Fv(0)1607 1605 y Fp(k)1685 1580 y Fs(=)27 b Fr(O)19 b Fs(\()p Fq(j)p Fr(g)1995 1595 y Fv(1)2034 1580 y Fq(j)j Fs(+)g Fq(j)p Fr(g)2257 1595 y Fv(2)2296 1580 y Fq(j)p Fs(\))48 b Fr(;)1372 1750 y Fq(r)1455 1765 y Fp(y)1497 1750 y Fr(f)1556 1708 y Fv(0)1545 1774 y Fp(k)1623 1750 y Fs(=)27 b Fr(O)19 b Fs(\()p Fq(jr)1969 1765 y Fp(y)2010 1750 y Fr(g)2057 1765 y Fv(1)2096 1750 y Fq(j)j Fs(+)g Fq(jr)2355 1765 y Fp(y)2396 1750 y Fr(g)2443 1765 y Fv(2)2482 1750 y Fq(j)p Fs(\))49 b Fr(;)1375 1919 y Fs(\001)1456 1934 y Fp(y)1498 1919 y Fr(f)1557 1878 y Fv(0)1546 1944 y Fp(k)1623 1919 y Fs(=)28 b Fr(O)19 b Fs(\()p Fq(j)p Fs(\001)1968 1934 y Fp(y)2009 1919 y Fr(g)2056 1934 y Fv(1)2095 1919 y Fq(j)j Fs(+)g Fq(j)p Fs(\001)2352 1934 y Fp(y)2394 1919 y Fr(g)2441 1934 y Fv(2)2480 1919 y Fq(j)p Fs(\))48 b Fr(;)1152 2119 y(f)1211 2078 y Fv(1)1200 2143 y Fp(k)1278 2119 y Fs(=)27 b Fr(O)1475 2008 y Fo(\020)q(\020)1594 2119 y Fq(j)p Fr(s)p Fq(j)22 b Fs(+)1826 2051 y Fr(")p 1826 2096 47 4 v 1826 2187 a(\016)1905 2119 y Fs(+)g Fq(k)p Fr(y)t Fq(k)2155 2008 y Fo(\021)2230 2119 y Fs(\()p Fq(j)p Fr(g)2343 2134 y Fv(1)2381 2119 y Fq(j)g Fs(+)g Fq(j)p Fr(g)2604 2134 y Fv(2)2643 2119 y Fq(j)p Fs(\))2709 2008 y Fo(\021)2817 2119 y Fr(;)666 2359 y Fq(r)749 2374 y Fp(y)790 2359 y Fr(f)849 2318 y Fv(1)838 2384 y Fp(k)916 2359 y Fs(=)28 b 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Fs(\))83 b(if)31 b Fr(\016)t(=")c Fq(!)h Fs(+)p Fq(1)167 3350 y Fs(T)-8 b(o)38 b(conclude)g(in)f(the)h Fr(\016)t(=")e Fq(!)g Fs(+)p Fq(1)h Fs(case,)j(w)m(e)e(just)g(drop)g(the)g(terms)2820 3311 y Fp(")p 2820 3327 34 4 v 2820 3385 a(\016)2863 3350 y Fr(f)2922 3314 y Fv(1)2911 3376 y Fp(k)2962 3350 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))35 b(but)j(add)21 3471 y(an)33 b(error)f(of)g(order)h Fr(O)855 3390 y Fo(\000)911 3431 y Fp(")p 910 3448 V 910 3505 a(\016)976 3471 y Fs(+)1084 3430 y Fp(t)1109 3440 y Fn(i)p 1084 3448 52 4 v 1093 3505 a Fp(\016)1146 3390 y Fo(\001)1192 3471 y Fs(.)43 b Ff(\003)21 3802 y Ft(6)161 b(Matc)l(hing)53 b(Pro)t(cedure)21 4021 y Fs(W)-8 b(e)35 b(no)m(w)g(try)g(to)f(matc)m(h) g(the)h(outer)g(and)g(inner)f(Ans\177)-49 b(atze.)50 b(W)-8 b(e)35 b(b)s(egin)f(with)g(the)h(incoming)d(outer)21 4141 y(Ansatz)j(\(17\))e(where)i(\003)909 4100 y Fl(\000)909 4168 y(A)970 4141 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))29 b(=)h(0)k(and)g(\003)1654 4100 y Fl(\000)1654 4168 y(B)1713 4141 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))30 b(=)f(1)34 b(and)g(w)m(e)h(ask)g(ho)m(w)f(to)g(c)m(ho)s(ose)h(\003)3351 4100 y Fv(+)3351 4168 y Fl(C)3410 4141 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))33 b(in)21 4262 y(the)j(outgoing)e(outer)i(Ansatz.)54 b(In)36 b(eac)m(h)g(matc)m(hing)f(\(incoming)e(outer)j(with)f(inner)g (Ans\177)-49 b(atze)37 b(and)21 4382 y(inner)28 b(with)h(outgoing)e (outer)h(Ans\177)-49 b(atze\),)31 b(w)m(e)e(mak)m(e)g(use)h(of)e(the)h (equalit)m(y)f(b)s(et)m(w)m(een)i(the)f(\014rst)g(terms)21 4502 y(in)h(the)i(asymptotic)e(expansion)i(of)e(eac)m(h)i(Ansatz)g(in)e (the)i(o)m(v)m(erlapping)e(region)g(where)i(b)s(oth)f(exist.)167 4623 y(Rigorous)h(statemen)m(t)h(of)f(the)h(pro)s(cedure)g(is)21 4816 y Fk(Lemma)k(6)49 b Fj(Supp)-5 b(ose)32 b Fr(H)8 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))31 b Fj(is)h(a)g(family)g(of)g (self-adjoint)f(op)-5 b(er)g(ators)32 b(in)g(any)g(sep)-5 b(ar)g(able)31 b(Hilb)-5 b(ert)21 4937 y(sp)g(ac)g(e)38 b Fq(H)q Fj(.)55 b(We)39 b(cho)-5 b(ose)37 b(thr)-5 b(e)g(e)39 b(times)f Fr(t)1484 4952 y Fp(l)1510 4937 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))34 b Fr(<)g(t)1896 4952 y Fp(m)1963 4937 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))34 b Fr(<)g(t)2349 4952 y Fp(r)2387 4937 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))38 b Fj(and)g(two)h(initial)f(c)-5 b(onditions)21 5057 y Fr(\013)83 5072 y Fp(l)109 5057 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))33 b Fj(and)g Fr(\013)599 5072 y Fp(r)637 5057 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))34 b Fj(of)f(or)-5 b(der)33 b Fr(O)s Fs(\(1\))f Fj(in)h(the)h(domain)e(of)h Fr(H)8 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))33 b Fj(when)g Fr(\016)k Fj(and)c Fr(")g Fj(tend)g(to)h Fs(0)p Fj(.)44 b(L)-5 b(et)21 5177 y Fr( )84 5192 y Fl(\003)124 5177 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))34 b Fj(denote)f(the)i (solution)f(of)f(\(4\))h(with)g(initial)g(c)-5 b(ondition)33 b Fr(\013)2545 5192 y Fl(\003)2584 5177 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))34 b Fj(at)h Fr(t)2977 5192 y Fl(\003)3016 5177 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))34 b Fj(for)g Fq(\003)28 b Fs(=)f Fr(l)r(;)34 b(r)21 5298 y Fj(and)g(we)h(supp)-5 b(ose)34 b(that)960 5506 y Fq(k)p Fr( )1073 5521 y Fp(l)1099 5506 y Fs(\()p Fr(t)1172 5521 y Fp(m)1239 5506 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fr(;)g(\016)n(;)g(")p Fs(\))k Fq(\000)i Fr( )1843 5521 y Fp(r)1881 5506 y Fs(\()p Fr(t)1954 5521 y Fp(m)2021 5506 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fr(;)g(\016)n(;)g(")p Fs(\))p Fq(k)27 b Fs(=)g Fr(o)p Fs(\(1\))1828 5755 y(21)p eop %%Page: 22 22 22 21 bop 21 219 a Fj(when)34 b Fr(\016)39 b Fj(and)34 b Fr(")h Fj(tend)g(to)g Fs(0)p Fj(.)167 339 y(Then)43 b(ther)-5 b(e)43 b(exists)g(a)g(function)g Fr(\014)6 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))42 b(=)i Fr(o)p Fs(\(1\))e Fj(in)h(the)h(domain)e(of)h Fr(H)8 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))42 b Fj(such)h(that,)j(if)41 433 y Fs(~)21 459 y Fr( )t Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))31 b Fj(denotes)f(the)h(solution)f(of)h(\(4\))f(with)h(initial)g (c)-5 b(ondition)29 b Fr(\013)2519 474 y Fp(r)2557 459 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))c(+)g Fr(\014)6 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))31 b Fj(in)f Fr(t)3315 474 y Fp(r)3354 459 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fj(,)31 b(we)21 580 y(have)j Fr( )308 595 y Fp(l)335 580 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))27 b(=)772 553 y(~)752 580 y Fr( )815 595 y Fp(r)853 580 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))35 b Fj(for)f(every)h Fr(t)g Fj(in)g(the)g(interval)f Fs([)p Fr(t)2358 595 y Fp(l)2384 580 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fr(;)g(t)2670 595 y Fp(r)2708 580 y Fs(\()p Fr(\016)n(;)g(")p Fs(\)])p Fj(.)167 748 y Fk(Remark)81 b Fs(This)38 b(lemma)d(remains)i (true)h(if)e(w)m(e)j(substitute)f Fr(O)s Fs(\()p Fr(\025)p Fs(\()p Fr(\016)n(;)17 b(")p Fs(\)\))36 b(for)h Fr(o)p Fs(\(1\))h(where)g Fr(\025)g Fs(is)21 869 y(an)m(y)33 b(function)f(tending)h(to)f(0)g(when)i Fr(\016)i Fs(and)d Fr(")f Fs(tend)h(to)g(0.)167 989 y Fk(Pro)s(of)95 b Fs(Self-adjoin)m (tness)44 b(of)g Fr(H)8 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))43 b(giv)m(es)i(us)g(the)f(existence)i(of)e(a)g(unitary)f (propagator)21 1110 y Fr(U)10 b Fs(\()p Fr(t;)17 b(t)249 1073 y Fl(0)273 1110 y Fr(;)g(\016)n(;)g(")p Fs(\))40 b(asso)s(ciated)g(to)g(the)g(Sc)m(hr\177)-49 b(odinger)40 b(equation)g(\(4\))g(\(see)h([18]\).)66 b(Th)m(us)42 b(left)d(and)h(righ)m(t)21 1230 y(solutions)32 b(are)g(giv)m(en)h(b)m (y)830 1409 y Fr( )893 1424 y Fp(l)919 1409 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))83 b(=)g Fr(U)10 b Fs(\()p Fr(t;)17 b(t)1675 1424 y Fp(l)1702 1409 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))p Fr(;)g(\016)n(;)g(")p Fs(\))p Fr(\013)2184 1424 y Fp(l)2209 1409 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))818 1555 y Fr( )881 1570 y Fp(r)919 1555 y Fs(\()p Fr(t;)g(\016)n(;)g(")p Fs(\))83 b(=)g Fr(U)10 b Fs(\()p Fr(t;)17 b(t)1675 1570 y Fp(r)1714 1555 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))p Fr(;)g(\016)n(;)g(")p Fs(\))p Fr(\013)2196 1570 y Fp(r)2233 1555 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))838 1674 y(~)818 1700 y Fr( )881 1715 y Fp(r)919 1700 y Fs(\()p Fr(t;)g(\016)n(;)g(")p Fs(\))83 b(=)g Fr( )1510 1715 y Fp(r)1548 1700 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))22 b(+)g Fr(U)10 b Fs(\()p Fr(t;)17 b(t)2182 1715 y Fp(r)2221 1700 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))p Fr(;)g(\016)n(;)g(")p Fs(\))p Fr(\014)6 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fr(:)21 1880 y Fs(Then)53 b Fr(\014)6 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))61 b(=)g Fr(U)10 b Fs(\()p Fr(t)910 1895 y Fp(r)948 1880 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fr(;)g(t)1234 1895 y Fp(m)1301 1880 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))p Fr(;)g(\016)n(;)g(")p Fs(\)[)p Fr( )1811 1895 y Fp(l)1836 1880 y Fs(\()p Fr(t)1909 1895 y Fp(m)1976 1880 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))p Fr(;)g(\016)n(;)g(")p Fs(\))35 b Fq(\000)h Fr( )2607 1895 y Fp(r)2645 1880 y Fs(\()p Fr(t)2718 1895 y Fp(m)2785 1880 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fr(;)g(\016)n(;)g(")p Fs(\)])51 b(mak)m(es)i(the)21 2000 y(lemma)30 b(true.)44 b Ff(\003)21 2283 y Fd(6.1)135 b(Narro)l(w)37 b(Av)l(oided)e(Crossing)h (\()p Fc(\016)t(=")d Fb(!)g Fz(0)p Fd(\):)55 b(w)l(e)36 b(use)f(the)g Fb(j)p Fc(t)p Fb(j)p Fc(=\016)j Fb(!)33 b Fz(+)p Fb(1)327 2432 y Fd(regime.)21 2617 y Fs(First,)f(b)m(y)h(prop) s(osition)e(1,)h(w)m(e)i(ha)m(v)m(e)g(the)f(follo)m(wing)d(asymptotics) 200 2850 y Fr(S)266 2809 y Fl(C)311 2850 y Fs(\()p Fr(t)p Fs(\))22 b Fq(\000)h Fr(S)6 b Fs(\()p Fr(t)p Fs(\))83 b(=)963 2740 y Fo(\020)1022 2850 y Fr(\021)1074 2809 y Fv(0)1109 2786 y Fh(C)1154 2850 y Fs(\()p Fr(\016)t Fs(\))22 b Fq(\000)g Fr(\021)1450 2809 y Fv(0)1489 2850 y Fs(\()p Fr(\016)t Fs(\))1612 2740 y Fo(\021)1688 2850 y Fr(:\021)1767 2809 y Fv(0)1807 2850 y Fr(t)g Fs(+)g Fr(O)2056 2710 y Fo(\022)2129 2850 y Fq(j)p Fr(t)p Fq(j)2220 2809 y Fv(3)2281 2850 y Fs(+)2389 2783 y Fr(\016)2436 2747 y Fv(4)p 2389 2827 87 4 v 2395 2918 a Fr(t)2430 2890 y Fv(2)2486 2710 y Fo(\023)963 3152 y Fq(\000)p Fr(r)s(\027)1141 3111 y Fl(C)1203 2982 y Fo(")1261 3152 y Fs(sgn)q(\()p Fr(t)p Fs(\))1531 2982 y Fo( )1609 3152 y Fr(\021)1661 3111 y Fv(0)1657 3177 y(1)1701 3152 y Fr(t)1736 3111 y Fv(2)1797 3152 y Fs(+)1932 3085 y Fr(\016)1979 3049 y Fv(2)p 1905 3129 140 4 v 1905 3221 a Fs(2)p Fr(\021)2006 3186 y Fv(0)2002 3245 y(1)2077 3152 y Fs(+)2269 3085 y Fr(\016)2316 3049 y Fv(2)p 2185 3129 255 4 v 2185 3229 a Fr(\021)2237 3194 y Fv(0)2272 3175 y Fh(C)2233 3253 y Fv(1)2317 3229 y Fs(\()p Fr(\016)t Fs(\))2466 3152 y(ln)2564 2982 y Fo( )2653 3085 y Fs(2)p Fr(\021)2754 3049 y Fv(0)2789 3025 y Fh(C)2750 3110 y Fv(1)2833 3085 y Fs(\()p Fr(\016)t Fs(\))p Fq(j)p Fr(t)p Fq(j)p 2653 3129 394 4 v 2826 3221 a Fr(\016)3056 2982 y Fo(!!)3236 3152 y Fq(\000)h Fr(\016)t(t)3418 2982 y Fo(#)3525 3152 y Fr(;)287 3697 y(a)338 3656 y Fl(C)383 3697 y Fs(\()p Fr(t)p Fs(\))f Fq(\000)h Fr(a)p Fs(\()p Fr(t)p Fs(\))83 b(=)1020 3587 y Fo(\020)1080 3697 y Fr(\021)1132 3656 y Fv(0)1167 3633 y Fh(C)1211 3697 y Fs(\()p Fr(\016)t Fs(\))22 b Fq(\000)g Fr(\021)1507 3656 y Fv(0)1547 3697 y Fs(\()p Fr(\016)t Fs(\))1670 3587 y Fo(\021)1746 3697 y Fr(t)g Fs(+)g Fr(O)1995 3557 y Fo(\022)2068 3697 y Fq(j)p Fr(t)p Fq(j)2159 3656 y Fv(3)2220 3697 y Fs(+)2328 3630 y Fr(\016)2375 3594 y Fv(4)p 2328 3674 87 4 v 2334 3766 a Fr(t)2369 3737 y Fv(2)2425 3557 y Fo(\023)2520 3697 y Fq(\000)h Fr(r)s(\027)2721 3656 y Fl(C)2782 3407 y Fo(0)2782 3583 y(B)2782 3643 y(B)2782 3702 y(B)2782 3766 y(@)2911 3495 y Fs(1)2911 3615 y(0)2922 3711 y(.)2922 3744 y(.)2922 3777 y(.)2911 3898 y(0)3001 3407 y Fo(1)3001 3583 y(C)3001 3643 y(C)3001 3702 y(C)3001 3766 y(A)1020 4149 y Fq(\002)1114 3979 y Fo(")1182 4082 y Fs(sgn)q(\()p Fr(t)p Fs(\))p 1182 4126 253 4 v 1284 4217 a(2)1461 3979 y Fo( )1540 4149 y Fr(t)1575 4108 y Fv(2)1637 4149 y Fs(+)1789 4082 y Fr(\016)1836 4045 y Fv(2)p 1745 4126 175 4 v 1745 4223 a Fs(2)p Fr(\021)1846 4189 y Fv(0)1881 4170 y Fi(2)1842 4248 y Fv(1)1952 4149 y Fs(+)2163 4082 y Fr(\016)2210 4045 y Fv(2)p 2060 4126 294 4 v 2060 4225 a Fr(\021)2112 4191 y Fv(0)2147 4172 y Fh(C)2108 4250 y Fv(1)2191 4225 y Fs(\()p Fr(\016)t Fs(\))2314 4196 y Fv(2)2380 4149 y Fs(ln)2478 3979 y Fo( )2566 4082 y Fs(2)p Fr(\021)2667 4045 y Fv(0)2702 4022 y Fh(C)2663 4106 y Fv(1)2747 4082 y Fs(\()p Fr(\016)t Fs(\))p Fq(j)p Fr(t)p Fq(j)p 2566 4126 394 4 v 2740 4217 a Fr(\016)2970 3979 y Fo(!!)3150 4149 y Fq(\000)f Fr(\016)t(t)3331 3979 y Fo(#)3439 4149 y Fr(;)372 4635 y(\021)424 4594 y Fl(C)468 4635 y Fs(\()p Fr(t)p Fs(\))h Fq(\000)f Fr(\021)t Fs(\()p Fr(t)p Fs(\))28 b(=)f Fr(\021)1047 4594 y Fv(0)1082 4570 y Fh(C)1127 4635 y Fs(\()p Fr(\016)t Fs(\))22 b Fq(\000)g Fr(\021)1423 4594 y Fv(0)1462 4635 y Fs(\()p Fr(\016)t Fs(\))g Fq(\000)1717 4567 y Fr(r)s(\027)1818 4531 y Fl(C)p 1717 4612 147 4 v 1744 4703 a Fr(\021)1796 4669 y Fv(0)1792 4728 y(1)1890 4345 y Fo(0)1890 4520 y(B)1890 4580 y(B)1890 4640 y(B)1890 4704 y(@)2018 4433 y Fs(1)2018 4553 y(0)2029 4649 y(.)2029 4682 y(.)2029 4715 y(.)2018 4835 y(0)2109 4345 y Fo(1)2109 4520 y(C)2109 4580 y(C)2109 4640 y(C)2109 4704 y(A)2212 4635 y Fs(\()p Fr(\021)2302 4594 y Fv(0)2298 4659 y(1)2342 4635 y