Content-Type: multipart/mixed; boundary="-------------0005030406366" This is a multi-part message in MIME format. ---------------0005030406366 Content-Type: text/plain; name="00-209.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-209.keywords" Born-Oppenheimer approximation, quantum dynamics, semiclassical methods, exponential asymptotics ---------------0005030406366 Content-Type: application/x-tex; name="hagjoy6.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="hagjoy6.tex" \documentclass[12pt,fleqn]{article} \textheight=9in \textwidth=6.5in \topmargin=-.75in \oddsidemargin=0mm \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \renewcommand{\baselinestretch}{1} \newcommand{\ra}{\rightarrow} \newcommand{\bra}{\langle} \newcommand{\ket}{\rangle} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\eps}{\epsilon} \newcommand{\E}{\mbox{e}} \newcommand{\e}{\mbox{\scriptsize e}} \newcommand{\ffi}{\varphi} \newcommand{\sign}{\mbox{sign}} %\newcommand{\ep}{\hfill {$\Box$}} \newcommand{\ep}{\qquad {\vrule height 10pt width 8pt depth 0pt}} \newcommand{\ode}{{\cal O}} \newcommand{\w}{\cal A} \newcommand{\z}{\cal B} \newcommand{\grintl}{[\kern-.18em [} \newcommand{\grintr}{]\kern-.18em ]} \newcommand{\ds}{\displaystyle} \newtheorem{lem}{Lemma}[section] \newtheorem{prop}{Proposition}[section] \newtheorem{thm}{Theorem}[section] \newtheorem{cor}{Corollary}[section] \renewcommand{\thefootnote}{\alph{footnote}} \def\smallR{\hbox{\scriptsize I\kern-.23em{R}}} \def\R{\hbox{$\mit I$\kern-.33em$\mit R$}} \def\C{\hbox{$\mit I$\kern-.6em$\mit C$}} \def\un{\hbox{$\mit I$\kern-.77em$\mit I$}} \def\0{\hbox{$\mit I$\kern-.70em$\mit O$}} %\def\r{\mbox{\bf \scriptsize R}} \def\r{I\kern-.277em R} \def\z{\mbox{\bf \scriptsize Z}} \def\N{\mbox{\bf N}} \def\dist{\mbox{\rm dist}} %\usepackage{showkeys} \begin{document} \title{A Time--Dependent Born--Oppenheimer Approximation with Exponentially Small Error Estimates} \author{George A. Hagedorn\thanks{Partially Supported by National Science Foundation Grant DMS--9703751.}\\ Department of Mathematics and\\ Center for Statistical Mechanics and Mathematical Physics\\ Virginia Polytechnic Institute and State University\\ Blacksburg, Virginia 24061-0123, U.S.A.\\[15pt] \and Alain Joye\\ Institut Fourier\\ Unit\'e Mixte de Recherche CNRS-UJF 5582\\ Universit\'e de Grenoble I\\ BP 74\\ F--38402 Saint Martin d'H\`eres Cedex, France} \date{1 May 2000} \maketitle \begin{abstract} We present the construction of an exponentially accurate time--dependent Born--Oppenheimer approximation for molecular quantum mechanics. We study molecular systems whose electron masses are held fixed and whose nuclear masses are proportional to $\epsilon^{-4}$, where $\epsilon$ is a small expansion parameter. By optimal truncation of an asymptotic expansion, we construct approximate solutions to the time--dependent Schr\"odinger equation that agree with exact normalized solutions up to errors whose norms are bounded by $\ds C\,\exp\left(\,-\gamma/\epsilon^2\,\right)$, for some $C$ and $\gamma>0$. \end{abstract} \newpage \section{Introduction}\label{intro} \setcounter{equation}{0} \vskip .5cm In this paper we construct exponentially accurate approximate solutions to the time--de\-pen\-dent Schr\"odinger equation for a molecular system. The small parameter that governs the approximation is the usual Born--Oppenheimer expansion parameter $\epsilon$, where $\epsilon^4$ is the ratio of the electron mass divided by the mean nuclear mass. The approximate solutions we construct agree with exact solutions up to errors whose norms are bounded by $\ds C\,\exp\left(\,-\gamma/\epsilon^2\,\right)$, for some $C$ and $\gamma>0$, under analyticity assumptions on the electron Hamiltonian. The Hamiltonian for a molecular system with $K$ nuclei and $N-K$ electrons moving in $l$ dimensions has the form $$H(\epsilon)\ =\ \sum_{j=1}^K\,-\,\frac{\epsilon^4}{2M_j}\,\Delta_{X_j}\,-\, \sum_{j=K+1}^N\,\frac{1}{2m_j}\,\Delta_{X_j}\,+\, \sum_{i0$, such that for every $\psi\in{\cal D}$, the vector $h(X)\psi$ is analytic in\\ $S_{\delta}=\{z\in\C^d\; :\; |\mbox{Im}(z_j)|<\delta,\;\; j=1,\dots, d\}$. \end{itemize} \item[{\bf H$_1$}] There exists an open set $\Xi\subset\R^d$, such that for all $X\in\Xi$, there exists an isolated, multiplicity one eigenvalue $E(X)$ of $h(X)$ associated with a normalized eigenvector $\Phi(X)\in {\cal H}_{\mbox{\scriptsize el}}$. We assume without loss that the origin belongs to $\Xi$. \end{itemize} \vspace{.3cm} \noindent {\bf Remarks:}\quad 1.\quad Hypothesis {\bf H$_0$} implies that the family of operators $\{h(X)\}_{X\in S_{\delta}}$ is a holomorphic family of type A. 2.