Fq(j)p Fr(t)p Fq(j)f(\000)i Fr(\016)t Fs(\))f(+)g Fr(O)2853 4494 y Fo(\022)2926 4635 y Fr(t)2961 4594 y Fv(2)3023 4635 y Fs(+)3133 4567 y Fr(\016)3180 4531 y Fv(2)p 3131 4612 91 4 v 3131 4703 a Fq(j)p Fr(t)p Fq(j)3231 4494 y Fo(\023)3354 4635 y Fr(;)28 5129 y(\021)80 5088 y Fl(C)125 5129 y Fs(\()p Fr(t)p Fs(\))p Fr(:)p Fs(\()p Fr(a)p Fs(\()p Fr(t)p Fs(\))h Fq(\000)f Fr(a)636 5088 y Fl(C)682 5129 y Fs(\()p Fr(t)p Fs(\)\))83 b(=)1072 5019 y Fo(\020)1132 5129 y Fr(\021)1184 5088 y Fv(0)1223 5129 y Fs(\()p Fr(\016)t Fs(\))22 b Fq(\000)h Fr(\021)1520 5088 y Fv(0)1555 5065 y Fh(C)1599 5129 y Fs(\()p Fr(\016)t Fs(\))1722 5019 y Fo(\021)1798 5129 y Fr(:\021)1877 5088 y Fv(0)1916 5129 y Fr(t)g Fs(+)f Fr(O)2165 4989 y Fo(\022)2239 5129 y Fq(j)p Fr(t)p Fq(j)2330 5088 y Fv(3)2391 5129 y Fs(+)2499 5062 y Fr(\016)2546 5026 y Fv(4)p 2499 5106 87 4 v 2505 5198 a Fr(t)2540 5169 y Fv(2)2595 4989 y Fo(\023)1095 5432 y Fs(+)g Fr(r)s(\027)1294 5390 y Fl(C)1355 5261 y Fo(")1424 5364 y Fs(sgn\()p Fr(t)p Fs(\))p 1424 5409 253 4 v 1525 5500 a(2)1703 5261 y Fo( )1782 5432 y Fr(\021)1834 5390 y Fv(0)1830 5456 y(1)1873 5432 y Fr(t)1908 5390 y Fv(2)1969 5432 y Fs(+)2104 5364 y Fr(\016)2151 5328 y Fv(2)p 2077 5409 140 4 v 2077 5500 a Fs(2)p Fr(\021)2178 5466 y Fv(0)2174 5524 y(1)2249 5432 y Fs(+)2441 5364 y Fr(\016)2488 5328 y Fv(2)p 2357 5409 255 4 v 2357 5508 a Fr(\021)2409 5474 y Fv(0)2444 5455 y Fh(C)2405 5532 y Fv(1)2489 5508 y Fs(\()p Fr(\016)t Fs(\))2638 5432 y(ln)2736 5261 y Fo( )2825 5364 y Fs(2)p Fr(\021)2926 5328 y Fv(0)2961 5305 y Fh(C)2922 5389 y Fv(1)3005 5364 y Fs(\()p Fr(\016)t Fs(\))p Fq(j)p Fr(t)p Fq(j)p 2825 5409 394 4 v 2998 5500 a Fr(\016)3228 5261 y Fo(!!)3408 5432 y Fq(\000)h Fr(\016)t(t)3590 5261 y Fo(#)3697 5432 y Fr(;)1828 5755 y Fs(22)p eop %%Page: 23 23 23 22 bop 76 138 a Fo(\002)118 219 y Fr(S)184 177 y Fl(C)229 219 y Fs(\()p Fr(t)p Fs(\))22 b(+)g Fr(\021)512 177 y Fl(C)557 219 y Fs(\()p Fr(t)p Fs(\))p Fr(:)p Fs(\()p Fr(x)h Fq(\000)f Fr(a)961 177 y Fl(C)1006 219 y Fs(\()p Fr(t)p Fs(\)\))1155 138 y Fo(\003)1219 219 y Fq(\000)h Fs([)p Fr(S)6 b Fs(\()p Fr(t)p Fs(\))22 b(+)g Fr(\021)t Fs(\()p Fr(t)p Fs(\))p Fr(:)p Fs(\()p Fr(x)g Fq(\000)h Fr(a)p Fs(\()p Fr(t)p Fs(\)\)])242 373 y(=)346 292 y Fo(\000)392 373 y Fr(S)458 332 y Fl(C)503 373 y Fs(\()p Fr(t)p Fs(\))f Fq(\000)g Fr(S)6 b Fs(\()p Fr(t)p Fs(\))912 292 y Fo(\001)980 373 y Fs(+)22 b Fr(\021)1130 332 y Fl(C)1175 373 y Fs(\()p Fr(t)p Fs(\))p Fr(:)1330 292 y Fo(\000)1375 373 y Fr(a)p Fs(\()p Fr(t)p Fs(\))h Fq(\000)f Fr(a)1710 332 y Fl(C)1756 373 y Fs(\()p Fr(t)p Fs(\))1867 292 y Fo(\001)1935 373 y Fs(+)2033 292 y Fo(\000)2078 373 y Fr(\021)2130 332 y Fl(C)2175 373 y Fs(\()p Fr(t)p Fs(\))g Fq(\000)h Fr(\021)t Fs(\()p Fr(t)p Fs(\))2571 292 y Fo(\001)2633 373 y Fr(:"y)242 616 y Fs(=)28 b Fq(\000)433 548 y Fr(r)s(\027)534 512 y Fl(C)580 548 y Fs(sgn\()p Fr(t)p Fs(\))p 433 593 399 4 v 608 684 a(2)859 445 y Fo( )938 616 y Fr(\021)990 574 y Fv(0)986 640 y(1)1029 616 y Fr(t)1064 574 y Fv(2)1126 616 y Fs(+)1260 548 y Fr(\016)1307 512 y Fv(2)p 1234 593 140 4 v 1234 684 a Fs(2)p Fr(\021)1335 650 y Fv(0)1331 708 y(1)1406 616 y Fs(+)1597 548 y Fr(\016)1644 512 y Fv(2)p 1514 593 255 4 v 1514 692 a Fr(\021)1566 658 y Fv(0)1601 639 y Fh(C)1562 716 y Fv(1)1645 692 y Fs(\()p Fr(\016)t Fs(\))1794 616 y(ln)1892 445 y Fo( )1981 548 y Fs(2)p Fr(\021)2082 512 y Fv(0)2117 489 y Fh(C)2078 573 y Fv(1)2161 548 y Fs(\()p Fr(\016)t Fs(\))p Fq(j)p Fr(t)p Fq(j)p 1981 593 394 4 v 2154 684 a Fr(\016)2384 445 y Fo(!!)3558 616 y Fs(\(26\))265 920 y(+)22 b Fr(O)456 779 y Fo(\022)530 920 y Fq(j)p Fr(t)p Fq(j)621 879 y Fv(3)682 920 y Fs(+)790 852 y Fr(\016)837 816 y Fv(4)p 790 897 87 4 v 796 988 a Fr(t)831 959 y Fv(2)908 920 y Fs(+)1006 779 y Fo(\022)1080 920 y Fr(t)1115 879 y Fv(2)1177 920 y Fs(+)1287 852 y Fr(\016)1334 816 y Fv(2)p 1285 897 91 4 v 1285 988 a Fq(j)p Fr(t)p Fq(j)1385 779 y Fo(\023)1475 920 y Fr(")p Fq(k)p Fr(y)t Fq(k)1673 779 y Fo(\023)1767 920 y Fs(+)1865 809 y Fo(\020)1924 920 y Fr(\021)1976 879 y Fv(0)2011 855 y Fh(C)2056 920 y Fs(\()p Fr(\016)t Fs(\))g Fq(\000)g Fr(\021)2352 879 y Fv(0)2391 920 y Fs(\()p Fr(\016)t Fs(\))2514 809 y Fo(\021)2590 920 y Fr(:"y)j Fs(+)d Fr(r)s(\027)2935 879 y Fl(C)2981 920 y Fr("y)3075 935 y Fv(1)3130 779 y Fo(\022)3235 852 y Fr(\016)p 3213 897 92 4 v 3213 988 a(\021)3265 954 y Fv(0)3261 1012 y(1)3336 920 y Fq(\000)h(j)p Fr(t)p Fq(j)3527 779 y Fo(\023)3649 920 y Fr(;)859 1308 y(A)932 1267 y Fl(C)977 1308 y Fs(\()p Fr(t)p Fs(\))84 b(=)e Fr(A)1403 1323 y Fv(0)1465 1308 y Fs(+)22 b Fr(O)s Fs(\()p Fr(t)p Fs(\))32 b Fr(;)853 1515 y(B)932 1474 y Fl(C)977 1515 y Fs(\()p Fr(t)p Fs(\))84 b(=)e Fr(B)1404 1530 y Fv(0)1466 1515 y Fs(+)1574 1448 y Fr(ir)s(\027)1708 1412 y Fl(C)1754 1448 y Fs(sgn\()p Fr(t)p Fs(\))p 1574 1493 433 4 v 1745 1584 a Fr(\021)1797 1549 y Fv(0)1793 1608 y(1)2016 1515 y Fr(P)14 b(A)2166 1530 y Fv(0)2227 1515 y Fs(+)22 b Fr(O)2419 1375 y Fo(\022)2492 1515 y Fq(j)p Fr(t)p Fq(j)g Fs(+)2713 1448 y Fr(\016)2760 1412 y Fv(2)p 2713 1493 87 4 v 2719 1584 a Fr(t)2754 1555 y Fv(2)2809 1375 y Fo(\023)1172 1790 y Fs(=)82 b Fr(B)1409 1748 y Fl(C)1404 1814 y Fv(0)1455 1790 y Fs(\(sgn\()p Fr(t)p Fs(\)\))22 b(+)g Fr(O)1997 1649 y Fo(\022)2070 1790 y Fq(j)p Fr(t)p Fq(j)g Fs(+)2291 1722 y Fr(\016)2338 1686 y Fv(2)p 2291 1767 V 2297 1858 a Fr(t)2332 1829 y Fv(2)2387 1649 y Fo(\023)2510 1790 y Fr(:)21 2074 y Fs(W)-8 b(e)33 b(note)f(that)g(\()p Fr(A)727 2089 y Fv(0)767 2074 y Fr(;)17 b(B)890 2038 y Fl(C)885 2098 y Fv(0)935 2074 y Fs(\()p Fq(\006)p Fs(1\)\))32 b(satis\014es)h(conditions)e (\(8\),)h(if)f(\()p Fr(A)2416 2089 y Fv(0)2456 2074 y Fr(;)17 b(B)2574 2089 y Fv(0)2613 2074 y Fs(\))32 b(do)s(es.)44 b(Moreo)m(v)m(er,)34 b(w)m(e)f(ha)m(v)m(e)21 2194 y(that)544 2207 y Fo(\015)544 2267 y(\015)544 2326 y(\015)544 2386 y(\015)600 2351 y Fr(\036)658 2366 y Fp(l)700 2211 y Fo(\022)773 2351 y Fr(A)846 2310 y Fl(C)891 2351 y Fs(\()p Fr(t)p Fs(\))p Fr(;)17 b(B)1125 2310 y Fl(C)1170 2351 y Fs(\()p Fr(t)p Fs(\))p Fr(;)g(y)25 b Fs(+)1506 2284 y Fr(a)p Fs(\()p Fr(t)p Fs(\))e Fq(\000)g Fr(a)1842 2248 y Fl(C)1887 2284 y Fs(\()p Fr(t)p Fs(\))p 1506 2328 492 4 v 1729 2420 a Fr(")2008 2211 y Fo(\023)2103 2351 y Fq(\000)g Fr(\036)2261 2366 y Fp(l)2287 2351 y Fs(\()p Fr(A)2398 2366 y Fv(0)2437 2351 y Fr(;)17 b(B)2560 2310 y Fl(C)2555 2376 y Fv(0)2605 2351 y Fs(\(sgn)q(\()p Fr(t)p Fs(\)\))p Fr(;)g(y)t Fs(\))3068 2207 y Fo(\015)3068 2267 y(\015)3068 2326 y(\015)3068 2386 y(\015)3122 2450 y Fp(L)3170 2431 y Fi(2)21 2607 y Fs(is)32 b(b)s(ounded)i(b)m(y)f(a)f (constan)m(t)i(times)d Fq(j)p Fr(t)p Fq(j)22 b Fs(+)1609 2568 y Fp(\016)1642 2545 y Fi(2)p 1609 2584 69 4 v 1613 2642 a Fp(t)1638 2623 y Fi(2)1710 2607 y Fs(+)1818 2568 y Fp(t)1843 2545 y Fi(2)p 1818 2584 61 4 v 1831 2642 a Fp(")1910 2607 y Fs(+)2018 2568 y Fp(\016)2051 2545 y Fi(2)p 2018 2584 69 4 v 2036 2642 a Fp(")2113 2607 y Fs(ln)2210 2523 y Fo(\014)2210 2582 y(\014)2258 2568 y Fp(t)p 2254 2584 34 4 v 2254 2642 a(\016)2297 2523 y Fo(\014)2297 2582 y(\014)2331 2607 y Fs(.)21 2867 y Fk(Incoming)36 b(Outer)i(Asymptotics.)27 3143 y Fr( )90 3158 y Fp(I)5 b(O)186 3143 y Fs(\()p Fr(t;)17 b(x;)g(\016)n(;)g(")p Fs(\))82 b(=)h Fr(F)906 3003 y Fo(\022)979 3143 y Fr(\015)1035 3102 y Fv(2)1084 3076 y Fq(k)p Fr(x)23 b Fq(\000)f Fr(a)1362 3040 y Fl(B)1415 3076 y Fs(\()p Fr(t)p Fs(\))p Fq(k)1576 3040 y Fv(2)p 1084 3120 531 4 v 1307 3212 a Fr(")1353 3183 y Fv(2)1625 3003 y Fo(\023)1715 3143 y Fr(')1779 3102 y Fl(B)1779 3168 y Fp(l)1847 3003 y Fo(\022)1921 3143 y Fr(t;)2010 3076 y(x)g Fq(\000)h Fr(a)2238 3040 y Fl(B)2290 3076 y Fs(\()p Fr(t)p Fs(\))p 2010 3120 392 4 v 2183 3212 a Fr(")2411 3143 y(;)17 b(")2501 3003 y Fo(\023)2591 3143 y Fs(\010)2661 3102 y Fl(\000)2661 3170 y(B)2720 3143 y Fs(\()p Fr(t;)g(x;)g(\016)t Fs(\))653 3396 y(=)83 b Fq(\000)p Fr(e)966 3324 y Fn(i)p 945 3336 64 3 v 945 3386 a(")974 3372 y Fi(2)1019 3351 y Fv(\()p Fp(S)t Fv(\()p Fp(t)p Fv(\)+)p Fp(\021)r Fv(\()p Fp(t)p Fv(\))p Fp(:)p Fv(\()p Fp(x)p Fl(\000)p Fp(a)p Fv(\()p Fp(t)p Fv(\)\)\))1665 3396 y Fr(e)1710 3351 y Fl(\000)p Fp(ir)r Fv(\()p Fp(\021)1887 3328 y Fi(0)1885 3372 y(1)1950 3324 y Fn(t)1974 3303 y Fi(2)p 1932 3336 95 3 v 1932 3386 a(2)p Fn(")1991 3372 y Fi(2)2037 3351 y Fv(+)2104 3324 y Fn(t)p 2102 3336 30 3 v 2102 3377 a(")2141 3351 y Fp(y)2176 3360 y Fi(1)2210 3351 y Fv(\))2242 3396 y Fr(e)2287 3355 y Fp(i!)2357 3325 y Fh(\000)2355 3378 y(B)2409 3355 y Fv(\()p Fp(t;a)2518 3332 y Fh(B)2565 3355 y Fv(\()p Fp(t)p Fv(\))p Fp(;\016)r Fv(\))812 3582 y Fq(\002)p Fr(\036)947 3597 y Fp(l)990 3442 y Fo(\022)1063 3582 y Fr(A)1136 3597 y Fv(0)1176 3582 y Fr(;)17 b(B)1299 3541 y Fl(B)1294 3607 y Fv(0)1351 3582 y Fs(\()p Fq(\000)p Fs(1\))p Fr(;)1607 3515 y(x)22 b Fq(\000)h Fr(a)p Fs(\()p Fr(t)p Fs(\))p 1607 3559 340 4 v 1754 3651 a Fr(")1956 3442 y Fo(\023)2046 3582 y Fr( )2109 3597 y Fv(1)2149 3582 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))812 3785 y Fq(\002)g Fs([1)22 b(+)g Fr(O)d Fs(\()p Fr(e)1279 3800 y Fp(')1330 3785 y Fs(\()p Fr(t;)e(\016)n(;)g(";)g(\015)5 b Fs(\))21 b(+)h Fr(e)1880 3800 y Fp(\036)1926 3785 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))22 b(+)g Fr(e)2377 3800 y Fv(\010)2433 3785 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))k(+)h Fr(e)2799 3800 y Fp(O)2859 3785 y Fs(\()p Fr(t;)17 b(")p Fs(\))22 b(+)g Fr(e)3225 3800 y Fp(!)3276 3785 y Fs(\()p Fr(t;)17 b(\016)n(;)g(";)g(\015)5 b Fs(\)\))o(])21 4005 y(where)167 4259 y Fr(e)212 4274 y Fp(')262 4259 y Fs(\()p Fr(t;)17 b(\016)n(;)g(";)g(\015)5 b Fs(\))27 b(=)789 4192 y Fq(j)p Fr(t)p Fq(j)880 4156 y Fv(3)p 789 4236 131 4 v 811 4327 a Fr(")857 4299 y Fv(2)951 4259 y Fs(+)1072 4192 y Fr(t)1107 4156 y Fv(2)p 1059 4236 102 4 v 1059 4327 a Fr("\015)1193 4259 y Fs(+)1301 4192 y Fr(\016)1348 4156 y Fv(2)p 1301 4236 87 4 v 1302 4327 a Fr(")1348 4299 y Fv(2)1414 4259 y Fs(ln)1512 4115 y Fo(\014)1512 4174 y(\014)1512 4234 y(\014)1512 4294 y(\014)1561 4192 y Fr(t)p 1555 4236 47 4 v 1555 4327 a(\016)1612 4115 y Fo(\014)1612 4174 y(\014)1612 4234 y(\014)1612 4294 y(\014)1667 4259 y Fs(+)1802 4192 y Fr(\016)p 1775 4236 102 4 v 1775 4327 a("\015)1919 4259 y(;)49 b(e)2040 4274 y Fp(\036)2087 4259 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))27 b(=)g Fq(j)p Fr(t)p Fq(j)22 b Fs(+)2724 4192 y Fr(\016)2771 4156 y Fv(2)p 2724 4236 87 4 v 2730 4327 a Fr(t)2765 4299 y Fv(2)2843 4259 y Fs(+)2951 4192 y Fr(t)2986 4156 y Fv(2)p 2951 4236 75 4 v 2965 4327 a Fr(")3057 4259 y Fs(+)3165 4192 y Fr(\016)3212 4156 y Fv(2)p 3165 4236 87 4 v 3186 4327 a Fr(")3278 4259 y Fs(ln)3376 4115 y Fo(\014)3376 4174 y(\014)3376 4234 y(\014)3376 4294 y(\014)3425 4192 y Fr(t)p 3419 4236 47 4 v 3419 4327 a(\016)3476 4115 y Fo(\014)3476 4174 y(\014)3476 4234 y(\014)3476 4294 y(\014)3559 4259 y Fr(;)165 4608 y(e)210 4623 y Fv(\010)265 4608 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))27 b(=)h Fq(j)p Fr(t)p Fq(j)22 b Fs(+)840 4541 y Fr(\016)p 819 4585 91 4 v 819 4676 a Fq(j)p Fr(t)p Fq(j)952 4608 y Fr(;)49 b(e)1073 4623 y Fp(O)1133 4608 y Fs(\()p Fr(t;)17 b(")p Fs(\))27 b(=)g Fr(")17 b Fs(ln)1624 4464 y Fo(\014)1624 4523 y(\014)1624 4583 y(\014)1624 4643 y(\014)1668 4541 y Fs(1)p 1668 4585 49 4 v 1675 4676 a Fr(t)1726 4464 y Fo(\014)1726 4523 y(\014)1726 4583 y(\014)1726 4643 y(\014)1782 4608 y Fs(+)1890 4541 y Fr(")1936 4504 y Fv(2)p 1890 4585 86 4 v 1895 4676 a Fr(t)1930 4648 y Fv(2)2007 4608 y Fs(+)2137 4541 y Fr(")2183 4504 y Fv(4)p 2115 4585 131 4 v 2115 4676 a Fq(j)p Fr(t)p Fq(j)2206 4648 y Fv(3)2287 4608 y Fr(;)50 b(e)2409 4623 y Fp(!)2459 4608 y Fs(\()p Fr(t;)17 b(\016)n(;)g(";)g(\015)5 b Fs(\))27 b(=)3013 4541 y Fr(\016)t(")p 2986 4585 147 4 v 2986 4676 a(\015)5 b Fq(j)p Fr(t)p Fq(j)3164 4608 y Fs(+)3278 4541 y Fr(")p 3272 4585 57 4 v 3272 4676 a(\015)3355 4608 y Fs(ln)3453 4464 y Fo(\014)3453 4523 y(\014)3453 4583 y(\014)3453 4643 y(\014)3496 4541 y Fs(1)p 3496 4585 49 4 v 3503 4676 a Fr(t)3555 4464 y Fo(\014)3555 4523 y(\014)3555 4583 y(\014)3555 4643 y(\014)21 4840 y Fs(are)25 b(errors)h(due)g(resp)s(ectiv)m(ely)g(to)f(the)h(phase)g Fr(S)6 b Fs(\()p Fr(t)p Fs(\))h(+)g Fr(\021)t Fs(\()p Fr(t)p Fs(\))p Fr(:)p Fs(\()p Fr(x)g Fq(\000)g Fr(a)p Fs(\()p Fr(t)p Fs(\)\))27 b(\(cf)e(\(26\)\),)h(the)g(Gaussian)f(w)m(a)m (v)m(e)21 4960 y(pac)m(k)m(et)35 b Fr(\036)386 4975 y Fp(l)445 4960 y Fs(\(see)f(the)f(end)h(of)f(the)g(preceding)h (paragraph\),)e(the)i(eigen)m(v)m(ector)g(\010)2991 4919 y Fl(\000)2991 4987 y(B)3051 4960 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))33 b(\(cf)g(lemma)21 5080 y(2\),)g(the)g(incoming)d (outer)j(Ansatz)g(\(cf)g(prop)s(osition)d(2\))j(and)f(the)h(corrected)h (phase)g Fr(!)3229 5039 y Fl(\000)3225 5107 y(B)3287 5080 y Fs(\()p Fr(t;)17 b(x;)g(\016)t Fs(\))32 b(\(cf)21 5201 y(lemma)e(4\).)1828 5755 y(23)p eop %%Page: 24 24 24 23 bop 21 219 a Fk(Inner)38 b(Asymptotics.)212 424 y Fs(^)192 450 y Fr( )255 465 y Fp(I)296 450 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))81 b(=)i Fr(e)1005 378 y Fn(i)p 984 390 64 3 v 984 440 a(")1013 426 y Fi(2)1058 405 y Fv(\()p Fp(S)t Fv(\()p Fp("s)p Fv(\)+)p Fp("\021)r Fv(\()p Fp("s)p Fv(\))p Fp(:y)r Fv(\))1622 356 y Fo(X)1603 568 y Fp(k)r Fv(=1)p Fp(;)p Fv(2)1803 450 y Fr(g)1850 465 y Fp(k)1892 450 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))p Fr(e)2240 409 y Fv(\()p Fl(\000)p Fv(1\))2384 386 y Fn(k)2422 409 y Fp(ir)r Fv(\()p Fp(\021)2544 386 y Fi(0)2542 430 y(1)2590 382 y Fn(s)2619 361 y Fi(2)p 2590 394 V 2606 435 a(2)2663 409 y Fv(+)p Fp(sy)2786 418 y Fi(1)2820 409 y Fv(\))2852 450 y Fr( )2915 465 y Fp(k)2958 450 y Fs(\()p Fr(a)p Fs(\()p Fr("s)p Fs(\))22 b(+)g Fr("y)t(;)17 b(\016)t Fs(\))929 699 y Fq(\002)g Fs([1)22 b(+)g Fr(O)d Fs(\()p Fr(e)1396 714 y Fp(I)1436 699 y Fs(\()p Fr("s;)e(\016)n(;)g(")p Fs(\)\))o(])21 918 y(where)34 b Fr(e)348 933 y Fp(I)388 918 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))27 b(=)h Fq(j)p Fr(t)p Fq(j)22 b Fs(+)1026 870 y Fl(j)p Fp(t)p Fl(j)1091 847 y Fi(3)p 1026 895 100 4 v 1042 952 a Fp(")1075 933 y Fi(2)1157 918 y Fs(+)1265 870 y Fl(j)p Fp(t)p Fl(j)p Fp(\016)p 1265 895 99 4 v 1281 952 a(")1314 933 y Fi(2)1396 918 y Fs(+)1504 878 y Fp(t)1529 855 y Fi(2)p 1504 895 61 4 v 1517 952 a Fp(")1606 918 y Fs(is)32 b(the)h(error)g(term)f(giv)m (en)g(b)m(y)i(prop)s(osition)d(3.)21 1175 y Fk(Matc)m(hing)37 b(for)h Fr(t)28 b(<)f Fs(0)p Fk(.)98 b Fs(W)-8 b(e)33 b(can)f(matc)m(h)h(those)g(t)m(w)m(o)g(Ans\177)-49 b(atze)34 b(with)e(an)h(error)f(of)g(order)333 1378 y Fr(O)18 b Fs(\()p Fr(e)509 1393 y Fp(')560 1378 y Fs(\()p Fr(t;)f(\016)n(;)g(";)g (\015)5 b Fs(\))21 b(+)h Fr(e)1110 1393 y Fp(\036)1157 1378 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))k(+)h Fr(e)1607 1393 y Fv(\010)1663 1378 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))22 b(+)g Fr(e)2030 1393 y Fp(E)2089 1378 y Fs(\()p Fr(t;)17 b(")p Fs(\))22 b(+)g Fr(e)2455 1393 y Fp(!)2506 1378 y Fs(\()p Fr(t;)17 b(\016)n(;)g(";)g(\015)5 b Fs(\))21 b(+)h Fr(e)3056 1393 y Fp(I)3096 1378 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\)\))21 1581 y(b)m(y)34 b(c)m(ho)s(osing)656 1794 y Fr(g)703 1809 y Fv(1)742 1794 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))82 b(=)h Fq(\000)p Fr(\036)1421 1809 y Fp(l)1447 1794 y Fs(\()p Fr(A)1558 1809 y Fv(0)1597 1794 y Fr(;)17 b(B)1720 1753 y Fl(B)1715 1819 y Fv(0)1772 1794 y Fs(\()p Fq(\000)p Fs(1\))p Fr(;)g(y)t Fs(\))p Fr(e)2153 1753 y Fp(i!)2223 1723 y Fh(\000)2221 1776 y(B)2275 1753 y Fv(\()p Fl(\000)p Fp(t)2382 1761 y Fn(m)2441 1753 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;a)2635 1730 y Fh(B)2681 1753 y Fv(\()p Fl(\000)p Fp(t)2788 1761 y Fn(m)2847 1753 y Fv(\()p Fp(\016)o(;")p Fv(\)\))p Fp(;\016)r Fv(\))656 1940 y Fr(g)703 1955 y Fv(2)742 1940 y Fs(\()p Fr(y)t(;)g(\016)n(;)g(")p Fs(\))82 b(=)h(0)32 b Fr(:)21 2197 y Fk(Outgoing)37 b(Outer)g(Asymptotics.)298 2400 y Fr( )361 2415 y Fp(O)r(O)477 2400 y Fs(\()p Fr(t;)17 b(x;)g(\016)n(;)g(")p Fs(\))27 b(=)464 2583 y Fr(e)540 2511 y Fn(i)p 520 2523 64 3 v 520 2573 a(")549 2559 y Fi(2)593 2538 y Fv(\()p Fp(S)t Fv(\()p Fp(t)p Fv(\)+)p Fp(\021)r Fv(\()p Fp(t)p Fv(\))p Fp(:)p Fv(\()p Fp(x)p Fl(\000)p Fp(a)p Fv(\()p Fp(t)p Fv(\)\)\))1460 2489 y Fo(X)1256 2705 y Fv(\()p Fl(C)t Fp(;k)r Fv(\)=\()p Fl(A)p Fp(;)p Fv(1\))p Fp(;)p Fv(\()p Fl(B)r Fp(;)p Fv(2\))1825 2583 y Fs(\003)1893 2542 y Fv(+)1893 2610 y Fl(C)1952 2583 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fr(e)2204 2538 y Fv(\()p Fl(\000)p Fv(1\))2348 2515 y Fn(k)2387 2538 y Fp(ir)r Fv(\()p Fp(\021)2509 2515 y Fi(0)2507 2559 y(1)2572 2511 y Fn(t)2596 2490 y Fi(2)p 2554 2523 95 3 v 2554 2573 a(2)p Fn(")2613 2559 y Fi(2)2659 2538 y Fv(+)2726 2511 y Fn(t)p 2723 2523 30 3 v 2723 2564 a(")2763 2538 y Fp(y)2798 2547 y Fi(1)2832 2538 y Fv(\))2864 2583 y Fr(e)2909 2542 y Fp(i!)2979 2512 y Fi(+)2977 2565 y Fh(C)3030 2542 y Fv(\()p Fp(t;a)3139 2519 y Fh(C)3180 2542 y Fv(\()p Fp(t)p Fv(\))p Fp(;\016)r Fv(\))464 2909 y Fq(\002)p Fr(\036)599 2924 y Fp(l)642 2768 y Fo(\022)716 2909 y Fr(A)789 2924 y Fv(0)828 2909 y Fr(;)g(B)951 2868 y Fl(C)946 2933 y Fv(0)996 2909 y Fs(\(+1\))p Fr(;)1250 2841 y(x)23 b Fq(\000)f Fr(a)p Fs(\()p Fr(t)p Fs(\))p 1250 2886 340 4 v 1397 2977 a Fr(")1600 2768 y Fo(\023)1690 2909 y Fr( )1753 2924 y Fp(k)1796 2909 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))464 3111 y Fq(\002)g Fs([)q(1)k(+)h Fr(O)d Fs(\()p Fr(e)931 3126 y Fp(')982 3111 y Fs(\()p Fr(t;)e(\016)n(;)g(";)g (\015)5 b Fs(\))21 b(+)h Fr(e)1532 3126 y Fp(\036)1578 3111 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))22 b(+)g Fr(e)2029 3126 y Fv(\010)2085 3111 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))22 b(+)g Fr(e)2452 3126 y Fp(O)2511 3111 y Fs(\()p Fr(t;)17 b(")p Fs(\))22 b(+)g Fr(e)2877 3126 y Fp(!)2928 3111 y Fs(\()p Fr(t;)17 b(\016)n(;)g(";)g(\015)5 b Fs(\)\))o(])49 b Fr(:)21 3369 y Fk(Matc)m(hing)35 b(for)f Fr(t)28 b(>)g Fs(0)p Fk(.)97 b Fs(W)-8 b(e)31 b(can)f(matc)m(h)g(the)h (t)m(w)m(o)g(preceding)f(Ans\177)-49 b(atze)32 b(with)e(an)g(error)g (of)g(same)21 3489 y(order)j(as)g(for)f Fr(t)c(<)f Fs(0)32 b(b)m(y)i(c)m(ho)s(osing)640 3702 y(\003)708 3661 y Fv(+)708 3729 y Fl(A)768 3702 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))83 b(=)g Fr(e)1262 3661 y Fp(i)p Fv([)p Fp(!)1352 3631 y Fh(\000)1350 3684 y(B)1404 3661 y Fv(\()p Fl(\000)p Fp(t)1511 3669 y Fn(m)1570 3661 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;a)1764 3638 y Fh(B)1810 3661 y Fv(\()p Fl(\000)p Fp(t)1917 3669 y Fn(m)1976 3661 y Fv(\()p Fp(\016)o(;")p Fv(\)\))p Fp(;\016)r Fv(\))p Fl(\000)p Fp(!)2321 3631 y Fi(+)2319 3684 y Fh(A)2374 3661 y Fv(\()p Fp(t)2426 3669 y Fn(m)2485 3661 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;a)2679 3638 y Fh(A)2733 3661 y Fv(\()p Fp(t)2785 3669 y Fn(m)2844 3661 y Fv(\()p Fp(\016)o(;")p Fv(\)\))p Fp(;\016)r Fv(\)])641 3848 y Fs(\003)709 3806 y Fv(+)709 3875 y Fl(B)768 3848 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))83 b(=)g(0)p Fr(:)167 4051 y Fs(Then,)34 b(the)f(error)f(term)g(is)g(of)g (order)g Fr(o)p Fs(\(1\))g(if)f(w)m(e)j(c)m(ho)s(ose)f Fr(t)28 b Fs(=)f Fr(t)2498 4066 y Fp(m)2565 4051 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))28 b Fq(2)g Fs([)p Fr(t)2956 4066 y Fp(o)2994 4051 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fr(;)g(t)3280 4066 y Fp(i)3308 4051 y Fs(\()p Fr(\016)n(;)g(")p Fs(\)])32 b(and)21 4171 y Fr(\015)h Fs(=)27 b Fr(\015)5 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))33 b(tending)f(to)g(0)g(with)380 4433 y(max)578 4293 y Fo(\022)652 4433 y Fr(\016)n(;)752 4366 y(")p 747 4411 57 4 v 747 4502 a(\015)813 4293 y Fo(\023)914 4433 y Fq(\034)27 b Fr(t)h Fq(\034)f Fs(min)1410 4293 y Fo(\022)1484 4433 y Fr(")1530 4392 y Fv(2)p Fp(=)p Fv(3)1639 4433 y Fr(;)1683 4367 y Fq(p)p 1766 4367 102 4 v 66 x Fr("\015)5 b(;)1922 4366 y(")1968 4330 y Fv(2)p 1922 4411 86 4 v 1941 4502 a Fr(\016)2017 4293 y Fo(\023)2140 4433 y Fs(and)50 b(max)2545 4293 y Fo(\022)2618 4433 y Fr(")2664 4392 y Fv(1)p Fp(=)p Fv(3)2774 4433 y Fr(;)2828 4366 y(\016)p 2828 4411 47 4 v 2829 4502 a(")2884 4293 y Fo(\023)2985 4433 y Fq(\034)28 b Fr(\015)k Fq(\034)c Fs(1)21 4694 y(\(whic)m(h)33 b(is)f(a)h(non-empt) m(y)f(zone\).)21 5158 y Fk(First)47 b(Order)h(Matc)m(hing.)98 b Fs(By)42 b(c)m(ho)s(osing)f Fr(\021)1852 5122 y Fv(0)1887 5098 y Fh(C)1931 5158 y Fs(\()p Fr(\016)t Fs(\))i(=)h Fr(\021)2269 5122 y Fv(0)2308 5158 y Fs(\()p Fr(\016)t Fs(\))28 b Fq(\000)2575 5119 y Fp(r)r(\027)2648 5095 y Fh(C)p 2575 5135 114 4 v 2595 5194 a Fp(\021)2632 5171 y Fi(0)2630 5215 y(1)2714 4868 y Fo(0)2714 5043 y(B)2714 5103 y(B)2714 5163 y(B)2714 5227 y(@)2843 4956 y Fs(1)2843 5076 y(0)2854 5172 y(.)2854 5205 y(.)2854 5238 y(.)2843 5359 y(0)2933 4868 y Fo(1)2933 5043 y(C)2933 5103 y(C)2933 5163 y(C)2933 5227 y(A)3037 5158 y Fr(\016)t Fs(,)44 b(w)m(e)f(substitute)21 5505 y(the)33 b(error)f(term)g(of)f(order)1054 5466 y Fp(\016)p 1035 5482 73 4 v 1035 5539 a("\015)1150 5505 y Fs(in)g Fr(e)1308 5520 y Fp(')1359 5505 y Fs(\()p Fr(t;)17 b(\016)n(;)g(";)g(\015)5 b Fs(\))31 b(b)m(y)1956 5466 y Fp(\016)1989 5442 y Fi(2)p 1921 5482 138 4 v 1921 5539 a Fp("\015)t Fl(j)p Fp(t)p Fl(j)2101 5505 y Fs(\(b)m(y)i(the)g(w)m (a)m(y)-8 b(,)33 b(total)e(energy)j(conserv)-5 b(ation)1828 5755 y(24)p eop %%Page: 25 25 25 24 bop 21 263 a Fs(at)35 b(0,)255 123 y Fo(\024)318 216 y Fl(k)p Fp(\021)390 192 y Fi(0)420 172 y Fh(A)479 216 y Fv(\()p Fp(\016)r Fv(\))p Fl(k)601 192 y Fi(2)p 318 240 320 4 v 460 298 a Fv(2)670 263 y Fq(\000)22 b Fr(E)841 278 y Fl(A)902 263 y Fs(\(0)p Fr(;)17 b(\016)t Fs(\))1118 123 y Fo(\025)1194 263 y Fq(\000)1295 123 y Fo(\024)1358 216 y Fl(k)p Fp(\021)1430 192 y Fi(0)1460 172 y Fh(B)1511 216 y Fv(\()p Fp(\016)r Fv(\))p Fl(k)1633 192 y Fi(2)p 1358 240 312 4 v 1496 298 a Fv(2)1702 263 y Fq(\000)23 b Fr(E)1874 278 y Fl(B)1926 263 y Fs(\(0)p Fr(;)17 b(\016)t Fs(\))2142 123 y Fo(\025)2226 263 y Fs(=)33 b Fr(O)s Fs(\()p Fr(\016)2498 227 y Fv(2)2536 263 y Fs(\),)k(is)d(no)m(w)j(ful\014lled)c(up)j(to)f(\014rst)21 441 y(order\))e(;)f(w)m(e)i(go)e(further)h(in)f(solving)f(\(23\))h(b)m (y)i(taking)384 695 y Fr(f)432 710 y Fv(1)471 695 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))26 b(=)i Fr(e)1039 654 y Fl(\000)p Fp(ir)r Fv(\()p Fp(\021)1216 631 y Fi(0)1214 675 y(1)1261 627 y Fn(s)1290 606 y Fi(2)p 1261 639 64 3 v 1278 680 a(2)1335 654 y Fv(+)p Fp(sy)1458 663 y Fi(1)1492 654 y Fv(\))550 898 y Fq(\002)644 757 y Fo(\024)696 898 y Fr(g)743 913 y Fv(1)783 898 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))j(+)1214 830 y Fr(\016)p 1214 875 47 4 v 1215 966 a(")1288 757 y Fo(\022)1361 898 y Fr(h)1417 913 y Fv(1)1457 898 y Fs(\()p Fr(y)t(;)d(\016)n(;)g(")p Fs(\))j Fq(\000)j Fr(ir)s(g)2007 913 y Fv(2)2046 898 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))2366 762 y Fo(Z)2464 788 y Fp(s)2420 988 y Fl(\0001)2566 898 y Fr(e)2611 857 y Fp(ir)r Fv(\()p Fp(\021)2733 833 y Fi(0)2731 877 y(1)2769 857 y Fp(\033)2811 833 y Fi(2)2846 857 y Fv(+2)p Fp(\033)r(y)3013 866 y Fi(1)3048 857 y Fv(\))3080 898 y Fr(d\033)3190 757 y Fo(\023\025)3558 898 y Fs(\(27\))384 1149 y Fr(f)432 1164 y Fv(2)471 1149 y Fs(\()p Fr(s;)g(y)t(;)g(\016)n(;)g(")p Fs(\))26 b(=)i Fr(e)1039 1108 y Fp(ir)r Fv(\()p Fp(\021)1161 1085 y Fi(0)1159 1129 y(1)1206 1081 y Fn(s)1235 1060 y Fi(2)p 1206 1093 64 3 v 1223 1134 a(2)1280 1108 y Fv(+)p Fp(sy)1403 1117 y Fi(1)1437 1108 y Fv(\))550 1352 y Fq(\002)644 1211 y Fo(\024)696 1352 y Fr(g)743 1367 y Fv(2)783 1352 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))j(+)1214 1284 y Fr(\016)p 1214 1329 47 4 v 1215 1420 a(")1288 1211 y Fo(\022)1361 1352 y Fr(h)1417 1367 y Fv(2)1457 1352 y Fs(\()p Fr(y)t(;)d(\016)n(;)g(")p Fs(\))j(+)i Fr(ir)s(g)2005 1367 y Fv(1)2045 1352 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))2365 1216 y Fo(Z)2462 1242 y Fp(s)2418 1442 y Fl(\0001)2564 1352 y Fr(e)2609 1311 y Fl(\000)p Fp(ir)r Fv(\()p Fp(\021)2786 1287 y Fi(0)2784 1331 y(1)2822 1311 y Fp(\033)2864 1287 y Fi(2)2899 1311 y Fv(+2)p Fp(\033)r(y)3066 1320 y Fi(1)3102 1311 y Fv(\))3133 1352 y Fr(d\033)3243 1211 y Fo(\023\025)3558 1352 y Fs(\(28\))21 1639 y(with)35 b Fr(g)293 1654 y Fp(k)335 1639 y Fr(;)17 b(h)435 1654 y Fp(k)509 1639 y Fq(2)33 b Fr(H)697 1603 y Fv(2)736 1639 y Fs(\()p Fm(R)839 1603 y Fp(d)886 1639 y Fs(\))24 b Fq(\\)g Fs(\(1)f(+)h Fq(k)p Fr(y)t Fq(k)1400 1603 y Fv(2)1438 1639 y Fs(\))1476 1603 y Fl(\000)p Fv(1)1570 1639 y Fr(L)1636 1603 y Fv(2)1676 1639 y Fs(\()p Fm(R)1780 1603 y Fp(d)1827 1639 y Fs(\),)35 b(th)m(us)h(w)m(e)g(substitute)g(the) f(error)g(term)g(of)f(order)31 1722 y Fl(j)p Fp(t)p Fl(j)p Fp(\016)p 31 1746 99 4 v 47 1804 a(")80 1785 y Fi(2)172 1769 y Fs(in)e Fr(e)331 1784 y Fp(I)371 1769 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))32 b(b)m(y)835 1730 y Fp(\016)868 1707 y Fi(2)p 835 1746 69 4 v 852 1804 a Fp(")945 1769 y Fs(;)h(matc)m(hing)e(for)h Fr(t)c(<)g Fs(0)k(can)h(b)s(e)g(p)s(erformed)f(b)m(y)h(c)m(ho)s(osing)661 2014 y Fr(g)708 2029 y Fv(1)747 2014 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))81 b(=)i Fq(\000)p Fr(\036)1425 2029 y Fp(l)1452 2014 y Fs(\()p Fr(A)1563 2029 y Fv(0)1602 2014 y Fr(;)17 b(B)1725 1973 y Fl(B)1720 2039 y Fv(0)1777 2014 y Fs(\()p Fq(\000)p Fs(1\))p Fr(;)g(y)t Fs(\))p Fr(e)2158 1973 y Fp(i!)2228 1943 y Fh(\000)2226 1996 y(B)2280 1973 y Fv(\()p Fl(\000)p Fp(t)2387 1981 y Fn(m)2446 1973 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;a)2640 1950 y Fh(B)2686 1973 y Fv(\()p Fl(\000)p Fp(t)2793 1981 y Fn(m)2852 1973 y Fv(\()p Fp(\016)o(;")p Fv(\)\))p Fp(;\016)r Fv(\))661 2160 y Fr(g)708 2175 y Fv(2)747 2160 y Fs(\()p Fr(y)t(;)g(\016)n(;)g(")p Fs(\))81 b(=)i(0)651 2305 y Fr(h)707 2320 y Fv(1)747 2305 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))81 b(=)i(0)651 2450 y Fr(h)707 2465 y Fv(2)747 2450 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))81 b(=)i(0)33 b(;)21 2670 y(matc)m(hing)45 b(for)g Fr(t)51 b(>)f Fs(0)45 b(can)h(b)s(e)g(p)s(erformed)g(to)s(o)f(\(w)m(e)i(use)f (the)h(iden)m(tit)m(y)e Fr(\036)2919 2685 y Fp(l)2945 2670 y Fs(\()p Fr(A)3056 2685 y Fv(0)3096 2670 y Fr(;)17 b(B)3219 2634 y Fl(B)3214 2695 y Fv(0)3270 2670 y Fs(\()p Fq(\000)p Fs(1\))p Fr(;)g(y)t