\quad It follows from {\bf H$_0$} and {\bf H$_1$} that there exists $\delta'\in(0,\,\delta)$ and $\Xi'\subset \Xi$ such that the complex and vector valued functions $E(\cdot)$ and $\Phi(\cdot)$ admit analytic continuations on the set\\ $\Sigma_{\delta'}=\{z\in\C^d\,:\,\mbox{Re}(z)\in\Xi'\quad\mbox{and}\quad |\mbox{Im}(z_j)|<\delta',\;\; j=1,\dots,d\}$. \vskip .5cm \subsection{Summary of the Main Results} Our main results are stated precisely as Theorem \ref{main} in Section \ref{mr}. Two generalizations of this result are presented in Section \ref{gene}. \vskip .2cm Roughly speaking, Theorem \ref{main} states the following: \vskip .5cm Under hypotheses {\bf H}$_0$ and {\bf H}$_1$, we construct $\Psi_*(X,t,\eps)$ (that depends on a parameter $g$) for $t\in [0,\,T]$. For small values of $g$, there exist $C(g)$ and $\Gamma(g)>0$, such that in the limit $\eps\ra 0$, $$ \left\|\,\E^{-itH(\eps)/\eps^2}\Psi_*(X,0,\eps)\,-\,\Psi_*(X,t,\eps)\, \right\|_{L^2({\smallR}^d,{\cal H}_{\mbox{\scriptsize el}})}\ \leq\ C(g)\ \E^{-\Gamma(g)/\eps^2} $$ In the state $\Psi_*(X, t, \eps)$, the electrons have a high probability of being in the electron state $\Phi(X)$. For any $b>0$ and sufficiently small values of $g$, the nuclei are localized near a classical path $a(t)$ in the sense that there exist $c(g)$ and $\gamma(g)>0$, such that in the limit $\eps\ra 0$, $$ \left(\,\int_{|X-a(t)|>b}\, \|\Psi_*(X, t, \eps)\|^2_{{\cal H}_{\mbox{\scriptsize el}}}\, dx\,\right)^{1/2}\ \leq\ c(g)\ \E^{-\gamma(g)/\eps^2}. $$ The mechanics of the nuclear configuration $a(t)$ is determined by classical dynamics in the effective potential $E(X)$. \vskip .3cm Two theorems in Section \ref{gene} generalize this result. The first allows the time interval to grow as $\epsilon$ tends to zero. The second allows more general initial conditions. \vskip .5cm \section{Coherent States and Classical Dynamics}\label{wp} \setcounter{equation}{0} In the construction of our approximation to the solution of the molecular Schr\"odinger equation, we need wave packets that describe the semiclassical dynamics of the heavy nuclei. In the present context, the semiclassical parameter is $\hbar=\eps^2$. We make use of a convenient set of coherent states (also called generalized squeezed states), that we express here in terms of the semiclassical parameter $\hbar$. \vskip .25cm We recall the definition of the coherent states $\ffi_j(A,\,B,\,\hbar,\,a,\,\eta,\,X)$ that are described in detail in \cite{raise}. A more explicit, but more complicated definition is given in \cite{semi4}. We adopt the standard multi-index notation. A multi-index $j=(j_1,\,j_2,\,\dots ,\,j_d)$ is a $d$-tuple of non-negative integers. We define % %$|j|=\sum_{k=1}^d\,j_k,\quad %j!=(j_1!)(j_2!)\cdots(j_d!)$,\\ %$x^j=x_1^{j_1}x_2^{j_2}\cdots x_d^{j_d},\quad\mbox{and}\quad %D^j=\frac{\partial^{|j|}}{(\partial x_1)^{j_1}(\partial x_2)^{j_2}\cdots %(\partial x_d)^{j_d}}$. % $|j|=\sum_{k=1}^d\,j_k$,\ \, $X^j=X_1^{j_1}X_2^{j_2}\cdots X_d^{j_d}$,\\ $j!=(j_1!)(j_2!)\cdots(j_d!)$,\ \, and\ \, $D^j=\frac{\partial^{|j|}}{(\partial X_1)^{j_1}(\partial X_2)^{j_2}\cdots (\partial X_d)^{j_d}}$. Throughout the paper we assume $a\in\R^d$, $\eta\in\R^d$ and $\hbar>0$. We also assume that $A$ and $B$ are $d\times d$ complex invertible matrices that satisfy \bea\label{cond1} A^t\,B\,-\,B^t\,A&=&0,\nonumber \\ A^*\,B\,+\,B^*\,A&=&2\,I. \eea These conditions guarantee that both the real and imaginary parts of $BA^{-1}$ are symmetric. Furthermore, $\mbox{Re}\,BA^{-1}$ is strictly positive definite, and $\left(\mbox{Re}\,BA^{-1}\right)^{-1}=\,A\,A^*$. Our definition of $\ffi_j(A,\,B,\,\hbar,\,a,\,\eta,\,X)$ is based on the following raising operators that are defined for $m=1,\,2,\,\dots ,\,d$. $$ \mathcal{A}_m(A,B,\hbar,a,\eta)^* \ =\ \frac{1}{\sqrt{2\hbar}}\,\left[\,\sum_{n=1}^d\,\overline{B}_{n\,m}\,(X_n-a_n) \ -\,i\ \sum_{n=1}^d\,\overline{A}_{n\,m}\, (-i\hbar\frac{\partial\phantom{X^n}}{\partial X_n}-\eta_n)\,\right] . $$ The corresponding lowering operators $\mathcal{A}_m(A,B,\hbar,a,\eta)$ are their formal adjoints. These operators satisfy commutation relations that lead to the properties of the\\ $\ffi_j(A,\,B,\,\hbar,\,a,\,\eta,\,X)$ that we list below. The raising operators $\mathcal{ A}_m(A,B,\hbar,a,\eta)^*$ for\\ $m=1,\,2,\,\dots,\,d$ commute with one another, and the lowering operators $\mathcal{A}_m(A,B,\hbar,a,\eta)$ commute with one another. However, $$\mathcal{A}_m(A,B,\hbar,a,\eta)\,\mathcal{A}_n(A,B,\hbar,a,\eta)^*\,-\, \mathcal{A}_n(A,B,\hbar,a,\eta)^*\,\mathcal{A}_m(A,B,\hbar,a,\eta) \ =\ \delta_{m,\,n}.$$ \vskip .25cm \noindent {\bf Definition}\quad For the multi-index $j=0$, we define the normalized complex Gaussian wave packet (modulo the sign of a square root) by \bea\nonumber &&\ffi_0(A,\,B,\,\hbar,\,a,\,\eta,\,X)\,=\,\pi^{-d/4}\, \hbar^{-d/4}\,(\det(A))^{-1/2}\\[6pt]\nonumber &&\qquad\qquad\quad \times\quad \exp\left\{\,-\,\langle\,(X-a),\,B\,A^{-1}\,(X-a)\,\rangle/(2\hbar)\, +\,i\,\langle\,\eta,\,(X-a)\,\rangle/\hbar\,\right\} . \eea Then, for any non-zero multi-index $j$, we define \bea\nonumber \ffi_j(A,\,B,\,\hbar,\,a,\,\eta,\,\cdot\,)&=& \frac{1}{\sqrt{j!}}\ \left(\,\mathcal{A}_1(A,B,\hbar,a,\eta)^*\right)^{j_1}\ \left(\,\mathcal{A}_2(A,B,\hbar,a,\eta)^*\right)^{j_2}\ \cdots \\[5pt] \nonumber &&\qquad\qquad\qquad\ \times\ \left(\,\mathcal{A}_d(A,B,\hbar,a,\eta)^*\right)^{j_d}\ \ffi_0(A\,,B,\,\hbar,\,a,\,\eta,\,\cdot\,). \eea \vskip .25cm \noindent {\bf Properties}\quad 1.\quad For $A=B=I$, $\hbar=1$, and $a=\eta=0$, the $\ffi_j(A,\,B,\,\hbar,\,a,\,\eta,\,\cdot\,)$ are just the standard Harmonic oscillator eigenstates with energies $|j|+d/2$.\\[7pt] 2.\quad For each admissible $A$, $B$, $\hbar$, $a$, and $\eta$, the set $\{\,\ffi_j(A,\,B,\,\hbar,\,a,\,\eta,\,\cdot\,)\,\}$ is an orthonormal basis for $L^2(\R^d)$.\\[7pt] 3.\quad In \cite{semi4}, the state $\ffi_j(A,\,B,\,\hbar,\,a,\,\eta,\,X)$ is defined as a normalization factor times $$ \mathcal{H}_j(A;\,\hbar^{-1/2}\,|A|^{-1}\,(X-a))\ \ffi_0(A,\,B,\,\hbar,\,a,\,\eta,\,X). $$ Here $\mathcal{H}_j(A;\,y)$ is a recursively defined $|j|^{\mbox{\scriptsize th}}$ order polynomial in $y$ that depends on $A$ only through $U_A$, where $A=|A|\,U_A$ is the polar decomposition of $A$.