Fs(\))50 b(=)21 2791 y(exp)q(\()p Fq(\000)p Fr(i)348 2751 y Fp(r)p 328 2768 73 4 v 328 2827 a(\021)365 2804 y Fi(0)363 2848 y(1)411 2791 y Fr(y)463 2754 y Fv(2)459 2815 y(1)502 2791 y Fs(\))p Fr(\036)598 2806 y Fp(l)623 2791 y Fs(\()p Fr(A)734 2806 y Fv(0)774 2791 y Fr(;)17 b(B)897 2754 y Fl(B)892 2815 y Fv(0)949 2791 y Fs(\(+1\))p Fr(;)g(y)t Fs(\)\))31 b(b)m(y)i(c)m(ho)s(osing)290 3086 y(\003)358 3045 y Fv(+)358 3113 y Fl(A)419 3086 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))82 b(=)h Fr(e)912 3045 y Fp(i)p Fv([)p Fp(!)1002 3015 y Fh(\000)1000 3068 y(B)1055 3045 y Fv(\()p Fl(\000)p Fp(t)1162 3053 y Fn(m)1221 3045 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;a)1415 3021 y Fh(B)1461 3045 y Fv(\()p Fl(\000)p Fp(t)1568 3053 y Fn(m)1627 3045 y Fv(\()p Fp(\016)o(;")p Fv(\)\))p Fp(;\016)r Fv(\))p Fl(\000)p Fp(!)1972 3015 y Fi(+)1970 3068 y Fh(A)2024 3045 y Fv(\()p Fp(t)2076 3053 y Fn(m)2136 3045 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;a)2330 3021 y Fh(A)2383 3045 y Fv(\()p Fp(t)2435 3053 y Fn(m)2495 3045 y Fv(\()p Fp(\016)o(;")p Fv(\)\))p Fp(;\016)r Fv(\)])2764 2964 y Fo(q)p 2863 2964 599 4 v 2863 3086 a Fs(1)22 b Fq(\000)h(j)p Fs(\003)3130 3045 y Fv(+)3130 3113 y Fl(B)3188 3086 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fq(j)3423 3057 y Fv(2)292 3314 y Fs(\003)360 3273 y Fv(+)360 3341 y Fl(B)419 3314 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))82 b(=)h Fq(\000)954 3246 y Fr(\016)p 954 3291 47 4 v 955 3382 a(")1012 3180 y Fo(r)p 1111 3180 126 4 v 1121 3246 a Fr(\031)t(r)p 1121 3291 106 4 v 1128 3382 a(\021)1180 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b Fr(\015)5 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))36 b(tending)21 3729 y(to)c(0)h(with)658 3891 y(max)856 3750 y Fo(\022)930 3891 y Fr(\016)n(;)1030 3823 y(")p 1025 3868 57 4 v 1025 3959 a(\015)1091 3891 y(;)1144 3823 y(")1190 3787 y Fv(4)p 1144 3868 86 4 v 1164 3959 a Fr(\016)1239 3891 y(;)1294 3823 y(")1340 3787 y Fv(3)p Fp(=)p Fv(2)p 1293 3868 157 4 v 1293 3959 a Fr(\016)1340 3930 y Fv(1)p Fp(=)p Fv(2)1460 3891 y Fr(;)1514 3823 y(")1560 3787 y Fv(5)p Fp(=)p Fv(3)p 1514 3868 V 1514 3959 a Fr(\016)1561 3930 y Fv(1)p Fp(=)p Fv(3)1681 3750 y Fo(\023)1782 3891 y Fq(\034)27 b Fr(t)h Fq(\034)f Fs(min)2278 3750 y Fo(\022)2352 3891 y Fr(\016)2399 3849 y Fv(1)p Fp(=)p Fv(3)2509 3891 y Fr(")2555 3849 y Fv(1)p Fp(=)p Fv(3)2664 3891 y Fr(;)2708 3801 y Fo(p)p 2808 3801 103 4 v 90 x Fr(\016)t(\015)5 b(;)2964 3823 y(\016)p 2964 3868 47 4 v 2965 3959 a(")3021 3750 y Fo(\023)21 4117 y Fs(and)1011 4278 y(max)1209 4138 y Fo(\022)1293 4211 y Fr(")1339 4175 y Fv(2)p Fp(=)p Fv(3)p 1293 4255 157 4 v 1293 4347 a Fr(\016)1340 4318 y Fv(1)p Fp(=)p Fv(3)1459 4278 y Fr(;)1513 4211 y(")1559 4175 y Fv(10)p Fp(=)p Fv(3)p 1513 4255 191 4 v 1530 4347 a Fr(\016)1577 4318 y Fv(5)p Fp(=)p Fv(3)1714 4278 y Fr(;)1768 4211 y(")1814 4175 y Fv(3)p 1768 4255 87 4 v 1768 4347 a Fr(\016)1815 4318 y Fv(2)1864 4278 y Fr(;)1918 4211 y(")1964 4175 y Fv(8)p 1918 4255 V 1918 4347 a Fr(\016)1965 4318 y Fv(3)2014 4278 y Fr(;)2068 4211 y(")2114 4175 y Fv(2)p 2068 4255 86 4 v 2087 4347 a Fr(\016)2163 4278 y(;)17 b(\016)2254 4138 y Fo(\023)2354 4278 y Fq(\034)28 b Fr(\015)k Fq(\034)c Fs(1)21 4521 y(whic)m(h)33 b(is)f(a)h(non-empt)m(y)g(zone)g (with)f(the)h(extra)g(condition)e Fr(\016)t(=")2402 4485 y Fv(7)p Fp(=)p Fv(5)2540 4521 y Fq(!)c Fs(+)p Fq(1)32 b Fs(\(a)g(natural)g(condition)21 4642 y(w)m(ould)i(b)s(e)h Fr(\016)t(=")584 4606 y Fv(2)653 4642 y Fq(!)30 b Fs(+)p Fq(1)p Fs(:)46 b(the)35 b(predicted)g(\014rst)f(order)h(term)e(is)h(of) g(order)g Fr(O)2888 4561 y Fo(\000)2944 4602 y Fp(\016)p 2944 4619 34 4 v 2945 4676 a(")2987 4561 y Fo(\001)3067 4642 y Fs(and)g(the)h(general)21 4762 y(Born-Opp)s(enheimer)26 b(error)g(is)g(of)g(order)h Fr(O)s Fs(\()p Fr(")p Fs(\))e(;)k(the)e (tec)m(hnical)f(condition)f(follo)m(ws)g(from)h(unkno)m(wn)21 4882 y(second)41 b(order)f(terms)f(of)g(the)h(op)s(erator)f Fr(h)1632 4897 y Fv(1)1672 4882 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))39 b(and)h(from)e(the)i(c)m(hoice)g(of)f(the)h(phase)g(in)f (\(21\):)21 5003 y(with)30 b(more)f(regularit)m(y)g(on)g Fr(h)p Fs(\()p Fr(x;)17 b(\016)t Fs(\),)31 b(one)f(can)g(impro)m(v)m(e) g(this)g(tec)m(hnical)f(condition)g(but)h(the)g(c)m(hoice)21 5123 y(of)i(the)h(phase)h(seems)f(to)f(b)s(e)h(the)g(limiting)28 b(factor)k(of)h(impro)m(v)m(emen)m(t\).)1828 5755 y(25)p eop %%Page: 26 26 26 25 bop 21 219 a Fd(6.2)135 b(Wide)55 b(Av)l(oided)g(Crossing)g(\()p Fc(\016)t(=")47 b Fb(!)g Fz(+)p Fb(1)p Fd(\):)79 b(w)l(e)55 b(use)f(the)h Fc(t=\016)c Fb(!)c Fz(0)327 368 y Fd(regime.)21 553 y Fs(Similar)29 b(calculations)i(lead)h(to)g(estimates)372 685 y Fo(\002)414 765 y Fr(S)480 724 y Fl(C)525 765 y Fs(\()p Fr(t)p Fs(\))22 b(+)g Fr(\021)808 724 y Fl(C)853 765 y Fs(\()p Fr(t)p Fs(\))p Fr(:)p Fs(\()p Fr(x)g Fq(\000)h Fr(a)1257 724 y Fl(C)1302 765 y Fs(\()p Fr(t)p Fs(\)\))1451 685 y Fo(\003)1515 765 y Fq(\000)f Fs([)q Fr(S)6 b Fs(\()p Fr(t)p Fs(\))22 b(+)g Fr(\021)t Fs(\()p Fr(t)p Fs(\))p Fr(:)p Fs(\()p Fr(x)g Fq(\000)h Fr(a)p Fs(\()p Fr(t)p Fs(\)\)])28 b(=)538 1008 y Fq(\000)p Fr(r)s(\027)716 967 y Fl(C)779 837 y Fo(")837 1008 y Fr(\016)t(t)22 b Fs(+)1049 940 y Fr(\021)1101 904 y Fv(0)1136 881 y Fi(2)1097 965 y Fv(1)1175 940 y Fr(t)1210 904 y Fv(3)p 1049 985 201 4 v 1101 1076 a Fs(6)p Fr(\016)1281 1008 y Fs(+)1389 940 y Fr(\021)1441 904 y Fv(0)1437 965 y(1)1481 940 y Fr(t)1516 904 y Fv(2)1555 940 y Fr(")p 1389 985 212 4 v 1447 1076 a Fs(2)p Fr(\016)1611 1008 y(y)1659 1023 y Fv(1)1698 837 y Fo(#)1778 1008 y Fs(+)g Fr(O)1970 867 y Fo(\022)2043 1008 y Fr(\016)2090 967 y Fv(2)2130 1008 y Fq(j)p Fr(t)p Fq(j)f Fs(+)2356 940 y Fr(t)2391 904 y Fv(4)p 2350 985 87 4 v 2350 1076 a Fr(\016)2397 1047 y Fv(2)2469 1008 y Fs(+)2567 867 y Fo(\022)2656 940 y Fr(t)2691 904 y Fv(4)p 2650 985 V 2650 1076 a Fr(\016)2697 1047 y Fv(3)2769 1008 y Fs(+)h Fr(\016)t Fq(j)p Fr(t)p Fq(j)3005 867 y Fo(\023)3094 1008 y Fr(")p Fq(k)p Fr(y)t Fq(k)3292 867 y Fo(\023)560 1280 y Fs(+)658 1169 y Fo(\020)718 1280 y Fr(\021)770 1239 y Fv(0)805 1215 y Fh(C)849 1280 y Fs(\()p Fr(\016)t Fs(\))g Fq(\000)h Fr(\021)1146 1239 y Fv(0)1185 1280 y Fs(\()p Fr(\016)t Fs(\))1308 1169 y Fo(\021)1384 1280 y Fr(:"y)35 b(;)21 1531 y Fs(w)m(e)50 b(remo)m(v)m(e)f(the)g(last)f(term)g(whic)m(h)i(w)m(ould)e(lead)g(to)h (an)f(error)h(of)f(order)h Fr(O)s Fs(\()3066 1492 y Fp(\016)p 3048 1508 73 4 v 3048 1565 a("\015)3130 1531 y Fs(\))f(b)m(y)i(c)m(ho)s (osing)21 1680 y Fr(\021)73 1644 y Fv(0)108 1620 y Fh(C)152 1680 y Fs(\()p Fr(\016)t Fs(\))28 b(=)g Fr(\021)459 1644 y Fv(0)498 1680 y Fs(\()p Fr(\016)t Fs(\))k(\(an)g(extra)h(c)m(hoice)g (compared)g(to)f(the)h(narro)m(w)g(a)m(v)m(oided)g(crossing)f(case\))i (;)e(and)714 1810 y Fo(\015)714 1870 y(\015)714 1929 y(\015)714 1989 y(\015)769 1954 y Fr(\036)827 1969 y Fp(l)869 1814 y Fo(\022)943 1954 y Fr(A)1016 1913 y Fl(C)1061 1954 y Fs(\()p Fr(t)p Fs(\))p Fr(;)17 b(B)1295 1913 y Fl(C)1340 1954 y Fs(\()p Fr(t)p Fs(\))p Fr(;)g(y)25 b Fs(+)1676 1887 y Fr(a)p Fs(\()p Fr(t)p Fs(\))d Fq(\000)h Fr(a)2011 1851 y Fl(C)2056 1887 y Fs(\()p Fr(t)p Fs(\))p 1676 1931 492 4 v 1899 2023 a Fr(")2177 1814 y Fo(\023)2273 1954 y Fq(\000)g Fr(\036)2431 1969 y Fp(l)2456 1954 y Fs(\()p Fr(A)2567 1969 y Fv(0)2607 1954 y Fr(;)17 b(B)2725 1969 y Fv(0)2764 1954 y Fr(;)g(y)t Fs(\))2898 1810 y Fo(\015)2898 1870 y(\015)2898 1929 y(\015)2898 1989 y(\015)2952 2053 y Fp(L)3000 2034 y Fi(2)21 2257 y Fs(is)32 b(b)s(ounded)i(b)m(y)f (a)f(constan)m(t)i(times)d Fq(j)p Fr(t)p Fq(j)22 b Fs(+)g Fr(\016)k Fs(+)1776 2209 y Fl(j)p Fp(t)p Fl(j)p 1776 2234 65 4 v 1792 2291 a Fp(\016)1873 2257 y Fs(+)1981 2209 y Fp(\016)2014 2186 y Fi(2)2049 2209 y Fl(j)p Fp(t)p Fl(j)p 1981 2234 133 4 v 2031 2291 a Fp(")2146 2257 y Fs(+)2254 2209 y Fl(j)p Fp(t)p Fl(j)2319 2186 y Fi(3)p 2254 2234 100 4 v 2271 2291 a Fp(\016)r(")2363 2257 y Fs(.)21 2515 y Fk(Incoming)36 b(Outer)i(Asymptotics.)329 2741 y Fr( )392 2756 y Fp(I)5 b(O)488 2741 y Fs(\()p Fr(t;)17 b(x;)g(\016)n(;)g(")p Fs(\))28 b(=)f Fr(e)1080 2668 y Fn(i)p 1059 2680 64 3 v 1059 2730 a(")1088 2716 y Fi(2)1133 2696 y Fv(\()p Fp(S)t Fv(\()p Fp(t)p Fv(\)+)p Fp(\021)r Fv(\()p Fp(t)p Fv(\))p Fp(:)p Fv(\()p Fp(x)p Fl(\000)p Fp(a)p Fv(\()p Fp(t)p Fv(\)\)\))1779 2741 y Fr(e)1824 2699 y Fp(ir)1892 2672 y Fn(\016)p 1892 2684 30 3 v 1892 2725 a(")1945 2672 y(t)p 1942 2684 V 1942 2725 a(")1985 2741 y Fr(e)2030 2699 y Fp(i!)2100 2670 y Fh(\000)2098 2723 y(B)2153 2699 y Fv(\()p Fp(t;a)2262 2676 y Fh(B)2309 2699 y Fv(\()p Fp(t)p Fv(\))p Fp(;\016)r Fv(\))495 2956 y Fq(\002)p Fr(\036)630 2971 y Fp(l)673 2816 y Fo(\022)746 2956 y Fr(A)819 2971 y Fv(0)859 2956 y Fr(;)17 b(B)977 2971 y Fv(0)1016 2956 y Fr(;)1070 2889 y(x)22 b Fq(\000)h Fr(a)p Fs(\()p Fr(t)p Fs(\))p 1070 2934 340 4 v 1217 3025 a Fr(")1419 2816 y Fo(\023)1519 2807 y Fq(p)p 1602 2807 49 4 v 82 x Fs(2)p 1519 2934 132 4 v 1561 3025 a(2)1661 2956 y([)p Fq(\000)p Fr( )1828 2971 y Fv(1)1868 2956 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))23 b(+)f Fr( )2274 2971 y Fv(2)2313 2956 y Fs(\()p Fr(x;)17 b(\016)t Fs(\)])495 3159 y Fq(\002)g Fs([1)22 b(+)g Fr(O)d Fs(\()p Fr(e)962 3174 y Fp(')1012 3159 y Fs(\()p Fr(t;)e(\016)n(;)g(";) g(\015)5 b Fs(\))22 b(+)g Fr(e)1563 3174 y Fp(\036)1609 3159 y Fs(\()p Fr(t;)17 b(";)g(\015)5 b Fs(\))22 b(+)g Fr(e)2075 3174 y Fv(\010)2130 3159 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))22 b(+)g Fr(e)2497 3174 y Fp(O)2557 3159 y Fs(\()p Fr(t;)17 b(")p Fs(\))k(+)h Fr(e)2922 3174 y Fp(!)2973 3159 y Fs(\()p Fr(t;)17 b(\016)n(;)g(";)g(\015)5 b Fs(\)\))o(])21 3372 y(where)421 3619 y Fr(e)466 3634 y Fp(\036)513 3619 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))27 b(=)g Fq(j)p Fr(t)p Fq(j)22 b Fs(+)g Fr(\016)k Fs(+)1317 3551 y Fq(j)p Fr(t)p Fq(j)p 1317 3596 91 4 v 1339 3687 a Fr(\016)1440 3619 y Fs(+)1548 3551 y Fr(\016)1595 3515 y Fv(2)1634 3551 y Fq(j)p Fr(t)p Fq(j)p 1548 3596 177 4 v 1614 3687 a Fr(")1757 3619 y Fs(+)1865 3551 y Fq(j)p Fr(t)p Fq(j)1956 3515 y Fv(3)p 1865 3596 131 4 v 1884 3687 a Fr(\016)t(")2037 3619 y(;)50 b(e)2159 3634 y Fp(!)2209 3619 y Fs(\()p Fr(t;)17 b(\016)n(;)g(";)g(\015)5 b Fs(\))27 b(=)2763 3551 y Fr(\016)t(")p 2736 3596 147 4 v 2736 3687 a(\015)5 b Fq(j)p Fr(t)p Fq(j)2915 3619 y Fs(+)3028 3551 y Fr(")p 3023 3596 57 4 v 3023 3687 a(\015)3105 3619 y Fs(ln)3213 3551 y(1)p 3213 3596 49 4 v 3214 3687 a Fr(\016)3304 3619 y(;)613 3959 y(e)658 3974 y Fp(')708 3959 y Fs(\()p Fr(t;)17 b(\016)n(;)g(";)g(\015)5 b Fs(\))27 b(=)1235 3891 y Fr(\016)1282 3855 y Fv(2)1321 3891 y Fq(j)p Fr(t)p Fq(j)p 1235 3936 177 4 v 1281 4027 a Fr(")1327 3998 y Fv(2)1444 3959 y Fs(+)1553 3891 y Fq(j)p Fr(t)p Fq(j)1644 3855 y Fv(3)p 1552 3936 133 4 v 1552 4027 a Fr(\016)t(")1645 3998 y Fv(2)1716 3959 y Fs(+)1861 3891 y Fr(t)1896 3855 y Fv(2)p 1824 3936 149 4 v 1824 4027 a Fr(\016)t("\015)2005 3959 y Fs(+)2113 3891 y Fr(\016)t Fq(j)p Fr(t)p Fq(j)p 2113 3936 138 4 v 2131 4027 a Fr("\015)2293 3959 y(;)49 b(e)2414 3974 y Fv(\010)2469 3959 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))28 b(=)f Fr(\016)f Fs(+)2979 3891 y Fq(j)p Fr(t)p Fq(j)p 2979 3936 91 4 v 3001 4027 a Fr(\016)3112 3959 y(;)1430 4252 y(e)1475 4267 y Fp(O)1535 4252 y Fs(\()p Fr(t;)17 b(")p Fs(\))27 b(=)h Fr(")17 b Fs(ln)2037 4185 y(1)p 2037 4229 49 4 v 2038 4320 a Fr(\016)2118 4252 y Fs(+)2227 4185 y Fr(")2273 4148 y Fv(2)p 2226 4229 87 4 v 2226 4320 a Fr(\016)2273 4291 y Fv(2)21 4449 y Fs(are)33 b(error)f(terms)h (analogous)e(to)h(the)h(narro)m(w)g(a)m(v)m(oided)g(crossing)g(case.)21 4707 y Fk(Inner)38 b(Asymptotics.)304 4906 y Fs(^)284 4932 y Fr( )347 4947 y Fp(I)388 4932 y Fs(\()p Fr(s;)17 b(y)t(;)g(\016)n(;)g(")p Fs(\))26 b(=)h Fr(e)986 4860 y Fn(i)p 965 4872 64 3 v 965 4922 a(")994 4908 y Fi(2)1039 4887 y Fp(S)t Fv(\()p Fp("s)p Fv(\)+)1274 4860 y Fn(i)p 1271 4872 30 3 v 1271 4913 a(")1310 4887 y Fp(\021)r Fv(\()p Fp("s)p Fv(\))p Fp(:y)450 5105 y Fq(\002)544 4994 y Fo(h)592 5105 y Fr(g)639 5120 y Fv(1)678 5105 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))p Fr(e)1026 5064 y Fl(\000)p Fp(ir)1147 5037 y Fn(\016)p 1147 5049 V 1147 5090 a(")1187 5064 y Fp(s)1224 5105 y Fs(\()p Fr( )1325 5120 y Fv(1)1387 5105 y Fs(+)22 b Fr( )1548 5120 y Fv(2)1588 5105 y Fs(\))g(+)g Fr(g)1793 5120 y Fv(2)1832 5105 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))p Fr(e)2180 5064 y Fp(ir)2247 5037 y Fn(\016)p 2247 5049 V 2247 5090 a(")2286 5064 y Fp(s)2323 5105 y Fs(\()p Fq(\000)p Fr( )2501 5120 y Fv(1)2564 5105 y Fs(+)22 b Fr( )2725 5120 y Fv(2)2764 5105 y Fs(\))2802 4994 y Fo(i)2866 5105 y Fs(\()p Fr(a)p Fs(\()p Fr("s)p Fs(\))g(+)g Fr("y)t(;)17 b(\016)t Fs(\))450 5278 y Fq(\002)g Fs([)q(1)22 b(+)g Fr(O)c Fs(\()p Fr(e)917 5293 y Fp(I)958 5278 y Fs(\()p Fr("s;)f(\016)n(;)g(")p Fs(\)\))n(])21 5506 y(where)34 b Fr(e)348 5521 y Fp(I)388 5506 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))27 b(=)815 5459 y Fp(\016)848 5435 y Fi(2)883 5459 y Fl(j)p Fp(t)p Fl(j)p 815 5483 133 4 v 848 5541 a Fp(")881 5522 y Fi(2)980 5506 y Fs(+)1088 5459 y Fl(j)p Fp(t)p Fl(j)p 1088 5483 65 4 v 1104 5541 a Fp(\016)1185 5506 y Fs(+)1296 5467 y Fp(t)1321 5444 y Fi(2)p 1293 5483 67 4 v 1293 5541 a Fp(\016)r(")1391 5506 y Fs(+)1500 5459 y Fl(j)p Fp(t)p Fl(j)1565 5435 y Fi(3)p 1499 5483 101 4 v 1499 5541 a Fp(\016)r(")1565 5522 y Fi(2)1643 5506 y Fs(is)32 b(the)h(error)f(term)g(giv)m(en)h(b)m (y)g(prop)s(osition)e(3.)1828 5755 y(26)p eop %%Page: 27 27 27 26 bop 21 219 a Fk(Matc)m(hing)37 b(for)h Fr(t)28 b(<)f Fs(0)p Fk(.)98 b Fs(W)-8 b(e)33 b(can)f(matc)m(h)h(those)g(t)m(w) m(o)g(Ans\177)-49 b(atze)34 b(with)e(an)h(error)f(of)g(order)333 439 y Fr(O)18 b Fs(\()p Fr(e)509 454 y Fp(')560 439 y Fs(\()p Fr(t;)f(\016)n(;)g(";)g(\015)5 b Fs(\))21 b(+)h Fr(e)1110 454 y Fp(\036)1157 439 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))k(+)h Fr(e)1607 454 y Fv(\010)1663 439 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))22 b(+)g Fr(e)2030 454 y Fp(E)2089 439 y Fs(\()p Fr(t;)17 b(")p Fs(\))22 b(+)g Fr(e)2455 454 y Fp(!)2506 439 y Fs(\()p Fr(t;)17 b(\016)n(;)g(";)g(\015)5 b Fs(\))21 b(+)h Fr(e)3056 454 y Fp(I)3096 439 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\)\))21 659 y(b)m(y)34 b(c)m(ho)s(osing) 699 879 y Fr(g)746 894 y Fv(1)785 879 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))82 b(=)h(0)699 1094 y Fr(g)746 1109 y Fv(2)785 1094 y Fs(\()p Fr(y)t(;)17 b(\016)n(;)g(")p Fs(\))82 b(=)1339 945 y Fq(p)p 1422 945 49 4 v 82 x Fs(2)p 1339 1072 132 4 v 1380 1163 a(2)1480 1094 y Fr(\036)1538 1109 y Fp(l)1564 1094 y Fs(\()p Fr(A)1675 1109 y Fv(0)1715 1094 y Fr(;)17 b(B)1833 1109 y Fv(0)1872 1094 y Fr(;)g(y)t Fs(\))p Fr(e)2051 1053 y Fp(i!)2121 1024 y Fh(\000)2119 1076 y(B)2172 1053 y Fv(\()p Fl(\000)p Fp(t)2279 1061 y Fn(m)2339 1053 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;a)2533 1030 y Fh(B)2578 1053 y Fv(\()p Fl(\000)p 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Fl(\000)p Fv(1\))1521 1872 y Fn(k)1560 1896 y Fp(ir)1628 1869 y Fn(\016)p 1628 1881 30 3 v 1628 1922 a(")1680 1869 y(t)p 1678 1881 V 1678 1922 a(")1721 1937 y Fr(e)1766 1896 y Fp(i!)1836 1866 y Fi(+)1834 1919 y Fh(C)1887 1896 y Fv(\()p Fp(t;a)1996 1872 y Fh(C)2037 1896 y Fv(\()p Fp(t)p Fv(\))p Fp(;\016)r Fv(\))2213 1787 y Fq(p)p 2296 1787 49 4 v 83 x Fs(2)p 2213 1914 132 4 v 2254 2005 a(2)2371 1856 y Fo(\000)2417 1937 y Fs(\()p Fq(\000)p Fs(1\))2619 1896 y Fp(k)r Fl(\000)p Fv(1)2752 1937 y Fr( )2815 1952 y Fv(1)2854 1937 y Fs(\()p Fr(x;)g(\016)t Fs(\))23 b(+)f Fr( )3260 1952 y Fv(2)3299 1937 y Fs(\()p Fr(x;)17 b(\016)t Fs(\))3521 1856 y Fo(\001)335 2193 y Fq(\002)g Fs([1)22 b(+)g Fr(O)d Fs(\()p Fr(e)802 2208 y Fp(')852 2193 y Fs(\()p Fr(t;)e(\016)n(;)g(";)g(\015)5 b Fs(\))22 b(+)g Fr(e)1403 2208 y Fp(\036)1449 2193 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))22 b(+)g Fr(e)1900 2208 y Fv(\010)1955 2193 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))22 b(+)g Fr(e)2322 2208 y Fp(O)2382 2193 y Fs(\()p Fr(t;)17 b(")p Fs(\))22 b(+)g Fr(e)2748 2208 y Fp(!)2798 2193 y Fs(\()p Fr(t;)17 b(\016)n(;)g(";)g(\015)5 b Fs(\)\)])49 b Fr(:)21 2453 y Fk(Matc)m(hing)40 b(for)h Fr(t)32 b(>)h Fs(0)p Fk(.)97 b Fs(W)-8 b(e)36 b(can)f(matc)m(h)g(the)h(t)m(w)m(o)g (preceding)g(Ans\177)-49 b(atze)36 b(with)f(an)g(error)g(of)g(the)21 2573 y(same)e(order)f(as)h(for)f Fr(t)c(<)g Fs(0)k(b)m(y)h(c)m(ho)s (osing)606 2793 y(\003)674 2752 y Fv(+)674 2820 y Fl(A)735 2793 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))82 b(=)h(0)608 2964 y(\003)676 2922 y Fv(+)676 2991 y Fl(B)735 2964 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))82 b(=)h Fr(e)1228 2920 y Fp(i)1252 2928 y Fs([)1280 2920 y Fp(!)1326 2890 y Fh(\000)1324 2943 y(B)1378 2920 y Fv(\()p Fl(\000)p Fp(t)1485 2928 y Fn(m)1544 2920 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;a)1738 2896 y Fh(B)1784 2920 y Fv(\()p Fl(\000)p Fp(t)1891 2928 y Fn(m)1951 2920 y Fv(\()p Fp(\016)o(;")p Fv(\)\))p Fp(;\016)r Fv(\))p Fl(\000)p Fp(!)2296 2890 y Fi(+)2294 2943 y Fh(B)2348 2920 y Fv(\()p Fp(t)2400 2928 y Fn(m)2459 2920 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;a)2653 2896 y Fh(B)2699 2920 y Fv(\()p Fp(t)2751 2928 y Fn(m)2810 2920 y Fv(\()p Fp(\016)o(;")p Fv(\)\))p Fp(;\016)r Fv(\))3054 2928 y Fs(])3119 2964 y Fr(:)167 3184 y Fs(Then,)34 b(the)f(error)f(term)g(is)g(of)g(order)g Fr(o)p Fs(\(1\))g(if)f(w)m(e)j(c)m(ho)s(ose)f Fr(t)28 b Fs(=)f Fr(t)2498 3199 y Fp(m)2565 3184 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))28 b Fq(2)g Fs([)p Fr(t)2956 3199 y Fp(o)2994 3184 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fr(;)g(t)3280 3199 y Fp(i)3308 3184 y Fs(\()p Fr(\016)n(;)g(")p Fs(\)])32 b(and)21 3304 y Fr(\015)h Fs(=)27 b Fr(\015)5 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))33 b(tending)f(to)g(0)g(with)310 3523 y Fr(")p 305 3567 57 4 v 305 3658 a(\015)399 3590 y Fq(\034)27 b Fr(t)h Fq(\034)f Fs(min)895 3450 y Fo(\022)969 3590 y Fr(\016)n(;)1054 3500 y Fo(p)p 1154 3500 149 4 v 90 x Fr(\016)t("\015)t(;)1356 3523 y("\015)p 1356 3567 102 4 v 1383 3658 a(\016)1467 3590 y(;)17 b(\016)1558 3549 y Fv(1)p Fp(=)p 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(satis\014es)g(the)g(hyp)-5 b(othesis)44 b(ab)-5 b(ove,)46 b(and)d(let)21 4888 y Fr( )t Fs(\()p Fr(t;)17 b(x;)g(\016)n(;)g(")p Fs(\))35 b Fj(denote)f(the)h(solution)g(of)f(\(4\))h(with)f(initial)h (c)-5 b(ondition)34 b(at)h Fr(t)28 b Fs(=)f Fq(\000)p Fr(T)315 5075 y Fo(X)277 5287 y Fl(C)t Fv(=)p Fl(A)p Fp(;)p Fl(B)513 5170 y Fs(\003)581 5129 y Fl(\000)581 5197 y(C)640 5170 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fr(F)940 5030 y Fo(\022)1014 5170 y Fr(\015)1070 5129 y Fv(2)1119 5103 y Fq(k)p Fr(x)22 b Fq(\000)h Fr(a)1397 5067 y Fl(C)1442 5103 y Fs(\()p Fq(\000)p Fr(T)14 b Fs(\))p Fq(k)1716 5067 y Fv(2)p 1119 5147 637 4 v 1395 5238 a Fr(")1441 5210 y Fv(2)1766 5030 y Fo(\023)1855 5170 y Fr(')1919 5129 y Fl(C)1919 5195 y Fp(l)1981 5030 y Fo(\022)2054 5170 y Fq(\000)p Fr(T)8 b(;)2251 5103 y(x)22 b Fq(\000)h Fr(a)2479 5067 y Fl(C)2524 5103 y Fs(\()p Fq(\000)p Fr(T)14 b Fs(\))p 2251 5147 498 4 v 2477 5238 a Fr(")2758 5170 y(;)j(")2848 5030 y Fo(\023)2938 5170 y Fs(\010)3008 5129 y Fl(\000)3008 5197 y(C)3067 5170 y Fs(\()p Fq(\000)p Fr(T)8 b(;)17 b(x;)g(\016)t Fs(\))1828 5755 y(27)p eop %%Page: 28 28 28 27 bop 21 219 a Fj(with)35 b Fq(j)p Fs(\003)329 177 y Fl(\000)329 246 y(A)389 219 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fq(j)624 182 y Fv(2)685 219 y Fs(+)22 b Fq(j)p Fs(\003)879 177 y Fl(\000)879 246 y(B)937 219 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fq(j)1172 182 y Fv(2)1239 219 y Fs(=)27 b(1)p Fj(,)35 b(then)g(we)f(have,)g(in)h(the) f(limit)h Fr(\016)k Fj(and)34 b Fr(")h Fj(tending)f(to)h Fs(0)p Fj(,)150 353 y Fo(\015)150 413 y(\015)150 473 y(\015)150 533 y(\015)150 593 y(\015)205 528 y Fr( )t Fs(\()p Fr(T)8 b(;)17 b(x;)g(\016)n(;)g(")p Fs(\))22 b Fq(\000)846 433 y Fo(X)809 645 y Fl(C)t Fv(=)p Fl(A)p Fp(;)p Fl(B)1045 528 y Fs(\003)1113 487 y Fv(+)1113 555 y Fl(C)1172 528 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fr(')1443 487 y Fl(C)1443 552 y Fp(l)1504 387 y Fo(\022)1577 528 y Fr(T)8 b(;)1696 460 y(x)23 b Fq(\000)f Fr(a)1924 424 y Fl(C)1970 460 y Fs(\()p Fr(T)14 b Fs(\))p 1696 505 421 4 v 1883 596 a Fr(")2126 528 y(;)j(")2216 387 y Fo(\023)2306 528 y Fs(\010)2376 487 y Fv(+)2376 555 y Fl(C)2435 528 y Fs(\()p Fr(T)8 b(;)17 b(x;)g(\016)t Fs(\))2766 353 y Fo(\015)2766 413 y(\015)2766 473 y(\015)2766 533 y(\015)2766 593 y(\015)2821 672 y Fp(L)2869 653 y Fi(2)2904 672 y Fv(\()p Fg(R)2979 653 y Fn(d)3016 672 y Fv(;)p Fl(H)p Fv(\))3155 528 y Fs(=)27 b Fr(o)p Fs(\(1\))128 b(\(29\))21 865 y Fj(wher)-5 b(e)1062 878 y Fo(\022)1177 958 y Fs(\003)1245 917 y Fv(+)1245 985 y Fl(A)1305 958 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))1178 1078 y(\003)1246 1037 y Fv(+)1246 1105 y Fl(B)1304 1078 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))1553 878 y Fo(\023)1655 1019 y Fs(=)27 b Fr(S)6 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))2048 878 y Fo(\022)2162 958 y Fs(\003)2230 917 y Fl(\000)2230 985 y(A)2290 958 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))2163 1078 y(\003)2231 1037 y Fl(\000)2231 1105 y(B)2290 1078 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))2539 878 y Fo(\023)2663 1019 y Fr(;)21 1246 y Fj(with,)166 1449 y Fq(\017)49 b Fj(if)35 b Fr(\016)t(=")27 b Fq(!)g Fs(0)p Fj(,)1252 1629 y Fr(S)6 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))27 b(=)1655 1489 y Fo(\022)1920 1568 y Fs(0)233 b Fr(e)2247 1532 y Fp(i!)2315 1543 y Fh(AB)2410 1532 y Fv(\()p Fp(\016)o(;")p Fv(\))1770 1691 y Fr(e)1815 1654 y Fp(i!)1883 1665 y Fh(B)q(A)1978 1654 y Fv(\()p Fp(\016)o(;")p Fv(\))2353 1691 y Fs(0)2593 1489 y Fo(\023)2718 1629 y Fr(;)166 1905 y Fq(\017)49 b Fj(if)35 b Fr(\016)t(=")27 b Fq(!)g Fs(+)p Fq(1)p Fj(,)1336 2085 y Fr(S)6 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))27 b(=)1740 1945 y Fo(\022)1855 2024 y Fr(e)1900 1988 y Fp(i!)1968 1999 y Fh(A)2021 1988 y Fv(\()p Fp(\016)o(;")p Fv(\))2371 2024 y Fs(0)1984 2146 y(0)212 b Fr(e)2290 2110 y Fp(i!)2358 2121 y Fh(B)2404 2110 y Fv(\()p Fp(\016)o(;")p Fv(\))2587 1945 y Fo(\023)21 2355 y Fj(wher)-5 b(e)28 b(e)-5 b(ach)27 b(phase)h(only)g(dep)-5 b(ends)27 b(on)h(the)g(choic)-5 b(e)28 b(of)g(an)g(initial)g(phase)f (for)h(dynamic)g(eigenve)-5 b(ctors)21 2476 y Fs(\010)91 2440 y Fl(\003)91 2501 y(C)137 2476 y Fs(\()p Fr(t;)17 b(x;)g(\016)t Fs(\))34 b Fj(\(the)h(matrix)g Fr(S)6 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))34 b Fj(is)h(unitary\).)167 2596 y(Mor)-5 b(e)g(over,)41 b(in)e(the)g(c)-5 b(ase)39 b Fr(\016)t(=")d Fq(!)f Fs(0)p Fj(,)41 b(with)e(the)g(extr)-5 b(a)40 b(c)-5 b(ondition)38 b Fr(\016)t(=")2771 2560 y Fv(7)p Fp(=)p Fv(5)2917 2596 y Fq(!)e Fs(+)p Fq(1)p Fj(,)j(\(29\))g(holds)21 2717 y(with)c Fr(o)297 2636 y Fo(\000)352 2677 y Fp(\016)p 352 2694 34 4 v 353 2751 a(")396 2636 y Fo(\001)476 2717 y Fj(on)g(the)g(right-hand)f(side)g (and)681 3079 y Fr(S)6 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))27 b(=)1084 2879 y Fo(0)1084 3058 y(@)1336 2935 y Fp(\016)p 1336 2951 V 1337 3008 a(")1380 2879 y Fo(q)p 1480 2879 97 4 v 1490 2935 a Fp(\031)r(r)p 1490 2951 77 4 v 1492 3010 a(\021)1529 2987 y Fi(0)1527 3031 y(1)1576 2974 y Fr(e)1621 2938 y Fp(i!)1689 2949 y Fh(A)1743 2938 y Fv(\()p Fp(\016)o(;")p Fv(\))2080 2867 y Fo(q)p 2180 2867 336 4 v 107 x 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b(ose)34 b Fr(H)8 b Fs(\()p Fr(t;)17 b(\016)t Fs(\))33 b Fj(is)h(a)g(family)g(of)g (self-adjoint)f(op)-5 b(er)g(ators)33 b(with)h(\014xe)-5 b(d)34 b(domain)f Fq(D)21 524 y Fj(\(in)j(any)g(sep)-5 b(ar)g(able)34 b(Hilb)-5 b(ert)37 b(sp)-5 b(ac)g(e)35 b Fq(H)q Fj(\))h(for)g Fs(])p Fr(t)1742 539 y Fv(0)1804 524 y Fq(\000)24 b Fs(2)p Fr(T)8 b(;)17 b(t)2098 539 y Fv(0)2160 524 y Fs(+)23 b(2)p Fr(T)14 b Fs([)p Fq(\002)p Fs(])23 b Fq(\000)h Fs(2)p Fr(\016)2726 539 y Fv(0)2765 524 y Fr(;)17 b Fs(2)p Fr(\016)2901 539 y Fv(0)2940 524 y Fs([)p Fj(.)48 b(Assume)36 b(that)h(for)21 644 y(every)29 b Fr(\016)j(>)c Fs(0)p Fj(,)i Fr(H)8 b Fs(\()p Fr(t;)17 b(\016)t Fs(\))28 b Fj(has)h(two)g(distinct)g(eigenvalues)f Fr(E)2145 659 y Fl(A)2206 644 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))29 b Fj(and)g Fr(E)2693 659 y Fl(B)2745 644 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))29 b Fj(uniformly)g(isolate)-5 b(d)21 764 y(fr)g(om)32 b(the)h(r)-5 b(est)32 b(of)h(the)f(sp)-5 b(e)g(ctrum)32 b(with)h Fr(E)1555 779 y Fl(A)1616 764 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))27 b Fr(<)h(E)2021 779 y Fl(B)2073 764 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))32 b Fj(and)g(that)h Fr(E)2763 779 y Fl(A)2824 764 y Fs(\()p Fr(t)2897 779 y Fv(0)2936 764 y Fr(;)17 b Fs(0\))27 b(=)h Fr(E)3270 779 y Fl(B)3322 764 y Fs(\()p Fr(t)3395 779 y Fv(0)3435 764 y Fr(;)17 b Fs(0\))p Fj(.)43 b(In)21 885 y(such)35 b(a)g(situation,)f(we)h(say)g(that)g Fr(H)8 b Fs(\()p Fr(t;)17 b(\016)t Fs(\))34 b Fj(has)g(an)h(avoide)-5 b(d)34 b(cr)-5 b(ossing)33 b(at)j Fr(t)2794 900 y Fv(0)2833 885 y Fj(.)167 1113 y Fs(F)-8 b(rom)33 b(no)m(w)h(on,)h(w)m(e)g(supp)s (ose)g(b)s(oth)f(eigen)m(v)-5 b(alues)34 b(ha)m(v)m(e)h(m)m(ultiplicit) m(y)30 b(one)k(and)g(w)m(e)h(repro)s(duce)21 1233 y(the)e(reduction)g (pro)s(cess)h(presen)m(ted)g(in)e([8].)167 1354 y(Let)j Fr(P)14 b Fs(\()p Fr(t;)j(\016)t Fs(\))34 b(b)s(e)h(the)h(sp)s(ectral)e (pro)5 b(jector)35 b(asso)s(ciated)g(to)g(the)g(t)m(w)m(o)g(eigen)m(v) -5 b(alues)35 b Fr(E)3277 1369 y Fl(A)3338 1354 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))34 b(and)21 1474 y Fr(E)93 1489 y Fl(B)145 1474 y Fs(\()p Fr(t;)17 b(\016)t Fs(\).)103 b(W)-8 b(e)52 b(set)h(successiv)m(ely)i Fr(H)1467 1490 y Fl(jj)1510 1474 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))61 b(=)g Fr(H)8 b Fs(\()p Fr(t;)17 b(\016)t Fs(\))p Fr(P)d Fs(\()p Fr(t;)j(\016)t Fs(\),)56 b Fr(E)6 b Fs(\()p Fr(t;)17 b(\016)t Fs(\))61 b(=)3051 1435 y Fv(1)p 3051 1451 36 4 v 3051 1509 a(2)3096 1474 y Fs(T)-8 b(r)p Fr(H)3277 1490 y Fl(jj)3320 1474 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))52 b(and)21 