\\[7pt] 4.\quad By scaling out the $|A|$ and $\hbar$ dependence and using Remark 3 above, one can show that $\mathcal{H}_j(A;\,y)\,\E^{-y^2/2}$ is an (unnormalized) eigenstate of the usual Harmonic oscillator with energy $|j|+d/2$.\\[7pt] 5.\quad When the dimension $d$ is $1$, the position and momentum uncertainties of the\newline $\ffi_j(A,\,B,\,\hbar,\,a,\,\eta,\,\cdot\,)$ are $\sqrt{(j+1/2)\hbar}\ |A|$ and $\sqrt{(j+1/2)\hbar}\ |B|$, respectively. In higher dimensions, they are bounded by $\sqrt{(|j|+d/2)\hbar}\ \|A\|$ and $\sqrt{(|j|+d/2)\hbar}\ \|B\|$, respectively.\\[7pt] 6.\quad When we approximately solve the Schr\"odinger equation, the choice of the sign of the square root in the definition of $\ffi_0(A,\,B,\,\hbar,\,a,\,\eta,\,\cdot\,)$ is determined by continuity in $t$ after an arbitrary initial choice. \vskip .5cm The following simple but very useful lemma is proven in \cite{hagjoy5}. \begin{lem}\label{ylem} Let $P_{|j|\le n}$ denote the projection onto the span of the $\ffi_j(A,\,B,\,\hbar,\,a,\,\eta,\,\cdot\,)$ with $|j|\le n$. \be\label{simple} (X-a)^m\ P_{|j|\le n}\ =\ P_{|j|\le n+|m|}\ (X-a)^m\ P_{|j|\le n}, \ee and \be\label{norm} \left\|\,(X-a)^m\,P_{|j|\le n}\,\right\|\ \le \ \left(\,\sqrt{2\hbar}\ d\,\|\,A\,\|\,\right)^{|m|}\ \left(\,\frac{(n+|m|)!}{n!}\,\right)^{1/2}. \ee \end{lem} In the Born-Oppenheimer approximation, the semiclassical dynamics of the nuclei is generated by an effective potential given by a chosen isolated electronic eigenvalue $E(X)$ of the electronic hamiltonian $h(X)$, $X\in\R^d$. For a given effective potential $E(X)$ we describe the semiclassical dynamics of the nuclei by means of the time dependent basis constructed as follows: By assumption {\bf H$_1$}, the potential $E:\Xi\subset\R^d\ra\R$ is smooth and bounded below. Associated to $E(X)$, we have the following classical equations of motion: \bea \dot{a}(t)&=&\eta(t) ,\nonumber \\ \dot{\eta}(t)&=&-\,\nabla E(a(t)),\nonumber \\ \dot{A}(t)&=&i\,B(t),\label{newton} \\ \dot{B}(t)&=&i\,E^{(2)}(a(t))\,A(t),\nonumber \\ \dot{S}(t)&=&\frac{\eta(t)^2}2\,-\,E(a(t)),\nonumber \eea where $E^{(2)}$ denotes the Hessian matrix for $E$. We always assume the initial conditions $A(0)$, $B(0)$, $a(0)$, $\eta(0)$, and $S(0)=0$ satisfy (\ref{cond1}). The matrices $A(t)$ and $B(t)$ are related to the linearization of the classical flow through the following identities: \bea A(t)&=&\frac{\partial a(t)}{\partial a(0)}\,A(0)\,+\, i\,\frac{\partial a(t)}{\partial \eta(0)}\,B(0),\nonumber\\[4pt]\nonumber B(t)&=&\frac{\partial\eta(t)}{\partial \eta(0)}\,B(0) \,-\,i\frac{\partial\eta(t)}{\partial a(0)}\,A(0). \eea Because $E$ is smooth and bounded below, there exist global solutions to the first two equations of the system (\ref{newton}) for any initial condition if $\Xi=\R^d$. From this, it follows immediately that the remaining three equations of the system (\ref{newton}) have global solutions. If $\Xi\neq\R^d$, for any initial conditions, there exists a $00$, there exist $C(g)>0$ and $\Gamma(g)>0$ such that, for $N(\eps)=\grintl g^2/\eps^2\grintr$, the vector $\Psi_*(X,t,\eps)\,=\,\hat{\Psi}_{N(\eps)}(X-a(t),(X-a(t))/\eps,t)$ satisfies $$ \left\|\,\E^{-itH(\eps)/\eps^2}\,\Psi_*(X,0,\eps)\,-\,\Psi_*(X,t,\eps)\, \right\|_{L^2({\smallR}^d,{\cal H}_{\mbox{\scriptsize el}})} \ \leq\ C(g)\ \E^{-\Gamma(g)/\eps^2}, $$ for all $t\in [0,\,T]$, as $\eps\ra 0$.\\ Moreover, we have the following exponential localization result. For any $b>0$ and a sufficiently small choice of $g>0$ (that depends on $b$), there exist $c(g)$ and\, $\gamma(g)>0$, such that $$ \left(\,\int_{|x-a(t)|>b}\, \|\Psi_*(X,t,\eps)\|^2_{{\cal H}_{\mbox{\scriptsize el}}}\ dx\,\right)^{1/2} \ \leq\ c(g)\ \E^{-\gamma(g)/\eps^2}, $$ for all $t\in [0,\,T]$, as $\eps\ra 0$. \end{thm} \vspace{.2cm} The strategy of the proof is as follows:\ We consider the approximation $\hat{\Psi}_N(X-a(t),(X-a(t))/\eps,t)$ and the exact solution to the Schr\"odinger equation with the same initial conditions. We estimate the norm of the error (that is the difference between these two quantities) as a function of both $N$ and $\eps$. Apart from some subtleties, the norm of the error is bounded by $C\eps^N (\tau N^{1/2})^N$, for some constants $C$ and $\tau>0$. We minimize the error estimate over all choices of $N$. This yields $N\simeq g^2/\eps^2$, for sufficiently small $g>0$, and an estimate of order $\E^{-\Gamma(g)/\eps^2}$ for the norm of the error. \vspace{.4cm} We prove two extensions of this result in Section \ref{gene}. In the first extension, we consider the validity of our approximation on the Ehrenfest time scale, {\it i.e.