1595 y Fr(H)102 1610 y Fv(1)141 1595 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))38 b(=)f Fr(H)8 b Fs(\()p Fr(t;)17 b(\016)t Fs(\))25 b Fq(\000)i Fr(E)6 b Fs(\()p Fr(t;)17 b(\016)t Fs(\))p Fr(I)8 b Fs(.)60 b(Let)39 b Fq(f)p Fr( )1626 1610 y Fv(1)1665 1595 y Fr(;)17 b( )1772 1610 y Fv(2)1812 1595 y Fq(g)38 b Fs(b)s(e)g(an)h(orthonormal)d(basis)i(for)g Fr(P)14 b Fs(\()p Fr(t)3294 1610 y Fv(0)3333 1595 y Fr(;)j Fs(0\))p Fq(H)q Fs(,)39 b(for)21 1715 y(\()p Fr(t;)17 b(\016)t Fs(\))32 b(around)h(\()p Fr(t)659 1730 y Fv(0)699 1715 y Fr(;)17 b Fs(0\).)42 b(W)-8 b(e)33 b(set)1324 1990 y Fr( )1387 2005 y Fv(1)1427 1990 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))27 b(=)1874 1923 y Fr(P)14 b Fs(\()p Fr(t;)j(\016)t Fs(\))p Fr( )2216 1938 y Fv(1)p 1770 1967 590 4 v 1770 2058 a Fq(h)p Fr( )1872 2073 y Fv(1)1911 2058 y Fq(j)p Fr(P)d Fs(\()p Fr(t;)j(\016)t Fs(\))p Fr( )2281 2073 y Fv(1)2320 2058 y Fq(i)2401 1990 y Fr(;)770 2333 y( )833 2348 y Fv(2)873 2333 y Fs(\()p Fr(t;)g(\016)t Fs(\))27 b(=)1265 2266 y Fr(P)14 b Fs(\()p Fr(t;)j(\016)t Fs(\))p Fr( )1607 2281 y Fv(2)1668 2266 y Fq(\000)23 b(h)p Fr( )1870 2281 y Fv(1)1910 2266 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))p Fq(j)p Fr(P)d Fs(\()p Fr(t;)j(\016)t Fs(\))p Fr( )2482 2281 y Fv(2)2520 2266 y Fq(i)p Fr( )2622 2281 y Fv(1)2662 2266 y Fs(\()p Fr(t;)g(\016)t Fs(\))p 1216 2310 1698 4 v 1216 2402 a Fq(k)p Fr(P)d Fs(\()p Fr(t;)j(\016)t Fs(\))p Fr( )1608 2417 y Fv(2)1668 2402 y Fq(\000)23 b(h)p Fr( )1870 2417 y Fv(1)1910 2402 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))p Fq(j)p Fr(P)d Fs(\()p Fr(t;)j(\016)t Fs(\))p Fr( )2482 2417 y Fv(2)2520 2402 y Fq(i)p Fr( )2622 2417 y Fv(1)2662 2402 y Fs(\()p Fr(t;)g(\016)t Fs(\))p Fq(k)2956 2333 y Fr(:)21 2563 y Fs(In)33 b(suc)m(h)h(an)f(orthonormal)d (basis)i(the)h(restriction)f(of)g Fr(H)2123 2578 y Fv(1)2162 2563 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))33 b(to)f Fr(P)14 b Fs(\()p Fr(t;)j(\016)t Fs(\))p Fq(H)33 b Fs(has)g(the)g(form)1350 2783 y Fr(A)p Fs(\()p Fr(t)22 b Fq(\000)h Fr(t)1653 2798 y Fv(0)1692 2783 y Fs(\))f(+)g Fr(B)5 b(\016)27 b Fs(+)22 b Fr(M)10 b Fs(\()p Fr(t;)17 b(\016)t Fs(\))21 3003 y(where)47 b Fr(M)56 b Fs(is)45 b(a)g Fq(C)729 2967 y Fv(3)814 3003 y Fs(matrix-v)-5 b(alued)44 b(function)h(with)g Fr(M)10 b Fs(\()p Fr(t;)17 b(\016)t Fs(\))46 b(self-adjoin)m(t)d(and)j(of)f (trace)h(zero,)21 3124 y Fr(M)10 b Fs(\()p Fr(t)198 3139 y Fv(0)239 3124 y Fr(;)17 b Fs(0\))27 b(=)g(0,)619 3085 y Fp(@)t(M)p 619 3101 116 4 v 643 3158 a(@)t(t)744 3124 y Fs(\()p Fr(t)817 3139 y Fv(0)857 3124 y Fr(;)17 b Fs(0\))27 b(=)h(0)k(and)1400 3085 y Fp(@)t(M)p 1400 3101 V 1420 3158 a(@)t(\016)1525 3124 y Fs(\()p Fr(t)1598 3139 y Fv(0)1638 3124 y Fr(;)17 b Fs(0\))27 b(=)h(0.)21 3352 y Fk(De\014nition)36 b(3)49 b Fj(We)31 b(say)f(that)h Fr(H)8 b Fs(\()p Fr(t;)17 b(\016)t Fs(\))30 b Fj(has)g(a)g(non-de)-5 b(gener)g(ate,)30 b(multiplicity)h(one)f(avoide)-5 b(d)29 b(cr)-5 b(oss-)21 3473 y(ing)35 b(at)g Fr(t)338 3488 y Fv(0)412 3473 y Fj(if)g Fq(f)p Fr(A;)17 b(B)5 b Fq(g)34 b Fj(is)h(a)g(set)f(of)h(indep)-5 b(endent)34 b(self-adjoint)f(matric) -5 b(es)34 b(with)h(tr)-5 b(ac)g(e)35 b(zer)-5 b(o.)167 3701 y Fs(Then,)47 b(b)m(y)d(a)e(\()p Fr(t;)17 b(\016)t Fs(\)-indep)s(enden)m(t)43 b(successiv)m(e)j(rotation)41 b(of)h(the)h(basis)g(and)g(rotation)e(of)h(the)21 3821 y(second)28 b(v)m(ector)g(of)e(the)h(basis,)h(w)m(e)g(can)f(assume)g (that)g Fr(A)g Fs(is)f(diagonal)e(and)j(that)g(b)s(oth)f(non-diagonal) 21 3942 y(co)s(e\016cien)m(ts)34 b(of)e Fr(B)38 b Fs(are)32 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Fs(,)f Fr(c)p Fs(,)g Fr(d)f Fs(and)h Fr(E)38 b Fs(are)33 b Fq(C)1110 182 y Fv(3)1182 219 y Fs(real-v)-5 b(alued)31 b(functions)h(and)h (satisfy)1201 307 y Fo(8)1201 397 y(>)1201 427 y(>)1201 457 y(<)1201 636 y(>)1201 666 y(>)1201 696 y(:)1367 389 y Fr(b)p Fs(\()p Fr(t)p 1446 405 36 4 v 1 w(;)17 b(\016)p 1526 405 47 4 v 4 w Fs(\))83 b(=)f Fr(r)s(t)p 1899 405 36 4 v 22 w Fs(+)22 b Fr(O)s Fs(\()p Fr(t)p 2170 405 V -36 x Fv(2)2266 389 y Fs(+)g Fr(\016)p 2364 405 47 4 v 2411 346 a Fv(2)2451 389 y Fs(\))1367 511 y Fr(c)p Fs(\()p Fr(t)p 1447 527 36 4 v(;)17 b(\016)p 1526 527 47 4 v 4 w Fs(\))83 b(=)f Fr(r)s(\016)p 1899 527 V 26 w Fs(+)22 b Fr(O)s Fs(\()p Fr(t)p 2182 527 36 4 v -36 x Fv(2)2278 511 y Fs(+)g Fr(\016)p 2376 527 47 4 v 2423 468 a Fv(2)2463 511 y Fs(\))1358 633 y Fr(d)p Fs(\()p Fr(t)p 1447 649 36 4 v(;)17 b(\016)p 1526 649 47 4 v 4 w Fs(\))83 b(=)f Fr(O)s Fs(\()p Fr(t)p 1968 649 36 4 v -36 x Fv(2)2064 633 y Fs(+)22 b Fr(\016)p 2162 649 47 4 v 2209 590 a Fv(2)2249 633 y Fs(\))1331 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2416 1825 V 2427 1866 a Fi(2)2524 1584 y Fo(!)2620 1754 y Fs(\()p Fq(\000)p Fs(\(1)g(+)g Fr(i)p Fs(\))3013 1683 y Fq(p)p 3096 1683 47 4 v 71 x Fr(r)s(s)p Fs(\))52 b Fj(if)34 b Fr(\016)e Fs(=)27 b Fr(")979 1885 y Fo(\022)1094 1965 y Fr(C)1164 1980 y Fv(1)1203 1965 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fr(e)1455 1929 y Fl(\000)p Fp(ir)r(\016)r(s=")1728 1965 y Fq(\000)23 b Fr(C)1898 1980 y Fv(2)1937 1965 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fr(e)2189 1929 y Fp(ir)r(\016)r(s=")1095 2087 y Fr(C)1165 2102 y Fv(1)1204 2087 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))p Fr(e)1456 2051 y Fl(\000)p Fp(ir)r(\016)r(s=")1729 2087 y Fs(+)22 b Fr(C)1897 2102 y Fv(2)1936 2087 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fr(e)2188 2051 y Fp(ir)r(\016)r(s=")2427 1885 y Fo(\023)2552 2026 y Fj(if)34 b Fr(\016)t(=")27 b Fq(!)h Fs(+)p Fq(1)21 2302 y Fj(wher)-5 b(e)41 b Fr(C)373 2317 y Fp(k)416 2302 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))41 b Fj(ar)-5 b(e)41 b(chosen)g(with)g(or)-5 b(der)42 b Fr(O)s Fs(\(1\))e Fj(and)h(if)g 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2796 a Fp(")2396 2777 y Fi(2)2463 2762 y Fs(+)2571 2721 y Fp(t)2596 2731 y Fn(i)2622 2721 y Fp(\016)p 2571 2739 86 4 v 2580 2796 a(")2613 2777 y Fi(2)2666 2651 y Fo(\021)2777 2762 y Fj(if)35 b Fr(\016)t(=")27 b Fq(!)g Fs(0)2190 2941 y Fr(C)2284 2830 y Fo(\020)2343 2941 y Fr(t)2378 2956 y Fp(i)2429 2941 y Fs(+)2540 2888 y Fp(t)2565 2865 y Fi(3)2565 2910 y Fn(i)p 2537 2918 68 4 v 2537 2975 a Fp(")2570 2956 y Fi(2)2614 2830 y Fo(\021)2725 2941 y Fj(if)35 b Fr(\016)c Fs(=)d Fr(")1997 3120 y(C)2090 3010 y Fo(\020)2163 3068 y Fp(t)2188 3044 y Fi(2)2188 3089 y Fn(i)p 2160 3097 V 2160 3155 a Fp(")2193 3136 y Fi(2)2259 3120 y Fs(+)2367 3079 y Fp(t)2392 3089 y Fn(i)2419 3079 y Fp(\016)2452 3056 y Fi(2)p 2367 3097 121 4 v 2394 3155 a Fp(")2427 3136 y Fi(2)2497 3010 y Fo(\021)2608 3120 y Fj(if)35 b Fr(\016)t(=")27 b Fq(!)g Fs(+)p Fq(1)35 b Fr(:)167 3372 y Fs(The)f(case)f Fr(\016)f Fs(=)c Fr(")k Fs(for)g Fr(H)1061 3387 y Fl(jj)1104 3372 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))32 b(real)g(symmetric)g(is)g(treated)h(in)f([8)o(].)21 3660 y Fd(8.4)135 b(Matc)l(hing)21 3845 y Fk(8.4.1)113 b(Narro)m(w)37 b(Av)m(oided)f(Crossing)h(\()p Fr(\016)t(=")27 b Fq(!)g Fs(0)p Fk(\):)50 b(w)m(e)37 b(use)h(the)f Fq(j)p Fr(t)p Fq(j)p Fr(=\016)31 b Fq(!)d Fs(+)p Fq(1)36 b Fk(regime.)21 4029 y Fs(W)-8 b(e)33 b(ha)m(v)m(e)703 4052 y Fo(Z)802 4079 y Fp(t)758 4278 y Fv(0)849 4188 y Fr(\032)p Fs(\()p Fr(\034)6 b(;)17 b(\016)t Fs(\))p Fr(d\034)38 b Fs(=)28 b(sgn\()p Fr(t)p Fs(\))p Fr(r)1665 4047 y Fo(\022)1748 4120 y Fr(t)1783 4084 y Fv(2)p 1748 4165 75 4 v 1761 4256 a Fs(2)1855 4188 y(+)1963 4120 y Fr(\016)2010 4084 y Fv(2)p 1963 4165 87 4 v 1982 4256 a Fs(2)2076 4188 y(ln)2174 4043 y Fo(\014)2174 4103 y(\014)2174 4163 y(\014)2174 4223 y(\014)2223 4120 y Fr(t)p 2217 4165 47 4 v 2217 4256 a(\016)2274 4043 y Fo(\014)2274 4103 y(\014)2274 4163 y(\014)2274 4223 y(\014)2307 4047 y(\023)2402 4188 y Fs(+)22 b Fr(O)s Fs(\()p Fq(j)p Fr(t)p Fq(j)2707 4147 y Fv(3)2768 4188 y Fs(+)g Fr(\016)2913 4147 y Fv(2)2952 4188 y Fs(\))33 b Fr(;)69 4589 y( )132 4604 y Fp(I)5 b(O)228 4589 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))27 b(=)h Fq(\000)17 b Fs(exp)905 4449 y Fo(\022)978 4589 y Fq(\000)1091 4522 y Fr(i)p 1065 4566 86 4 v 1065 4658 a(")1111 4629 y Fv(2)1177 4454 y Fo(Z)1277 4480 y Fp(t)1232 4679 y Fv(0)1323 4589 y Fr(E)6 b Fs(\()p Fr(\034)g(;)17 b(\016)t Fs(\))p Fr(d\034)1720 4449 y Fo(\023)235 4863 y Fq(\002)g Fs(exp)495 4723 y Fo(\022)568 4863 y Fq(\000)p Fr(i)694 4796 y(r)s(t)776 4760 y Fv(2)p 688 4841 134 4 v 688 4932 a Fs(2)p Fr(")783 4903 y Fv(2)833 4723 y Fo(\023)923 4863 y Fs(exp)q(\()p Fr(i!)1208 4822 y Fl(\000)1204 4890 y(B)1266 4863 y Fs(\()p Fr(t;)g(\016)t Fs(\)\))p Fr( )1569 4878 y Fv(1)1609 4863 y Fs(\()p Fr(t;)g(\016)t Fs(\))g([)o(1)22 b(+)g Fr(O)s Fs(\()p Fr(e)2184 4878 y Fp(O)2243 4863 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\)\)])49 b Fr(;)69 5141 y( )132 5156 y Fp(I)172 5141 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))28 b(=)f(exp)755 5001 y Fo(\022)828 5141 y Fq(\000)942 5074 y Fr(i)p 915 5119 86 4 v 915 5210 a(")961 5181 y Fv(2)1027 5006 y Fo(Z)1127 5032 y Fp(t)1083 5231 y Fv(0)1173 5141 y Fr(E)6 b Fs(\()p Fr(\034)g(;)17 b(\016)t Fs(\))p Fr(d\034)1570 5001 y Fo(\023)235 5416 y Fq(\002)349 5321 y Fo(X)329 5533 y Fp(k)r Fv(=1)p Fp(;)p Fv(2)529 5416 y Fr(C)599 5431 y Fp(k)642 5416 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))g(exp)1031 5275 y Fo(\022)1104 5416 y Fs(\()p Fq(\000)p Fs(1\))1306 5375 y Fp(k)1349 5416 y Fr(i)1398 5348 y(r)s(t)1480 5312 y Fv(2)p 1392 5393 134 4 v 1392 5484 a Fs(2)p Fr(")1487 5455 y Fv(2)1536 5275 y Fo(\023)1626 5416 y Fr( )1689 5431 y Fp(k)1732 5416 y Fs(\()p Fr(t;)g(\016)t Fs(\))f([)q(1)21 b(+)h Fr(O)s Fs(\()p Fr(e)2307 5431 y Fp(I)2347 5416 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\)\)])49 b Fr(;)1828 5755 y Fs(31)p eop %%Page: 32 32 32 31 bop 69 269 a Fr( )132 284 y Fp(O)r(O)248 269 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))27 b(=)h(exp)830 129 y Fo(\022)904 269 y Fq(\000)1017 202 y Fr(i)p 991 246 86 4 v 991 337 a(")1037 309 y Fv(2)1103 133 y Fo(Z)1202 160 y Fp(t)1158 359 y Fv(0)1248 269 y Fr(E)6 b Fs(\()p Fr(\034)g(;)17 b(\016)t Fs(\))p Fr(d\034)1645 129 y Fo(\023)235 543 y Fq(\002)533 449 y Fo(X)329 664 y Fv(\()p Fl(C)t Fp(;k)r Fv(\)=\()p Fl(A)p Fp(;)p Fv(1\))p Fp(;)p Fv(\()p Fl(B)r Fp(;)p Fv(2\))898 543 y Fs(\003)966 502 y Fv(+)966 570 y Fl(C)1025 543 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))g(exp)1414 403 y Fo(\022)1487 543 y Fs(\()p Fq(\000)p Fs(1\))1689 502 y Fp(k)1732 543 y Fr(i)1781 476 y(r)s(t)1863 440 y Fv(2)p 1775 520 134 4 v 1775 612 a Fs(2)p Fr(")1870 583 y Fv(2)1919 403 y Fo(\023)2009 543 y Fs(exp)q(\()p Fr(i!)2294 502 y Fv(+)2290 570 y Fl(C)2353 543 y Fs(\()p Fr(t;)g(\016)t Fs(\)\))p Fr( )2656 558 y Fp(k)2698 543 y Fs(\()p Fr(t;)g(\016)t Fs(\))g([1)22 b(+)g Fr(O)s Fs(\()p Fr(e)3274 558 y Fp(O)3333 543 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\)\)])21 886 y(with)32 b Fr(e)288 901 y Fp(O)348 886 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))28 b(=)775 847 y Fp(\016)808 823 y Fi(2)p 775 863 69 4 v 776 920 a Fp(")809 901 y Fi(2)870 886 y Fs(ln)968 801 y Fo(\014)968 861 y(\014)1015 847 y Fp(t)p 1011 863 34 4 v 1011 920 a(\016)1055 801 y Fo(\014)1055 861 y(\014)1110 886 y Fs(+)1218 839 y Fl(j)p Fp(t)p Fl(j)1283 815 y Fi(3)p 1218 863 100 4 v 1234 920 a Fp(")1267 901 y Fi(2)1349 886 y Fs(+)1457 847 y Fp(\016)1490 823 y Fi(2)p 1457 863 69 4 v 1458 920 a Fp(")1491 901 y Fi(2)1558 886 y Fs(+)22 b Fq(j)p Fr(t)p Fq(j)g Fs(+)1867 801 y Fo(\014)1867 861 y(\014)1910 847 y Fp(\016)p 1910 863 34 4 v 1914 920 a(t)1953 801 y Fo(\014)1953 861 y(\014)2009 886 y Fs(+)2117 847 y Fp(")2150 823 y Fi(2)p 2117 863 68 4 v 2121 920 a Fp(t)2146 901 y Fi(2)2226 886 y Fs(and)33 b Fr(e)2461 901 y Fp(I)2501 886 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))28 b(=)f Fq(j)p Fr(t)p Fq(j)22 b Fs(+)3139 839 y Fl(j)p Fp(t)p Fl(j)3204 815 y Fi(3)p 3139 863 100 4 v 3155 920 a Fp(")3188 901 y Fi(2)3270 886 y Fs(+)3378 839 y Fl(j)p Fp(t)p Fl(j)p Fp(\016)p 3378 863 99 4 v 3394 920 a(")3427 901 y Fi(2)3486 886 y Fs(.)167 1006 y(Matc)m(hings)50 b(with)f(an)g(error)g(of)g(order)g Fr(O)19 b Fs(\()p Fr(e)1878 1021 y Fp(O)1937 1006 y Fs(\()p Fr(t;)e(\016)n(;)g(")p Fs(\))22 b(+)g Fr(e)2388 1021 y Fp(I)2428 1006 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\)\))49 b(can)h(b)s(e)f(p)s(erformed)f(b)m(y)21 1127 y(c)m(ho)s(osing)32 b(successiv)m(ely)1243 1341 y Fr(C)1313 1356 y Fv(1)1352 1341 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))83 b(=)f Fq(\000)p Fr(e)1922 1300 y Fp(i!)1992 1270 y Fh(\000)1990 1323 y(B)2046 1300 y Fv(\()p Fl(\000)p Fp(t)2153 1308 y Fn(m)2212 1300 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;\016)r Fv(\))2466 1341 y Fr(;)1243 1486 y(C)1313 1501 y Fv(2)1352 1486 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))83 b(=)f(0)21 1690 y(and)973 1894 y(\003)1041 1853 y Fv(+)1041 1921 y Fl(A)1101 1894 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))83 b(=)g Fq(\000)p Fr(e)1672 1853 y Fp(i)p Fv([)p Fp(!)1762 1823 y Fh(\000)1760 1876 y(B)1814 1853 y Fv(\()p Fl(\000)p Fp(t)1921 1861 y Fn(m)1981 1853 