}, when $T=T(\eps)\simeq \ln(1/\eps)$. In the second extension, we study the dependence of our construction on $J$, in order to extend our main result to a wider class of initial conditions. We refer the reader to Section \ref{gene} for the precise statements. \vskip .5cm \section{Analyticity Properties}\label{AP} \setcounter{equation}{0} Our estimates depend on analyticity in $t\in\Omega$ of the vectors $c_{n}(w,t)\in l^2(\N^d,\C)$ and $d_{n}(w,t)\in l^2(\N^d,{\cal H}_{\mbox{\scriptsize el}})$, where $\Omega$ is the particular simply connected open complex neighborhood of the real interval $[0,\,T]$ mentioned at the beginning of Section \ref{mr}. To construct $\Omega$, we begin with several observations. Our hypotheses imply that the eigenvalue $E(X)$ is analytic in $\Sigma_{\delta'}$, so the solutions $a(t)$, $\eta(t)$, $A(t)$, $B(t)$, and $S(t)$ are well defined for all $t\in [0,\,T]$. Moreover, by standard arguments \cite{dieudonne}, these functions all have analytic continuations from $[0,\,T]$ to a simply connected open set $\Omega_1$ that contains $[0,\,T]$. We assume without loss of generality that $\Omega_1=\overline{\Omega_1}$, where $\overline{\Omega_1}$ denotes the conjugate of $\Omega_1$. We note that $A^*(t)$ and $B^*(t)$ also have analytic continuations from $[0,\,T]$ to $\Omega_1$. To see this for $A^*(t)$, note that for $t\in[0,\,T]$, $A^*(t)\,=\,A^*(\overline{t})$, and $A^*(\overline{t})$ has an analytic continutation to $\overline{\Omega_1}$. The argument for $B^*(t)$ is similar. It now follows easily from the definitions that for each $X$, $\ffi_j(A(t),B(t),\eps^2,a(t),\eta(t),X)$ and $\overline{\ffi_j(A(t),B(t),\eps^2,a(t),\eta(t),X)}$ have analytic continuations from $[0,\,T]$ to some simply connected open set $\Omega_2$. For $t\in[0,\,T]$, the real part of $B(t)A(t)^{-1}$ is strictly positive. This positivity will remain true for the real part of the analytic continuation of $B(t)A(t)^{-1}$ on some simply connected subset $\Omega\subset\Omega_1\cap\Omega_2$ that contains $[0,\,T]$. We assume without loss of generality that $\Omega=\overline{\Omega}$ and we can assume that $\Omega$ has the form $\{\,t\,:\,-a<\mbox{Re}\,tc>0$ and $b>T+c$. It follows that for $t\in\Omega$, both $\ffi_j(A(t),B(t),\eps^2,a(t),\eta(t),x)$ and $\overline{\ffi_j(A(t),B(t),\eps^2,a(t),\eta(t),x)}$ have analytic continuations from $[0,\,T]$ to $\Omega$ as elements of $L^2(\R^d)$. Using these results and carefully examining the constructions of the vectors $c_n(w,t)$ and $d_n(w,t)$, we see that they are analytic in $t$ for $t\in\Omega$, also. \\ Our hypotheses on $h(\cdot)$ and the above results also show that each of the following quantities is analytic in $t$ for $t\in \Omega$ and each fixed $w\in\Sigma_\delta\subset\C^d$, for sufficiently small $\delta$: \bea\nonumber &&r(w,t)\ =\ \left[\,h(a(t)+w)\,-\,E(a(t)+w)\,\right]_r^{-1},\\[4pt] \nonumber &&{\Phi}(w,t),\\[4pt] \nonumber &&(D_w^\alpha{\Phi})(w,t),\quad\mbox{for } |\alpha|\le 2,\\[4pt] \nonumber &&D_w^\alpha E(a(t)), \quad\mbox{for all}\quad\alpha\\[4pt] \nonumber &&P_\perp(w,t). \eea By explicit computation of the phase corresponding to (\ref{phasechoice}) it is easy to check that $\Phi(w,t)$ and its derivatives are also analytic for $t\in\Omega$. Moreover, if $f_i(w,t)$, ($i$ in some finite set) represents any of these quantities, $f_i$ is analytic in $w\in \Sigma_\delta$, for any fixed $t\in \Omega$. Thus, by the Cauchy integral formula, we can assume that the following bounds hold (with the appropriate norm in each case): \be\label{bound} \left\|\,(D^\alpha_wf_i)(w,t)\,\right\|\ \le\ c_i\ G_i^{|\alpha|}\ \frac{\alpha !}{(1+|\alpha|)^{d+1}}, \ee for some $c_i$, $G_i$, $w\in\Sigma_\delta$, and $\alpha$ ranges over all multi-indices. We can assume here that all $G_{i}\leq D_{2}$ for some constant $$ D_{2}\ \geq\ 1, $$ and we associate the prefactors $c_{i}$ in (\ref{bound}) with the different functions according to the rules $$\matrix{ c_{1} & \leftrightarrow & rP_{\perp}&\qquad\qquad&c_{2} & \leftrightarrow & \Phi\cr\vspace{4pt} c_{3} & \leftrightarrow & \dot{\Phi}&&c_{4} & \leftrightarrow & \nabla_{w}\Phi\cr\vspace{4pt} c_{5} & \leftrightarrow & \Delta_{w}\Phi&&c_{6}&\leftrightarrow & E\cr \vspace{4pt} c_{4}' & \leftrightarrow & \bra \Phi,\,\nabla_{w}\Phi\ket&\qquad& c_{5}' & \leftrightarrow & \bra \Phi,\,\Delta_{w}\Phi\ket .} $$ \vskip .5cm \section{Structure and Estimates of the $c_{n}(w,t)$ and $d_n(w,t)$} \setcounter{equation}{0} In this section, we decompose the functions $g_n$ and $\phi_n^{\perp}$ of Section \ref{expand} into pieces, each of which satisfies various estimates. Throughout this section, all $w$--dependent quantities are defined for $w$ in the support of the cut-off function $F$. Furthermore, all the results of this section are claimed to hold only on the support of $F$. Our decompositions of $g_n(w,y,t)$ and $\phi_n^{\perp}(w,y,t)$ have the following forms: \bea\label{messg} \!\!\!\!\!&&g_n(w,y,t)\\[5pt] \nonumber \!\!\!\!\!&=&\eps^{-d/2}\ \sum_{\beta\in {\cal B}_{n,1}}\ \sum_{p\le n}\ \sum_{|l|+k\le p+\frac n2}\ \sum_{|j|\le J+n+2(p-|l|-k)}\ c_{n,p,l,k,\beta,j}(w,t)\ \ffi_j(A(t),B(t),1,0,0,y). \eea and \bea\label{messperp} &&\!\!\!\!\!\!\!\!\!\!\!\!\phi_n^\perp(w,y,t)\ =\\[5pt] \nonumber &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\eps^{-d/2} \sum_{\beta\in {\cal B}_{n,2}}\, \sum_{p\le n-1}\, \sum_{|l|+k\le p+\frac{n-1}2}\ \sum_{|j|\le J+(n-1)+2(p-|l|-k)}\ d_{n,p,l,k,\beta,j}(w,t)\ \ffi_j(A(t),B(t),1,0,0,y). \eea In (\ref{messg}), $n$, $k$ and $p$ are non-negative integers; $j$ and $l$ are multi-indices; and the index $\beta$ runs over a finite set ${\cal B}_{n,1}$. The number $J$ is fixed by the initial conditions. Each $c_{n,p,l,k,\beta,j}$ is a complex valued function.\\ In (\ref{messperp}), $n\ge 2$, $k$ and $p$ are non-negative integers; $j$ and $l$ are multi-indices; and the index $\beta$ runs over a finite set ${\cal B}_{n,2}$. Each $d_{n,p,l,k,\beta,j}(w,t)$ takes values in ${\cal H}_{el}$. We let $c_{n,p,l,k,\beta}(w,t)$ and $d_{n,p,l,k,\beta}(w,t)$ respectively denote vectors in $l^2(\N^d,\C)$ and\\ $l^2(\N^d,{\cal H}_{el})$ whose components are $c_{n,p,l,k,\beta,j}(w,t)$ and $d_{n,p,l,k,\beta,j}(w,t)$. \vskip .3cm The crucial step in the proof of Theorem \ref{main} is the following: \begin{prop}\label{mainprop} There is a recursive construction of the coefficients $c_{n,p,l,k,\beta,j}(w,t)$ and $d_{n,p,l,k,\beta,j}(w,t)$ for $w$ on the support of $F$.