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;\016)r Fv(\))p Fl(\000)p Fp(!)2299 1823 y Fi(+)2297 1876 y Fh(A)2351 1853 y Fv(\()p Fp(t)2403 1861 y Fn(m)2462 1853 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;\016)r Fv(\)])2736 1894 y Fr(;)974 2039 y Fs(\003)1042 1998 y Fv(+)1042 2066 y Fl(B)1101 2039 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))83 b(=)g(0)32 b Fr(:)21 2253 y Fs(The)26 b(global)c(error)j(is)f(no)m(w)h(of)g(order)f Fr(o)p Fs(\(1\))h(if)e Fr(t)28 b Fs(=)g Fr(t)1839 2268 y Fp(m)1905 2253 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))25 b(tends)h(to)e(0)g(with)h Fr(")i Fq(\034)g Fr(t)h Fq(\034)f Fs(min)o(\()p Fr(")3426 2217 y Fv(2)p Fp(=)p Fv(3)3536 2253 y Fr(;)3589 2214 y Fp(")3622 2190 y Fi(2)p 3589 2230 68 4 v 3606 2287 a Fp(\016)3666 2253 y Fs(\).)167 2374 y(T)-8 b(o)33 b(mak)m(e)g(a)f(\014rst)h(order)g(matc)m(hing,)e(w)m (e)j(c)m(ho)s(ose)323 2629 y Fr(f)371 2644 y Fv(1)411 2629 y Fs(\()p Fr(s;)17 b(\016)n(;)g(")p Fs(\))82 b(=)h Fr(e)994 2588 y Fl(\000)p Fp(i)1083 2561 y Fn(r)r(s)1143 2540 y Fi(2)p 1083 2573 95 3 v 1115 2614 a(2)1208 2489 y Fo(\024)1261 2629 y Fr(C)1331 2644 y Fv(1)1370 2629 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))22 b(+)1707 2562 y Fr(\016)p 1707 2606 47 4 v 1708 2698 a(")1780 2489 y Fo(\022)1853 2629 y Fr(D)1934 2644 y Fv(1)1974 2629 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))22 b Fq(\000)g Fr(ir)s(C)2452 2644 y Fv(2)2492 2629 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))2716 2494 y Fo(Z)2815 2520 y Fp(s)2770 2719 y Fl(\0001)2917 2629 y Fr(e)2962 2588 y Fp(ir)r(\033)3062 2565 y Fi(2)3101 2629 y Fr(d\033)3211 2489 y Fo(\023\025)3386 2629 y Fr(;)323 2904 y(f)371 2919 y Fv(2)411 2904 y Fs(\()p Fr(s;)g(\016)n(;)g(")p Fs(\))82 b(=)h Fr(e)994 2863 y Fp(i)1028 2836 y Fn(r)r(s)1088 2815 y Fi(2)p 1028 2848 95 3 v 1060 2889 a(2)1153 2763 y Fo(\024)1206 2904 y Fr(C)1276 2919 y Fv(2)1315 2904 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))22 b(+)1652 2836 y Fr(\016)p 1652 2881 47 4 v 1653 2972 a(")1725 2763 y Fo(\022)1799 2904 y Fr(D)1880 2919 y Fv(2)1919 2904 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))22 b(+)g Fr(ir)s(C)2396 2919 y Fv(1)2435 2904 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))2659 2768 y Fo(Z)2758 2795 y Fp(s)2714 2994 y Fl(\0001)2860 2904 y Fr(e)2905 2863 y Fl(\000)p Fp(ir)r(\033)3060 2839 y Fi(2)3099 2904 y Fr(d\033)3209 2763 y Fo(\023\025)21 3207 y Fs(with)41 b Fr(C)322 3222 y Fp(k)364 3207 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))41 b(and)g Fr(D)891 3222 y Fp(k)934 3207 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))40 b(of)h(order)g Fr(O)s Fs(\(1\))f(in)g(\(30\).)68 b(The)42 b(error)f(term)f(of)h(order)g Fr(O)3372 3097 y Fo(\020)3441 3160 y Fl(j)p Fp(t)p Fl(j)p Fp(\016)p 3441 3184 99 4 v 3457 3242 a(")3490 3223 y Fi(2)3550 3097 y Fo(\021)3650 3207 y Fs(in)21 3395 y Fr(e)66 3410 y Fp(I)106 3395 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))32 b(no)m(w)i(turns)f(out)f(to) h(b)s(e)f Fr(O)1402 3284 y Fo(\020)1472 3347 y Fl(j)p Fp(t)p Fl(j)p Fp(\016)1570 3324 y Fi(2)p 1472 3372 133 4 v 1505 3429 a Fp(")1538 3410 y Fi(3)1615 3284 y Fo(\021)1674 3395 y Fs(.)44 b(W)-8 b(e)33 b(obtain)624 3682 y(\003)692 3640 y Fv(+)692 3709 y Fl(A)752 3682 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))83 b(=)f Fq(\000)p Fr(e)1322 3641 y Fp(i)p Fv([)p Fp(!)1412 3611 y Fh(\000)1410 3664 y(B)1465 3641 y Fv(\()p Fl(\000)p Fp(t)1572 3649 y Fn(m)1632 3641 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;\016)r Fv(\))p Fl(\000)p Fp(!)1950 3611 y Fi(+)1948 3664 y Fh(A)2001 3641 y Fv(\()p Fp(t)2053 3649 y Fn(m)2113 3641 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;\016)r Fv(\)])2354 3559 y Fo(q)p 2454 3559 599 4 v 123 x Fs(1)22 b Fq(\000)h(j)p Fs(\003)2721 3640 y Fv(+)2721 3709 y Fl(B)2779 3682 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fq(j)3014 3653 y Fv(2)3085 3682 y Fr(;)625 3905 y Fs(\003)693 3863 y Fv(+)693 3932 y Fl(B)752 3905 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))83 b(=)f Fq(\000)1287 3837 y Fr(\016)p 1287 3882 47 4 v 1288 3973 a(")1345 3828 y Fq(p)p 1428 3828 106 4 v 77 x Fr(\031)t(r)r(e)1578 3863 y Fp(i)1612 3836 y Fn(\031)p 1613 3848 38 3 v 1617 3890 a Fi(4)1665 3905 y Fr(e)1710 3863 y Fp(i)p Fv([)p Fp(!)1800 3834 y Fh(\000)1798 3887 y(B)1852 3863 y Fv(\()p Fl(\000)p Fp(t)1959 3871 y Fn(m)2018 3863 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;\016)r Fv(\))p Fl(\000)p Fp(!)2336 3834 y Fi(+)2334 3887 y Fh(B)2388 3863 y Fv(\()p Fp(t)2440 3871 y Fn(m)2499 3863 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;\016)r Fv(\)])21 4150 y Fs(with)32 b(an)h(error)f(of)g(order)h Fr(o)1047 4070 y Fo(\000)1102 4111 y Fp(\016)p 1102 4127 34 4 v 1103 4185 a(")1146 4070 y Fo(\001)1224 4150 y Fs(if)f Fr(t)27 b Fs(=)h Fr(t)1515 4165 y Fp(m)1582 4150 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))32 b(tends)i(to)e(0)g(with)966 4430 y(max)1165 4290 y Fo(\022)1248 4363 y Fr(")1294 4327 y Fv(3)p Fp(=)p Fv(2)p 1248 4407 157 4 v 1248 4499 a Fr(\016)1295 4470 y Fv(1)p Fp(=)p Fv(2)1415 4430 y Fr(;)17 b(")1505 4290 y Fo(\023)1605 4430 y Fq(\034)27 b Fr(t)h Fq(\034)g Fs(min)2102 4290 y Fo(\022)2175 4430 y Fs(\()p Fr(\016)t(")p Fs(\))2344 4389 y Fv(1)p Fp(=)p Fv(3)2454 4430 y Fr(;)2507 4363 y(\016)p 2507 4407 47 4 v 2508 4499 a(")2564 4430 y(;)2618 4363 y(")2664 4327 y Fv(2)p 2618 4407 86 4 v 2637 4499 a Fr(\016)2713 4290 y Fo(\023)21 4703 y Fs(whic)m(h)42 b(implies)e(the)i(extra)g(tec)m (hnical)f(condition)f Fr(\016)t(=")2085 4667 y Fv(7)p Fp(=)p Fv(5)2238 4703 y Fq(!)j Fs(+)p Fq(1)e Fs(\(the)h(exp)s(ected)i (condition)c(is)21 4823 y Fr(\016)t(=")163 4787 y Fv(3)230 4823 y Fq(!)27 b Fs(+)p Fq(1)p Fs(,)j(the)h(tec)m(hnical)g(condition)e (can)i(b)s(e)f(impro)m(v)m(ed)h(b)m(y)g(in)m(tro)s(ducing)f(more)g (terms)g(in)g(the)21 4944 y(asymptotics)i(but)h(the)g(calculations)e (are)i(length)m(y\).)21 5201 y Fk(8.4.2)113 b(Critical)34 b(Av)m(oided)j(Crossing)g(\()p Fr(\016)31 b Fs(=)d Fr(")p Fk(\):)49 b(w)m(e)37 b(use)h(the)f Fq(j)p Fr(t)p Fq(j)p Fr(=")27 b Fq(!)g Fs(+)p Fq(1)37 b Fk(regime.)21 5386 y Fs(The)43 b(computations)e(for)h Fr(H)1088 5401 y Fl(jj)1131 5386 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))42 b(real)f(symmetric)h(are)g (made)g(in)f([8],)k(w)m(e)e(only)f(add)g(that)g(the)21 5506 y(error)33 b(is)f(of)g(order)h Fr(O)s Fs(\()p Fr(")886 5470 y Fv(1)p Fp(=)p Fv(4)994 5506 y Fs(\))g(if)e(w)m(e)j(c)m(ho)s(ose) f Fr(t)1642 5521 y Fp(m)1709 5506 y Fs(\()p Fr(")p Fs(\))27 b(=)h Fr(")2008 5470 y Fv(3)p Fp(=)p Fv(4)2117 5506 y Fs(.)1828 5755 y(32)p eop %%Page: 33 33 33 32 bop 21 219 a Fk(8.4.3)113 b(Wide)36 b(Av)m(oided)h(Crossing)g(\() p Fr(\016)t(=")27 b Fq(!)g Fs(+)p Fq(1)p Fk(\):)49 b(w)m(e)37 b(use)h(the)g Fq(j)p Fr(t)p Fq(j)p Fr(=\016)31 b Fq(!)c Fs(0)37 b Fk(regime.)21 403 y Fs(W)-8 b(e)33 b(ha)m(v)m(e)1217 426 y Fo(Z)1317 452 y Fp(t)1273 652 y Fv(0)1363 562 y Fr(\032)p Fs(\()p Fr(\034)6 b(;)17 b(\016)t Fs(\))p Fr(d\034)39 b Fs(=)27 b Fr(r)s(t\016)f Fs(+)c Fr(O)2206 421 y Fo(\022)2295 494 y Fr(t)2330 458 y Fv(4)p 2290 539 87 4 v 2290 630 a Fr(\016)2337 601 y Fv(2)2386 421 y Fo(\023)2508 562 y Fr(;)107 968 y( )170 983 y Fp(I)5 b(O)266 968 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))27 b(=)h(exp)848 828 y Fo(\022)958 901 y Fr(i)p 932 946 86 4 v 932 1037 a(")978 1008 y Fv(2)1043 833 y Fo(Z)1143 859 y Fp(t)1099 1058 y Fv(0)1189 968 y Fr(E)6 b Fs(\()p Fr(\034)g(;)17 b(\016)t Fs(\))p Fr(d\034)1586 828 y Fo(\023)273 1254 y Fq(\002)p Fr(e)395 1213 y Fp(ir)r(t\016)r(=")579 1189 y Fi(2)636 1254 y Fs(exp)q(\()p Fr(i!)921 1213 y Fl(\000)917 1281 y(B)980 1254 y Fs(\()p Fr(t;)g(\016)t Fs(\)\))1230 1104 y Fq(p)p 1312 1104 49 4 v 1312 1187 a Fs(2)p 1230 1231 132 4 v 1271 1322 a(2)1371 1254 y([)p Fq(\000)p Fr( )1538 1269 y Fv(1)1579 1254 y Fs(\()p Fr(t;)g(\016)t Fs(\))k(+)h Fr( )1963 1269 y Fv(2)2003 1254 y Fs(\()p Fr(t;)17 b(\016)t Fs(\)])g([1)22 b(+)g Fr(O)s Fs(\()p Fr(e)2606 1269 y Fp(O)2665 1254 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\)\)])49 b Fr(;)107 1506 y( )170 1521 y Fp(I)210 1506 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))28 b(=)f(exp)793 1365 y Fo(\022)902 1438 y Fr(i)p 876 1483 86 4 v 876 1574 a(")922 1545 y Fv(2)988 1370 y Fo(Z)1088 1396 y Fp(t)1043 1596 y Fv(0)1134 1506 y Fr(E)6 b Fs(\()p Fr(\034)g(;)17 b(\016)t Fs(\))p Fr(d\034)1531 1365 y Fo(\023)1621 1506 y Fs([1)22 b(+)g Fr(O)s Fs(\()p Fr(e)1978 1521 y Fp(I)2017 1506 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\)\)])273 1748 y Fq(\002)367 1638 y Fo(h)414 1748 y Fr(C)484 1763 y Fv(1)523 1748 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))p Fr(e)775 1707 y Fl(\000)p Fp(ir)r(\016)r(t=")1014 1684 y Fi(2)1071 1748 y Fs(\()p Fr( )1172 1763 y Fv(1)1211 1748 y Fs(\()p Fr(t;)g(\016)t Fs(\))22 b(+)g Fr( )1596 1763 y Fv(2)1636 1748 y Fs(\()p Fr(t;)17 b(\016)t Fs(\)\))22 b(+)g Fr(C)2066 1763 y Fv(2)2105 1748 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fr(e)2357 1707 y Fp(ir)r(\016)r(t=")2541 1684 y Fi(2)2597 1748 y Fs(\()p Fq(\000)p Fr( )2775 1763 y Fv(1)2816 1748 y Fs(\()p Fr(t;)g(\016)t Fs(\))22 b(+)g Fr( )3201 1763 y Fv(2)3240 1748 y Fs(\()p Fr(t;)17 b(\016)t Fs(\)\))3480 1638 y Fo(i)3576 1748 y Fr(;)107 1996 y( )170 2011 y Fp(O)r(O)286 1996 y Fs(\()p Fr(t;)g(\016)n(;)g(")p Fs(\))27 b(=)h(exp)868 1856 y Fo(\022)977 1929 y Fr(i)p 951 1973 V 951 2065 a(")997 2036 y Fv(2)1063 1861 y Fo(Z)1163 1887 y Fp(t)1118 2086 y Fv(0)1209 1996 y Fr(E)6 b Fs(\()p Fr(\034)g(;)17 b(\016)t Fs(\))p Fr(d\034)1606 1856 y Fo(\023)1696 1996 y Fs([1)22 b(+)g Fr(O)s Fs(\()p Fr(e)2053 2011 y Fp(O)2112 1996 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\)\)])273 2282 y Fq(\002)571 2187 y Fo(X)367 2403 y Fv(\()p Fl(C)t Fp(;k)r Fv(\)=\()p Fl(A)p Fp(;)p Fv(1\))p Fp(;)p Fv(\()p Fl(B)r Fp(;)p Fv(2\))936 2282 y Fs(\003)1004 2241 y Fv(+)1004 2309 y Fl(C)1063 2282 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))p Fr(e)1315 2241 y Fv(\()p Fl(\000)p Fv(1\))1459 2217 y Fn(k)1498 2241 y Fp(ir)r(t\016)r(=")1682 2217 y Fi(2)1738 2282 y Fs(exp)q(\()p Fr(i!)2023 2241 y Fv(+)2019 2309 y Fl(C)2082 2282 y Fs(\()p Fr(t;)g(\016)t Fs(\)\))2332 2132 y Fq(p)p 2415 2132 49 4 v 83 x Fs(2)p 2332 2259 132 4 v 2373 2350 a(2)2490 2201 y Fo(\000)2536 2282 y Fs(\()p Fq(\000)p Fs(1\))2738 2241 y Fp(k)r Fl(\000)p Fv(1)2871 2282 y Fr( )2934 2297 y Fv(1)2973 2282 y Fs(\()p Fr(t;)g(\016)t Fs(\))22 b(+)g Fr( )3358 2297 y Fv(2)3398 2282 y Fs(\()p Fr(t;)17 b(\016)t Fs(\))3600 2201 y Fo(\001)21 2641 y Fs(with)32 b Fr(e)288 2656 y Fp(O)348 2641 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))28 b(=)813 2601 y Fp(t)838 2578 y Fi(4)p 775 2618 136 4 v 775 2675 a Fp(\016)808 2656 y Fi(2)843 2675 y Fp(")876 2656 y Fi(2)943 2641 y Fs(+)22 b Fr(\016)k Fs(+)1208 2556 y Fo(\014)1208 2616 y(\014)1255 2601 y Fp(t)p 1251 2618 34 4 v 1251 2675 a(\016)1295 2556 y Fo(\014)1295 2616 y(\014)1350 2641 y Fs(+)1458 2601 y Fp(")1491 2578 y Fi(2)p 1458 2618 69 4 v 1458 2675 a Fp(\016)1491 2656 y Fi(2)1569 2641 y Fs(and)32 b Fr(e)1803 2656 y Fp(I)1844 2641 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))27 b(=)2274 2601 y Fp(t)2299 2578 y Fi(2)p 2270 2618 68 4 v 2270 2675 a Fp(")2303 2656 y Fi(2)2370 2641 y Fs(+)2478 2593 y Fl(j)p Fp(t)p Fl(j)p Fp(\016)2576 2570 y Fi(2)p 2478 2618 133 4 v 2511 2675 a Fp(")2544 2656 y Fi(2)2621 2641 y Fs(.)167 2761 y(Matc)m(hings)50 b(with)f(an)g(error)g(of)g(order)g Fr(O)19 b Fs(\()p Fr(e)1878 2776 y Fp(O)1937 2761 y Fs(\()p Fr(t;)e(\016)n(;)g(")p Fs(\))22 b(+)g Fr(e)2388 2776 y Fp(I)2428 2761 y Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\)\))49 b(can)h(b)s(e)f(p)s(erformed)f(b)m(y)21 2881 y(c)m(ho)s(osing)32 b(successiv)m(ely)1244 3101 y Fr(C)1314 3116 y Fv(1)1353 3101 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))83 b(=)f(0)33 b Fr(;)1244 3317 y(C)1314 3332 y Fv(2)1353 3317 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))83 b(=)1811 3168 y Fq(p)p 1894 3168 49 4 v 82 x Fs(2)p 1811 3294 132 4 v 1853 3386 a(2)1953 3317 y Fr(e)1998 3276 y Fp(i!)2068 3246 y Fh(\000)2066 3299 y(B)2121 3276 y Fv(\()p Fl(\000)p Fp(t)2228 3284 y Fn(m)2287 3276 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;\016)r Fv(\))21 3563 y Fs(and)1020 3783 y(\003)1088 3742 y Fv(+)1088 3810 y Fl(A)1148 3783 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))83 b(=)g(0)32 b Fr(;)1021 3946 y Fs(\003)1089 3905 y Fv(+)1089 3973 y Fl(B)1148 3946 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))83 b(=)g Fr(e)1642 3905 y Fp(i)p Fv([)p Fp(!)1732 3875 y Fh(\000)1730 3928 y(B)1784 3905 y Fv(\()p Fl(\000)p Fp(t)1891 3913 y Fn(m)1951 3905 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;\016)r Fv(\))p Fl(\000)p Fp(!)2269 3875 y Fi(+)2267 3928 y Fh(B)2320 3905 y Fv(\()p Fp(t)2372 3913 y Fn(m)2432 3905 y Fv(\()p Fp(\016)o(;")p Fv(\))p Fp(;\016)r Fv(\)])2706 3946 y Fr(:)21 4195 y Fs(The)34 b(global)c(error)i(is)g(no)m(w)i(of)e (order)g Fr(o)p Fs(\(1\))h(if)e Fr(t)d Fs(=)f Fr(t)1909 4210 y Fp(m)1976 4195 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))32 