\\ The indices for $c_{n,p,l,k,\beta,j}(w,t)$ are non-negative and satisfy \bea\nonumber \beta&\in&{\cal B}_{n,1}\\[3pt] \nonumber p&\le&n,\\[3pt] \nonumber |l|+k&\le&p+\frac{n}2,\\[3pt] \nonumber |j|&\le&J+n+2(p-|l|-k). \eea The indices for $d_{n,p,l,k,\beta,j}(w,t)$ are non-negative and satisfy \bea\nonumber n&\ge&2\\[3pt] \nonumber \beta&\in&{\cal B}_{n,2},\\[3pt] \nonumber p&\le&n-1\\[3pt] \nonumber |l|+k&\le&p+\frac{n-1}2\\[3pt] \nonumber |j|&\le&J+(n-1)+2(p-|l|-k). \eea Moreover, the following conditions are satisfied:\\ i)\quad For any $n>0$,\quad $c_{n,0,l,k,\beta,j}(w,t)\,=\,0$.\\ ii)\quad There exists $K_0>0$, such that the number of terms in both of the sums (\ref{messg}) and (\ref{messperp}) is bounded by $\E^{K_0 n}$.\\ iii)\quad For $t\in\Omega$, let $\dist(t)$ be the distance from $t$ to the complement of $\Omega$. The coefficients $c_{n,p,l,k,\beta}(w,t)$ and $d_{n,p,l,k,\beta}(w,t)$ are analytic for $t\in\Omega$, and there exist constants $D_1$ and $D_2$, such that \bea\label{cestimate} &&\|(D_w^\alpha c_{n,p,l,k,\beta})(w,t)\|\\[7pt] \nonumber &\le&D_1\ D_2^{|\alpha|+|l|+4n}\ \frac{(\alpha+l)!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^p}{p!}\ \frac{k^k}{\dist(t)^k}\ \left[\,\frac{(J+n+2(p-|l|-k))!}{J!}\,\right]^{1/2}, \eea and \bea\label{destimate} &&\!\!\!\!\!\!\!\!\!\!\!\!\|(D_w^\alpha d_{n,p,l,k,\beta})(w,t)\|\\ \nonumber &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\le\ D_1D_2^{|\alpha|+|l|+4(n-1)}\,\frac{(\alpha+l)!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^p}{p!}\ \frac{k^k}{\dist(t)^k}\ \left[\frac{(J+(n-1)+2(p-|l|-k))!}{J!}\right]^{1/2}. \eea \end{prop} \vskip .3cm \noindent {\bf Remark}\quad The complicated estimates (\ref{cestimate}) and (\ref{destimate}) are motivated by estimates used in semiclassical approximations and adiabatic approximations. The factors on the right hand sides that explicitly involve $J$, $n$, and $p$ occur in the semiclassical paper \cite{hagjoy5}. The factors that involve $\alpha$ and $l$ appear in the adiabatic paper \cite{nenciu}. The factors that involve $k$ occur in a proof of the adiabatic results of \cite{nenciu} that are based on Cauchy estimates instead of Nenciu's lemma \cite{nenciu} (that we generalize below as Lemma \ref{nenciuprop2}). We were unable to prove Proposition \ref{mainprop} without using a combination of all of these techniques. We estimate adiabatic error terms by using Nenciu's approach in the $w$ variable and Cauchy estimates in the $t$ variable. \vskip .5cm \subsection{The Toolbox}\label{tool} To prove Proposition \ref{mainprop}, we repeatedly use the following very handy lemmas, whose proofs are given in Section \ref{techie}. The first two lemmas deal with basic properties of analytic functions of one variable and are consequences of the Cauchy integral formula. \vskip .5cm \begin{lem}\label{cauchylem} For $k=0$, define $k^k=1$. Suppose $g$ is an analytic vector--valued function on the strip $S_\delta\ =\ \{\,t\,:\,|\mbox{\rm Im}\,t|<\delta\,\}$. If $g$ satisfies $$\|g(t)\|\ \le\ C\,k^k\,(\delta\,-\,|\mbox{\rm Im}\,t|)^{-k},$$ for some $k\ge 0$, then $g'$ satisfies $$\|g'(t)\|\ \le\ C\,(k+1)^{k+1}\,(\delta\,-\,|\mbox{\rm Im}\,t|)^{-k-1},$$ for all $t\in S_\delta$. \end{lem} \vskip .5cm Lemma \ref{cauchylem} has a generalization to regions other than infinite strips. The generalization is needed if one wishes to study problems where analyticity holds only in a neighborhood of a finite time interval. The proof of the generalized lemma is similar to that of Lemma \ref{cauchylem}, but involves slightly more complicated geometry. The precise statement is the following: \vskip .5cm \begin{lem}\label{upgrade} For $k=0$, define $k^k=1$. Suppose $g$ is an analytic vector--valued function in an open region $\Omega\subset\C$. For $t\in\Omega$, let $\dist(t)$ be the distance from $t$ to $\Omega^C$, the complement of $\Omega$. If $g$ satisfies $$\|g(t)\|\ \le\ C\,k^k\,(\dist(t))^{-k},$$ for all $t\in\Omega$ and some $k\ge 0$, then $g'$ satisfies $$\|g'(t)\|\ \le\ C\,(k+1)^{k+1}\,(\dist(t))^{-k-1},$$ for all $t\in\Omega$. \end{lem} The next lemma gives estimates on indefinite integrals of certain analytic functions under stronger assumptions on the domain $\Omega$. \vskip .5cm \begin{lem}\label{integrate} Suppose $f$ is an analytic vector--valued function in an open region $\Omega\subset\C$. For $t\in\Omega$, let $\dist(t)$ be the distance from $t$ to $\Omega^C$. We assume the domain is star-shaped with respect to the origin and that the origin is the most distant point to $\Omega^C$, {\it i.e.}, $\dist(0)\geq\dist(t)$, for all $t\in \Omega$. Moreover, we assume that $\dist(t)$ is monotone decreasing along any line emanating from the origin. If $f$ satisfies $$\|f(t)\|\ \le\ C\ |t|^p\ (\dist(t))^{-k},$$ for all $t\in\Omega$ and some $k\ge 0$, then $\left\|\,\int_0^t\,f(s)\,ds\,\right\|$ satisfies $$\left\|\,\int_0^t\,f(s)\,ds\,\right\|\ \le\ C\ \frac{|t|^{p+1}}{p+1}\ (\dist(t))^{-k},$$ for all $t\in\Omega$. \end{lem} \vskip .3cm \noindent {\bf Remark:}\quad In our situation, examples of sets $\Omega$ we can use that satisfy the conditions of Lemma \ref{integrate} are infinite symmetrical horizontal strips or the rectangular regions chosen in Section \ref{AP}. \vskip .5cm A fourth tool we repeatedly use below is a multidimensional generalization of a lemma used in \cite{nenciu}. We warn the reader that the symbol for a norm means different things in different contexts, {\it e.g.