b(tends)i(to)e(0)h(with)f Fr(t)c Fq(\034)f Fs(min)3268 4085 y Fo(\020)3328 4195 y Fr(";)3428 4156 y Fp(")3461 4133 y Fi(2)p 3427 4172 69 4 v 3427 4230 a Fp(\016)3460 4211 y Fi(2)3505 4085 y Fo(\021)3565 4195 y Fs(.)167 4338 y(First)40 b(order)h(matc)m(hing)e(can)i(b)s(e)g(p)s(erformed)f (without)g(extra)h(calculations)d(if)i(the)h(tec)m(hnical)21 4458 y(condition)31 b Fr(\016)t(=")591 4422 y Fv(1)p Fp(=)p Fv(2)728 4458 y Fq(!)d Fs(0)k(is)g(satis\014ed.)44 b(As)33 b(exp)s(ected,)i(no)d(extra)h(term)f(\(of)g(order)3060 4419 y Fp(")p 3060 4435 34 4 v 3060 4492 a(\016)3104 4458 y Fs(\))g(app)s(ears.)21 4747 y Fd(8.5)135 b(Main)45 b(Result)21 4932 y Fk(Theorem)37 b(2)49 b Fj(L)-5 b(et)44 b Fr(H)8 b Fs(\()p Fr(t;)17 b(\016)t Fs(\))42 b Fj(b)-5 b(e)43 b(a)g(Hamiltonian)f(that)i(satis\014es)f(the)g(hyp)-5 b(othesis)42 b(ab)-5 b(ove,)45 b(and)d(let)21 5052 y Fr( )t Fs(\()p Fr(t;)17 b(\016)n(;)g(")p Fs(\))35 b Fj(denote)f(the)h (solution)g(of)f(\(7\))g(with)h(initial)g(c)-5 b(ondition)473 5339 y Fr( )t Fs(\()p Fq(\000)p Fr(T)8 b(;)17 b(\016)n(;)g(")p Fs(\))28 b(=)1102 5245 y Fo(X)1064 5456 y Fl(C)t Fv(=)p Fl(A)p Fp(;)p Fl(B)1301 5339 y Fs(\003)1369 5298 y Fl(\000)1369 5366 y(C)1427 5339 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))g(exp)1816 5199 y Fo(\022)1890 5339 y Fq(\000)2003 5272 y Fr(i)p 1977 5317 86 4 v 1977 5408 a(")2023 5379 y Fv(2)2089 5204 y Fo(Z)2188 5230 y Fl(\000)p Fp(T)2144 5429 y Fv(0)2315 5339 y Fr(E)2387 5354 y Fl(C)2432 5339 y Fs(\()p Fr(\034)6 b(;)17 b(\016)t Fs(\))p Fr(d\034)2751 5199 y Fo(\023)2841 5339 y Fs(\010)p 2841 5355 71 4 v -41 x Fl(\000)2911 5366 y(C)2970 5339 y Fs(\()p Fq(\000)p Fr(T)8 b(;)17 b(\016)t Fs(\))1828 5755 y(33)p eop %%Page: 34 34 34 33 bop 21 219 a Fj(wher)-5 b(e)34 b Fq(j)p Fs(\003)392 177 y Fl(\000)392 246 y(A)452 219 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fq(j)687 182 y Fv(2)748 219 y Fs(+)22 b Fq(j)p Fs(\003)942 177 y Fl(\000)942 246 y(B)1001 219 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fq(j)1236 182 y Fv(2)1302 219 y Fs(=)28 b(1)p Fj(,)34 b(then)h(we)g(have)376 345 y Fo(\015)376 405 y(\015)376 465 y(\015)376 525 y(\015)376 584 y(\015)431 520 y Fr( )t Fs(\()p Fr(T)8 b(;)17 b(\016)n(;)g(")p Fs(\))22 b Fq(\000)974 425 y Fo(X)936 636 y Fl(C)t Fv(=)p Fl(A)p Fp(;)p Fl(B)1172 520 y Fs(\003)1240 478 y Fv(+)1240 547 y Fl(C)1299 520 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))g(exp)1688 379 y Fo(\022)1761 520 y Fq(\000)1874 452 y Fr(i)p 1848 497 86 4 v 1848 588 a(")1894 559 y Fv(2)1960 384 y Fo(Z)2060 410 y Fp(T)2016 610 y Fv(0)2131 520 y Fr(E)2203 535 y Fl(C)2249 520 y Fs(\()p Fr(\034)6 b(;)17 b(\016)t Fs(\))p Fr(d\034)2567 379 y Fo(\023)2657 520 y Fs(\010)p 2657 536 71 4 v 2728 478 a Fv(+)2728 546 y Fl(C)2787 520 y Fs(\()p Fr(T)8 b(;)17 b(\016)t Fs(\))3019 345 y Fo(\015)3019 405 y(\015)3019 465 y(\015)3019 525 y(\015)3019 584 y(\015)3102 520 y Fs(=)27 b Fr(o)p Fs(\(1\))181 b(\(31\))21 828 y Fj(wher)-5 b(e)1080 841 y Fo(\022)1194 921 y Fs(\003)1262 879 y Fv(+)1262 948 y Fl(A)1323 921 y Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))1195 1041 y(\003)1263 1000 y Fv(+)1263 1068 y Fl(B)1322 1041 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))1571 841 y Fo(\023)1672 982 y Fs(=)27 b Fr(S)6 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))2065 841 y Fo(\022)2179 921 y Fs(\003)2247 879 y Fl(\000)2247 948 y(A)2308 921 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))2180 1041 y(\003)2248 1000 y Fl(\000)2248 1068 y(B)2307 1041 y Fs(\()p Fr(\016)n(;)g(")p Fs(\))2556 841 y Fo(\023)2646 982 y Fr(;)21 1205 y Fj(with,)166 1401 y Fq(\017)49 b Fj(if)35 b Fr(\016)t(=")27 b Fq(!)g Fs(0)p Fj(,)1252 1582 y Fr(S)6 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))27 b(=)1655 1441 y Fo(\022)1920 1521 y Fs(0)233 b Fr(e)2247 1484 y Fp(i!)2315 1495 y Fh(AB)2410 1484 y Fv(\()p 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4275 y(the)c(same)f Fr(S)6 b Fs(\()p Fr(\016)n(;)17 b(")p Fs(\))p Fj(.)21 4594 y Ft(References)67 4797 y Fx([1])49 b(J.)39 b(D.)g(Cole,)i(P)m (erturbation)d(Metho)s(ds)h(in)e(Applied)f(Mathematics,)42 b(W)-8 b(altham,)42 b(Mass.,)g(T)-8 b(oron)m(to,)211 4910 y(London)30 b(:)41 b(Blaisdell)27 b(1968.)67 5095 y([2])49 b(Y.)c(Colin)e(de)i(V)-8 b(erdi)m(\022)-43 b(ere,)49 b(The)44 b(Lev)m(el)h(Crossing)e(Problem)h(in)f(Semi-Classical)f (Analysis)h(I.)i(The)211 5208 y(Symmetric)30 b(Case,)h Fa(to)i(app)-5 b(e)g(ar)35 b(in)e(the)g(Pr)-5 b(o)g(c)g(c)g(e)g(dings) 34 b(of)f(F)-7 b(r)n(\023)-44 b(ed)n(\023)g(eric)33 b(Pham's)h(Congr)-5 b(ess)32 b Fx(\(2002\).)67 5393 y([3])49 b(Y.)41 b(Colin)d(de)i(V)-8 b(erdi)m(\022)-43 b(ere,)44 b(J.)c(P)m(ollet,)j(M.)e(Lom)m(bardi,)g (The)f(Microlo)s(cal)g(Landau-Zener)f(F)-8 b(orm)m(ula,)211 5506 y Fa(A)n(nn.)32 b(Inst.)h(H.)f(Poinc)-5 b(ar)n(\023)-44 b(e)31 b Fx(\(1999\),)i Fy(71)p Fx(,)f(p)d(95.)1828 5755 y Fs(34)p eop %%Page: 35 35 35 34 bop 67 219 a Fx([4])49 b(C.)30 b(F)-8 b(ermanian)29 b(Kammerer,)g(Wigner)g(Measures)h(and)e(Molecular)h(Propagation)h (through)f(Generic)211 331 y(Energy)h(Lev)m(el)h(Crossings,)e Fa(Pr)n(\023)-44 b(epublic)-5 b(ation)34 b(de)f(l'Universit)n(\023)-44 b(e)32 b(de)h(Cer)-5 b(gy)33 b(Pontoise)e Fx(\(2002\).)67 519 y([5])49 b(G.)32 b(A.)g(Hagedorn,)h(Semi-Classical)c(Quan)m(tum)h (Mec)m(hanics)i(IV)g(:)f(Large)h(Order)f(Asymptotics)g(and)211 632 y(More)24 b(General)f(States)h(in)e(More)i(than)f(One)g(Dimension,) g Fa(A)n(nn.)i(Inst.)i(H.)e(Poinc)-5 b(ar)n(\023)-44 b(e)27 b(Se)-5 b(ct.)26 b(A)d Fx(\(1985\),)211 745 y Fy(42)p Fx(,)31 b(p)f(363.)67 933 y([6])49 b(G.)27 b(A.)g(Hagedorn,)h (High)e(Order)f(Corrections)h(to)h(the)g(Time-Dep)s(enden)m(t)f (Born-Opp)s(enheimer)d(Ap-)211 1045 y(pro)m(ximation)j(I)h(:)g(Smo)s (oth)g(P)m(oten)m(tials,)h Fa(A)n(nn.)h(Math.)f Fx(\(1986\),)i Fy(124)p Fx(,)f(p)d(571.)j(Erratum)d(\(1987\),)k Fy(126)p Fx(,)211 1158 y(p)g(219.)67 1346 y([7])49 b(G.)27 b(A.)g(Hagedorn,)h (High)e(Order)f(Corrections)h(to)h(the)g(Time-Dep)s(enden)m(t)f (Born-Opp)s(enheimer)d(Ap-)211 1459 y(pro)m(ximation)30 b(I)s(I)f(:)i(Diatomic)f(Coulom)m(b)g(Systems,)g Fa(Commun.)k(Math.)f (Phys.)d Fx(\(1988\),)j Fy(117)p Fx(,)f(p)d(387.)67 1647 y([8])49 b(G.)35 b(A.)g(Hagedorn,)i(Pro)s(of)d(of)h(the)g(Landau-Zener) e(F)-8 b(orm)m(ula)35 b(in)e(an)i(Adiabatic)f(Limit)f(with)g(Small)211 1759 y(Eigen)m(v)-5 b(alue)30 b(Gaps,)h Fa(Commun.)i(Math.)g(Phys.)e Fx(\(1991\),)i Fy(136)p Fx(,)e(p)f(433.)67 1947 y([9])49 b(G.)25 b(A.)g(Hagedorn,)h(Molecular)e(Propagation)h(Through)e (Electron)h(Energy)g(Lev)m(el)h(Crossings,)f Fa(Mem-)211 2060 y(oirs)34 b(A)n(mer.)e(Math.)h(So)-5 b(c.)31 b Fx(\(1994\),)i Fy(111)p Fx(,)e(No.)g(536.)21 2248 y([10])50 b(G.)29 b(A.)g(Hagedorn,)h(Raising)d(and)h(Lo)m(w)m(ering)h(Op)s(erators)f(for) g(Semi-Classical)e(W)-8 b(a)m(v)m(e)31 b(P)m(ac)m(k)m(ets,)h Fa(A)n(nn.)211 2360 y(Phys.)f Fx(\(1998\),)i Fy(269)p Fx(,)e(p)f(77.)21 2548 y([11])50 b(G.)43 b(A.)g(Hagedorn,)j (Classi\014cation)40 b(and)i(Normal)g(F)-8 b(orms)42 b(for)g(Av)m(oided)g(Crossings,)i Fa(J.)f(Phys.)h(A)211 2661 y Fx(\(1998\),)33 b Fy(31)p Fx(,)f(p)d(369.)21 2849 y([12])50 b(G.)25 b(A.)f(Hagedorn,)i(A.)f(Jo)m(y)m(e,)i(Landau-Zener)c (T)-8 b(ransitions)22 b(Through)g(Small)g(Electronic)i(Eigen)m(v)-5 b(alue)211 2962 y(Gaps)34 b(in)d(the)j(Born-Opp)s(enheimer)c(Appro)m (ximation,)j Fa(A)n(nn.)i(Inst.)g(H.)g(Poinc)-5 b(ar)n(\023)-44 b(e)36 b(Se)-5 b(ct.)35 b(A)e Fx(\(1998\),)211 3074 y Fy(68)p Fx(,)e(p)f(85.)21 3262 y([13])50 b(G.)40 b(A.)g(Hagedorn,)j(A.) d(Jo)m(y)m(e,)k(A)c(Time-Dep)s(enden)m(t)e(Born-Opp)s(enheimer)f(Appro) m(ximation)i(with)211 3375 y(Exp)s(onen)m(tially)28 b(Small)h(Error)g (Estimates,)i Fa(Commun.)j(Math.)f(Phys.)d Fx(\(2001\),)j Fy(223)p Fx(,)f(p)e(583.)21 3563 y([14])50 b(G.)d(A.)g(Hagedorn,)j(S.)c (L.)h(Robinson,)h(Bohr-Sommerfeld)d(Quan)m(tization)h(Rules)f(in)g(the) h(Semi-)211 3676 y(Classical)29 b(Limit,)g Fa(J.)j(Phys.)h(A)d 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8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin %%Page: 1 1 10 setmiterlimit 0.06000 0.06000 sc % Polyline 7.500 slw n 1580 3025 m 1580 3125 l gs col0 s gr % Polyline n 1625 3069 m 1534 3069 l gs col0 s gr % Polyline n 2829 4066 m 2829 4166 l gs col0 s gr % Polyline n 2874 4110 m 2783 4110 l gs col0 s gr % Polyline n 9869 5871 m 9869 5971 l gs col0 s gr % Polyline n 9914 5915 m 9823 5915 l gs col0 s gr % Polyline n 9572 3255 m 9572 3355 l gs col0 s gr % Polyline n 9617 3299 m 9526 3299 l gs col0 s gr % Polyline n 2579 4481 m 2579 4581 l gs col0 s gr % Polyline n 2624 4525 m 2533 4525 l gs col0 s gr % Ellipse n 1575 3075 237 237 0 360 DrawEllipse gs col0 s gr % Ellipse n 2833 4107 237 237 0 360 DrawEllipse gs col0 s gr % Polyline gs clippath 3089 4092 m 3082 4048 l 2980 4064 l 3058 4075 l 2987 4108 l cp eoclip n 2832 4110 m 3071 4073 l gs col0 s gr gr % arrowhead n 2987 4108 m 3058 4075 l 2980 4064 l 2987 4108 l cp gs -0.00 setgray ef gr col0 s % Polyline n 6525 675 m 6525 676 l 6524 677 l 6523 679 l 6521 683 l 6518 688 l 6515 695 l 6510 705 l 6504 717 l 6496 731 l 6488 748 l 6478 768 l 6466 791 l 6453 817 l 6439 846 l 6423 879 l 6406 914 l 6387 952 l 6367 994 l 6346 1038 l 6324 1086 l 6300 1136 l 6276 1188 l 6250 1244 l 6224 1301 l 6197 1361 l 6169 1423 l 6141 1487 l 6113 1553 l 6084 1620 l 6055 1689 l 6026 1760 l 5997 1832 l 5967 1906 l 5938 1981 l 5910 2057 l 5881 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l 6596 8693 l 6598 8696 l 6599 8698 l 6600 8700 l gs col0 s gr % Polyline n 825 4125 m 826 4125 l 828 4126 l 831 4127 l 836 4128 l 841 4130 l 849 4132 l 859 4135 l 871 4138 l 885 4142 l 901 4147 l 921 4152 l 942 4158 l 967 4165 l 994 4172 l 1024 4180 l 1056 4189 l 1092 4199 l 1130 4209 l 1170 4220 l 1214 4231 l 1260 4243 l 1308 4255 l 1359 4268 l 1411 4282 l 1467 4296 l 1524 4310 l 1583 4324 l 1644 4339 l 1707 4354 l 1771 4369 l 1838 4384 l 1905 4399 l 1975 4415 l 2046 4430 l 2118 4445 l 2192 4461 l 2267 4476 l 2344 4491 l 2422 4506 l 2502 4521 l 2584 4536 l 2667 4550 l 2752 4565 l 2838 4579 l 2927 4593 l 3018 4607 l 3110 4620 l 3205 4634 l 3303 4647 l 3403 4659 l 3505 4672 l 3611 4684 l 3719 4695 l 3830 4707 l 3944 4717 l 4062 4728 l 4182 4738 l 4306 4747 l 4433 4756 l 4564 4765 l 4697 4773 l 4833 4780 l 4972 4786 l 5113 4792 l 5256 4796 l 5400 4800 l 5537 4803 l 5673 4805 l 5809 4806 l 5944 4806 l 6077 4805 l 6209 4804 l 6339 4802 l 6467 4800 l 6593 4797 l 6716 4793 l 6837 4789 l 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5698 l 9450 5730 l 9522 5762 l 9592 5793 l 9660 5823 l 9726 5853 l 9789 5882 l 9850 5910 l 9909 5938 l 9966 5964 l 10019 5990 l 10070 6014 l 10119 6037 l 10164 6059 l 10206 6079 l 10245 6098 l 10281 6116 l 10314 6132 l 10344 6147 l 10371 6160 l 10395 6172 l 10416 6182 l 10434 6192 l 10450 6199 l 10463 6206 l 10473 6211 l 10482 6216 l 10488 6219 l 10493 6222 l 10496 6223 l 10498 6224 l 10500 6225 l gs col0 s gr % Polyline n 1413 3251 m 1414 3252 l 1415 3254 l 1418 3258 l 1423 3263 l 1429 3271 l 1437 3281 l 1447 3294 l 1459 3308 l 1473 3325 l 1489 3344 l 1506 3364 l 1524 3385 l 1544 3407 l 1565 3431 l 1588 3456 l 1613 3481 l 1639 3509 l 1668 3537 l 1699 3567 l 1732 3599 l 1769 3633 l 1810 3670 l 1854 3708 l 1901 3748 l 1951 3790 l 1996 3826 l 2040 3861 l 2084 3896 l 2127 3928 l 2168 3959 l 2208 3988 l 2246 4016 l 2284 4043 l 2320 4068 l 2355 4092 l 2389 4116 l 2423 4138 l 2455 4159 l 2487 4180 l 2518 4200 l 2548 4219 l 2576 4237 l 2603 4254 l 2628 4270 l 2650 4284 l 2671 4297 l 2688 4308 l 2703 4317 l 2715 4324 l 2724 4330 l 2731 4334 l 2735 4336 l 2737 4337 l 2738 4338 l gs col0 s gr % Polyline n 1705 2862 m 1706 2863 l 1707 2865 l 1710 2869 l 1715 2874 l 1721 2882 l 1729 2892 l 1739 2905 l 1751 2919 l 1765 2936 l 1781 2955 l 1798 2975 l 1816 2996 l 1836 3018 l 1857 3042 l 1880 3067 l 1905 3092 l 1931 3120 l 1960 3148 l 1991 3178 l 2024 3210 l 2061 3244 l 2102 3281 l 2146 3319 l 2193 3359 l 2243 3401 l 2288 3437 l 2332 3472 l 2376 3507 l 2419 3539 l 2460 3570 l 2500 3599 l 2538 3627 l 2576 3654 l 2612 3679 l 2647 3703 l 2681 3727 l 2715 3749 l 2747 3770 l 2779 3791 l 2810 3811 l 2840 3830 l 2868 3848 l 2895 3865 l 2920 3881 l 2942 3895 l 2963 3908 l 2980 3919 l 2995 3928 l 3007 3935 l 3016 3941 l 3023 3945 l 3027 3947 l 3029 3948 l 3030 3949 l gs col0 s gr % Polyline n 1425 2925 m 1426 2926 l 1428 2928 l 1431 2931 l 1434 2935 l 1440 2941 l 1447 2948 l 1455 2958 l 1466 2969 l 1479 2982 l 1494 2997 l 1511 3015 l 1530 3035 l 1552 3057 l 1576 3082 l 1603 3109 l 1632 3138 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9021 3647 l 9097 3601 l 9172 3556 l 9246 3511 l 9318 3467 l 9388 3422 l 9457 3379 l 9523 3336 l 9588 3295 l 9650 3254 l 9711 3214 l 9769 3176 l 9824 3139 l 9877 3104 l 9927 3070 l 9974 3038 l 10019 3007 l 10060 2979 l 10099 2952 l 10134 2928 l 10167 2905 l 10196 2884 l 10223 2866 l 10246 2849 l 10267 2834 l 10285 2822 l 10300 2811 l 10313 2801 l 10324 2794 l 10332 2788 l 10339 2783 l 10343 2780 l 10347 2778 l 10349 2776 l 10350 2775 l gs col0 s gr $F2psEnd rs ---------------0301140855301--