}, for scalar--valued, operator--valued, and vector--valued functions, it respectively means absolute value, operator norm, and vector space norm. \vskip .5cm \begin{lem}\label{nenciuprop2} The quantity \be\label{nudef} \nu\ =\ \sup_{\alpha}\quad (1+|\alpha|)^{d+1}\ \sum_{\{\,l\,:\,0\,\le\,l_i\,\le\,\alpha_i\,\}}\ \frac{1}{(1+|l|)^{d+1}}\ \frac{1}{(1+|\alpha-l|)^{d+1}}. \ee is finite.\\ Let $\Sigma$ be an open subset of $\C^d$. Suppose $M(\cdot)\in C^\infty(\Sigma)$ is scalar--valued or operator--valued, and $N(\cdot)\in C^\infty(\Sigma)$ is either operator--valued or vector--valued. Assume these functions satisfy \bea \left\|\,\left(D^\alpha\,M\right)(x)\,\right\|&\le&m(x)\ a(x)^{|\alpha+p|}\quad \frac{(\alpha+p)!}{(1+|\alpha|)^{d+1}}\\[5pt] \left\|\,\left(D^\alpha\,N\right)(x)\,\right\|&\le&n(x)\ a(x)^{|\alpha+q|}\quad \frac{(\alpha+q)!}{(1+|\alpha|)^{d+1}} \eea for $x\in\Sigma$, all multi-indices $\alpha$, and some fixed multi-indices $p$ and $q$. Then \be \left\|\,\left(\,D^\alpha\,(M\,N)\,\right)(x)\,\right\|\ \le\ m(x)\ n(x)\ \nu\ a(x)^{|\alpha+p+q|}\quad \frac{(\alpha+p+q)!}{(1+|\alpha|)^{d+1}} \ee for each multi-index $\alpha$, where $\nu$ is defined by (\ref{nudef}). \end{lem} \vskip .5cm \subsection{Proof of Proposition \ref{mainprop}} We prove Proposition \ref{mainprop} by induction and begin with the case $n=0$. We construct $c_{0,0,0,0,\beta,j}\equiv c_{0,j}$ with $\beta=1\in{\cal B}_{0,1}\equiv\{1\}$. We note that there is no $d_{n,p,l,k,\beta}(w,t)$ for $n\le 1$; the inequalities for its indices in the conclusion to the proposition cannot be satisfied by non-negative integers. Whenever $d_{n,p,l,k,\beta}(w,t)$ with $n\le 1$ appears in any of the formal calculations below, it is understood to be zero. We now assume that the estimates (\ref{cestimate}) and (\ref{destimate}) on $c_{m,p,l,k,\beta}(w,t)$ and $d_{m,p,l,k,\beta }(w,t)$ are true for all $m\leq n-1$ and prove they still hold for $m=n$. Our strategy is to show that each contribution $\Delta_{i}$ and $\Gamma_{i}$ consists of a finite sum of terms that satisfy the required estimate. We estimate the number of terms by a separate argument. Our main tools are Lemmas \ref{ylem}, \ref{upgrade}, \ref{integrate}, and \ref{nenciuprop2}. The index $\beta$ must be considered when counting the number of terms, but it plays no role in the estimates of the individual terms. To simplify the notation, we drop it while estimating the terms. % Alain, please note: % I removed the next two paragraphs because of the complications associated with % the term $\Delta_2$. It is more complicated, so I simply decided to drop an % explanation at this point. I simply stuck the explanations with the individual % terms. %To facilitate the counting of %terms later, we note that for $\Delta_2$, $Delta_3$, \dots $\Delta_8$, each %relevant $c_{m,p,l,k,\beta}(w,t)$ or $d_{m,p,l,k,\beta}(w,t)$ with $m|m|$. We adopt the analogous notation for the infinite matrix $\bra\ffi,\,D_y^m\ffi\ket$ that represents the operator $D_{y}^{m}$ in the basis of semiclassical wave packets. We define $d_{0}=\sqrt{2}\,d$. Then, using (\ref{bound}), Lemmas \ref{ylem}, \ref{integrate}, \ref{upgrade}, and \ref{nenciuprop2}, and some algebra, we obtain \bea\nonumber &&\left\|\,D_{w}^{\alpha}\,\sum_{\tilde{m}=3}^{n}\,\sum_{|m|=\tilde{m}} \frac{D^{m}E(a(t))}{m!}\,\bra\ffi,\,y^{m}\ffi\ket\,(rP_{\perp})(w,t)\, d_{n-\tilde{m},p,l,k}(w,t)\,\right\|\\[5pt] \nonumber &\leq& \sum_{\tilde{m}=3}^{n}\ \sum_{|m|=\tilde{m}}(d_{0}\|A\|)^{\tilde{m}}\ \frac{c_{6}\,c_{1}\,\nu\,D_{2}^{\tilde{m}}\,m!}{(1+{\tilde{m}})^{d+1}\,m!}\ \sqrt{\frac{(J+(n-1)+2(p-|l|-k))!}{J!}}\\[5pt] \nonumber &&\qquad\qquad\qquad\qquad\qquad \times\quad D_{1}\,D_{2}^{|\alpha |+|l|+4(n-1-\tilde{m})}\ \frac{(\alpha+l)!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p}}{p!}\ \frac{{k}^{k}}{\dist(t)^{k}}\\[5pt] \nonumber &\leq&\sum_{\tilde{m}=3}^{n}\ \sum_{|m|=\tilde{m}}D_{1}\,c_{6}\,c_{1}\nu\, \frac{(d_{0}\|A\|)^{\tilde{m}}}{D_{2}^{3\tilde{m}}}\, D_{2}^{|\alpha |+|l|+4(n-1)}\, \frac{(\alpha+l)!}{(1+|\alpha|)^{d+1}}\,\frac{|t|^{p}}{p!}\, \frac{{k}^{k}}{\dist(t)^{k}}\\[5pt] \nonumber &&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \times\quad\sqrt{\frac{(J+n-1+2(p-|l|-k))!}{J!}}\,. \eea We also verify the constraints on the parameters and components of the vectors: \bea\nonumber p&\leq&n-1- \tilde{m}\quad\leq\quad n-1,\\[3pt] \nonumber |l|+k&\leq&p+(n-1-\tilde{m})/2\quad\leq\quad p+(n-1)/2,\\[3pt] \nonumber |j|&\leq&J+(n-\tilde{m}-1)+2(p-|l|-k)+\tilde{m}\quad\le\quad J+(n-1)+2(p-|l|-k). \eea Hence, we see that each contribution from $\Delta_{2}(w,t)$ satisfies the required bound, provided the following two conditions are fulfilled \bea\nonumber D_{2}^{3}&\geq&d_{0}\,\|A\|,\\[4pt]\nonumber D_{2}^{9}&\geq&(d_{0}\|A\|)^{3}\,c_{6}\,c_{1}\,\nu . \eea There are $$ \sum_{\tilde{m}=3}^{n}\ \sum_{|m|=\tilde{m}}\ 1\ \leq\ \sum_{|m|\leq n}\ 1 \ =\ \pmatrix{n+d\cr d}\ \leq\ \sigma_{0}\,\E^{\sigma n} $$ such contributions, where $\sigma>0$ can be chosen arbitrarily small (see \cite{hagjoy3}). \vspace{.5cm} \noindent {\bf The Term $\Delta_{3}$}\\ For this term we make the laplacian explicit and write $$ \Delta_{w}d_{n-4,p,l,k}(w,t)\ =\ \sum_{i=1}^{d}\ (D^{2}_{w_{i}}d_{n-4,p,l,k})(w,t). $$ We introduce\quad $\ds l_{i,2}\ =\ l+(0,0,\dots 0,2,0,\dots,0)$, where the $2$ sits in the $i$th column.\\ We then estimate \bea\nonumber &&\left\|\,D_{w}^{\alpha}\ \frac{1}{2}\ \left( rP_{\perp}(w,t) \Delta_{w}{d}_{n-4,p,l,k}(w,t)\, \right)\,\right\|\\[5pt] \nonumber &\leq&\frac{1}{2}\,\sum_{i=1}^{d}\,c_{1}\,\nu\,D_{1}\, D_{2}^{|\alpha|+|l|+2+4(n-5)}\ \frac{(\alpha+l_{i,2})!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p}}{p!}\ \frac{{k}^{k}}{\dist(t)^{k}} \\[5pt] \nonumber &&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\times\quad\sqrt{\frac{(J+(n-5)+2(p-|l|-k))!}{J!}}\\[5pt] \nonumber &=&\sum_{i=1}^{d}\ \frac{c_{1}\,\nu\,D_{1}}{2\,D_{2}^{16}}\ D_{2}^{|\alpha|+|l_{i,2}|+4(n-1)}\ \frac{(\alpha+l_{i,2})!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p}}{p!}\ \frac{{k}^{k}}{\dist(t)^{k}}\\[5pt]\nonumber &&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\times\quad\sqrt{\frac{(J+(n-1)+2(p-|l_{i,2}|-k))!}{J!}}. \eea Again, the constraints are satisfied since \bea\nonumber p&\leq&n-5\quad <\quad n-1\\[3pt] \nonumber |l_{i,2}|+k&\leq&p+(n-5)/2+2\quad=\quad p+(n-1)/2\\[3pt] \nonumber |j|&\leq&J+(n-5)+2(p-|l|-k)\quad=\quad J+(n-1)+2(p-|l_{i,2}|-k) \eea and each of the $d$ contributions stemming from $\Delta_{3}(w,t)$ satisfies the required estimate, provided $$ D_{2}^{16}\ \geq\ c_{1}\nu/2. $$ We estimate each of the remaining terms $\Delta_{i}(w,t)$,\ $i=4,\dots, 8$, in the same fashion, using the same tools. Since this is straightforward, we only outline the arguments. \vspace{.5cm} \noindent {\bf The Term $\Delta_{4}$}\\ We expand the dot product $$ (\nabla_{w}\Phi)\cdot(\nabla_{w}c_{n-4,p,l,k})\ =\ \sum_{i=1}^{d}\ (D_{w_{i}}\Phi)\,(D_{w_{i}}c_{n-4,p,l,k}) $$ and use the definition $$ l_{i,1}=l+(0,0,\dots 0,1,0,\dots,0), $$ where the $1$ sits at the $i$th column. Recall that the estimates on the $c_{m,p,l,k}$'s differ from those on the $d_{m,p,l,k}$'s by a shift of $1$ in the $m$ dependence. We have \bea\nonumber &&\left\|\,D_{w}^{\alpha}\left( rP_{\perp}(w,t)\, \nabla_{w}\Phi(w,t) \cdot \nabla_{w}{c}_{n-4,p,l,k}(w,t) \right)\,\right\|\\[5pt] \nonumber &\leq&\sum_{i=1}^{d}\ \frac{c_{1}\,c_{4}\,\nu^{2}\,D_{1}}{D_{2}^{12}}\ D_{2}^{|\alpha|+|l_{i,1}|+4(n-1)}\ \frac{(\alpha+l_{i,1})!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p}}{p!}\ \frac{{k}^{k}}{\dist(t)^{k}}\\[5pt] \nonumber &&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\quad \sqrt{\frac{(J+(n-1)+2(p-|l_{i, 1}|-k))!}{J!}}, \eea with all constraints on $|j|,p,|l_{i,1}|,k$ satisfied. Thus each of the $d$ contributions stemming from $\Delta_{4}(w,t)$ satisfies the required estimate, provided $$ D_{2}^{12}\ \geq\ c_{1}\,c_{4}\,\nu^{2}. $$ \vspace{.5cm} \noindent {\bf The Term $\Delta_{5}$}\\ This term is similar to the previous one. We obtain \bea\nonumber &&\left\|\,D_{w}^{\alpha}\left(\,\frac 12\, rP_{\perp}(w,t)\, (\Delta_{w}\Phi)(w,t)\,c_{n-4,p,l,k}(w,t) \right)\,\right\|\\[5pt] \nonumber &\leq&\frac{c_{1}\,c_{5}\,\nu^{2}\,D_{1}}{2D_{2}^{12}}\ D_{2}^{|\alpha |+|l|+4(n-1)}\ \frac{(\alpha+l)!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p}}{p!}\ \frac{{k}^{k}}{\dist(t)^{k}}\\[5pt]\nonumber &&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\quad \sqrt{\frac{(J+(n-1)+2(p-|l|-k))!}{J!}}, \eea with all constraints on $|j|,p,|l|,k$ satisfied. Thus the contribution stemming from $\Delta_{5}(w,t)$ satisfies the required estimate, provided $$ D_{2}^{12}\ \geq\ c_{1}\,c_{5}\,\nu^{2}/2. $$ \vspace{.5cm} \noindent {\bf The Term $\Delta_{6}$}\\ At this point the matrices $\bra\ffi,\,D_{y_{i}}\ffi\ket$ play a role that we control by the momentum space analog of Lemma \ref{ylem}. Expanding the dot product and introducing the matrices $\bra\ffi,\,D_{y_{i}}\ffi\ket$,\ $i=1,\dots,d$ we have the following estimate for this term: \bea\nonumber &&\left\|\,D_{w}^{\alpha}\ \sum_{i=1}^{d}\ (rP_{\perp})(w,t)\ \bra\ffi,\,D_{y_{i}}\ffi\ket\ D_{w_{i}}d_{n-3,p,l,k}(w,t)\,\right\| \\[5pt] \nonumber &\leq&\sum_{i=1}^{d}\ \frac{d_{0}\,c_{1}\,\nu\,\|B\|\,D_{1}}{D_{2}^{12}}\ D_{2}^{|\alpha|+|l_{i,1}|+4(n-1)}\ \frac{(\alpha+l_{i,1})!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p}}{p!}\ \frac{{k}^{k}}{\dist(t)^{k}}\\[5pt] \nonumber &&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\quad \sqrt{\frac{(J+(n-1)+2(p-|l_{i,1}|-k))!}{J!}} \eea with all constraints on $|j|,p,|l_{i,1}|,k$ satisfied. Thus, each of the $d$ contributions stemming from $\Delta_{6}(w,t)$ satisfies the required estimate, provided $$ D_{2}^{12}\ \geq\ d_{0}\,c_{1}\,\nu \|B\|. $$ \vspace{.5cm} \noindent {\bf The Term $\Delta_{7}$}\\ Similarly, \bea\nonumber &&\left\|\,D_{w}^{\alpha}\ \sum_{i=1}^{d}\ (rP_{\perp})(w,t)\ \bra\ffi,\,D_{y_{i}}\ffi\ket\ (D_{w_{i}}\Phi)(w,t)\ c_{n-3,p,l,k}(w,t)\,\right\|\\[5pt] \nonumber &\leq&\sum_{i=1}^{d}\ \frac{d_{0}\,c_{1}\,c_{4}\,\nu^{2}\,\|B\|\,D_{1}}{D_{2}^{8}}\ D_{2}^{|\alpha|+|l|+4(n-1)}\ \frac{(\alpha+l)!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p}}{p!}\ \frac{{k}^{k}}{\dist(t)^{k}}\\[5pt] \nonumber &&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\quad \sqrt{\frac{(J+(n-1)+2(p-|l|-k))!}{J!}}, \eea with all constraints on $|j|,p,|l|,k$ satisfied. Thus, each of the $d$ contributions stemming from $\Delta_{7}(w,t)$ satisfies the required estimate, provided $$ D_{2}^{12}\ \geq\ d_{0}\,c_{1}\,c_{4}\,\nu^{2}\,\|B\|. $$ \vspace{.5cm} \noindent {\bf The Term $\Delta_{8}$}\\ Finally, \bea\nonumber &&\left\|\,D_{w}^{\alpha}\,\left( rP_{\perp}(w,t)\, \dot{\Phi}(w,t)\,c_{n-2,p,l,k}(w,t)\right)\,\right\|\\[5pt] \nonumber &&\!\!\!\!\!\!\!\!\!\!\!\!\!\! \leq\ \frac{c_{1}\,c_{3}\,\nu^{2}\,D_{1}}{D_{2}^{4}}\ D_{2}^{|\alpha|+|l|+4(n-1)}\ \frac{(\alpha+l)!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p}}{p!}\ \frac{{k}^{k}}{\dist(t)^{k}}\ \sqrt{\frac{(J+(n-1)+2(p-|l|-k))!}{J!}}, \eea with all constraints on $|j|,p,|l|,k$ satisfied. Thus the contribution stemming from $\Delta_{8}(w,t)$ satisfies the required estimate, provided $$ D_{2}^{4}\ \geq\ c_{1}\,c_{3}\,\nu^{2}. $$ We now perform a similar analysis for the quantities $\Gamma_{i}(w,t)$ that appear in the expression for $\dot{c}_{n,p,l,k}(w,t)$. We integrate these terms with respect to $t$ and apply Lemma \ref{integrate}. According to the lemma, integration of a term with a given value of $p$ gives rise to a term with $p'=p+1$ in the estimates. We also note that the estimates we want to prove for the $c$'s differ from those for the $d$'s by the replacement of $n-1$ by $n$. \vspace{.5cm} \noindent {\bf The Term $\int_{0}^{t}\,\Gamma_{1}$}\\ We use the same techniques above to obtain \bea\nonumber &&\left\|\,D_{w}^{\alpha}\ \int_{0}^{t}\ \frac 12\ (\Delta_{w} {c}_{n-2, p, l, k})(w,s)\, ds\,\right\|\\[5pt] \nonumber &\leq&\sum_{i=1}^{d}\ \frac{D_{1}}{2D_{2}^{8}}\ D_{2}^{|\alpha |+|l_{i, 2}|+4n}\ \frac{(\alpha+l_{i,2})!}{(1+|\alpha|)^{d+1}}\ \frac{t^{p'}}{p'!}\ \frac{{k}^{k}}{\dist(t)^{k}}\ \sqrt{\frac{(J+n+2(p'-|l_{i, 2}|-k))!}{J!}}. \eea We check that the constraints are satisfied: \bea\nonumber p'&\leq&n-1\quad<\quad n,\\[3pt] \nonumber |l_{i, 2}|+k&\leq&p+(n-2)/2+2\quad =\quad p'+n/2\\[3pt] \nonumber |j|&\leq&J+(n-2)+2(p-|l|-k)\quad =\quad J+n+2(p'-|l_{i,2}|-k). \eea Thus, each of the $d$ contributions stemming from $\Gamma_{1}(w,t)$ satisfies the required estimate provided $$ D_{2}^{8}\ \geq\ 1/2. $$ \vspace{.5cm} \noindent {\bf The Term $\int_{0}^{t}\,\Gamma_{2}$}\\ Similarly, with $p'=p+1$, \bea\nonumber &&\left\|\,D_{w}^{\alpha}\ \int_{0}^{t}\ \sum_{i=1}^{d}\ \bra\Phi,\,D_{w_{i}}\Phi\ket(w,s)\ D_{w_{i}}{c}_{n-2,p,l,k}(w,s)\,ds\,\right\| \\[5pt] \nonumber &\leq&\sum_{i=1}^{d}\ \frac{c'_{4}\,\nu\,D_{1}}{D_{2}^{8}}\ D_{2}^{|\alpha|+|l_{i,1}|+4n}\ \frac{(\alpha+l_{i,1})!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p'}}{p'!}\ \frac{{k}^{k}}{\dist(t)^{k}}\ \sqrt{\frac{(J+n+2(p'-|l_{i, 1}|-k))!}{J!}}, \eea with all constraints on $|j|,p',|l_{i,1}|,k$ satisfied. Thus each of the $d$ contributions stemming from $\Gamma_{2}(w,t)$ satisfies the required estimate, provided $$ D_{2}^{8}\ \geq\ c'_{4}\,\nu. $$ \vspace{.5cm} \noindent {\bf The Term $\int_{0}^{t}\,\Gamma_{3}$}\\ Again, with $p'=p+1$, \bea\nonumber &&\left\|\,D_{w}^{\alpha}\ \int_{0}^{t}\ \frac{1}{2} \bra\Phi,\,(\Delta_{w}\Phi)\ket(w,s)\ c_{n-2,p,l,k}(w,s)\,ds\, \right\| \\[5pt] \nonumber &\leq&\frac{c'_{5}\,\nu\,D_{1}}{2\,D_{2}^{8}}\ D_{2}^{|\alpha|+|l|+4n}\ \frac{(\alpha+l)!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p'}}{p'!}\ \frac{{k}^{k}}{\dist(t)^{k}}\ \sqrt{\frac{(J+n+2(p'-|l|-k))!}{J!}}, \eea with all constraints on $|j|,p',|l|,k$ satisfied. Thus, the contribution stemming from $\Gamma_{3}(w,t)$ satisfies the required estimate, provided $$ D_{2}^{8}\ \geq\ c'_{5}\,\nu/2. $$ \vspace{.5cm} \noindent {\bf The Term $\int_{0}^{t}\,\Gamma_{4}$}\\ Recall that the matrices $\bra\ffi,\,D_{y_{i}}\ffi\ket$ are controlled by an analog of Lemma \ref{ylem}. \bea\nonumber &&\left\|\,D_{w}^{\alpha}\ \int_{0}^{t}\ \sum_{i=1}^{d}\ \bra\ffi,\,D_{y_{i}}\ffi\ket\ D_{w_{i}}{c}_{n-1,p,l,k}(w,s)\,ds\, \right\|\\[5pt] \nonumber &&\!\!\!\!\!\!\!\!\!\!\!\! \leq\ \sum_{i=1}^{d}\ \frac{d_{0}\,\|B\|\,D_{1}}{D_{2}^{4}}\ D_{2}^{|\alpha|+|l_{i,1}|+4n}\ \frac{(\alpha+l_{i,1})!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p'}}{p'!}\ \frac{{k}^{k}}{\dist(t)^{k}}\ \sqrt{\frac{(J+n+2(p'-|l_{i,1}|-k))!}{J!}}, \eea with all constraints on $|j|,p',|l_{i,1}|,k$ satisfied. Thus each of the $d$ contributions stemming from $\Gamma_{4}(w,t)$ satisfies the required estimate, provided $$ D_{2}^{4}\ \geq\ d_{0}\,\|B\|. $$ \vspace{.5cm} \noindent {\bf The Term $\int_{0}^{t}\,\Gamma_{5}$}\\ For this term we obtain \bea\nonumber &&\left\|\,D_{w}^{\alpha}\ \int_{0}^{t}\ \sum_{i=1}^{d}\ \bra\Phi,\,(D_{w_{i}}\Phi)(w,s)\ket\,\bra\ffi,\,D_{y_{i}}\ffi\ket\ c_{n-1,p,l,k}(w,s)\,ds\,\right\|\\[5pt] \nonumber &&\!\!\!\!\!\!\!\!\!\!\!\!\! \leq\ \sum_{i=1}^{d}\ \frac{c'_{4}\,\nu\,d_{0}\,\|B\|\,D_{1}}{D_{2}^{4}}\ D_{2}^{|\alpha|+|l|+4n}\ \frac{(\alpha+l)!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p'}}{p'!}\ \frac{{k}^{k}}{\dist(t)^{k}}\ \sqrt{\frac{(J+n+2(p'-|l|-k))!}{J!}}, \eea with all constraints on $|j|,p',|l|,k$ satisfied. Thus each of the $d$ contributions stemming from $\Gamma_{5}(w,t)$ satisfies the required estimate, provided $$ D_{2}^{4}\ \geq\ c'_{4}\,\nu\,d_{0}\,\|B\|. $$ \vspace{.5cm} \noindent {\bf The Term $\int_{0}^{t}\,\Gamma_{6}$}\\ In this term, we encounter the sum over all previous $c$'s. As in the similar contribution from $\Delta_{2}$, we obtain \bea\nonumber &&\left\|\,D_{w}^{\alpha}\ \sum_{\tilde{m}=3}^{n+2}\ \sum_{|m|=\tilde{m}}\ \int_{0}^{t}\,\frac{D^{m}E(a(t))}{m!}\ \bra\ffi,\,y^{m}\ffi\ket\ c_{n+2-\tilde{m},p,l,k}(w,s)\, \right\| \\[5pt] \nonumber &\leq&\sum_{\tilde{m}=3}^{n}\ \sum_{|m|=\tilde{m}}\ D_{1}\ c_{6}\ D_{2}^{8}\ \frac{(d_{0}\,\|A\|)^{\tilde{m}}}{D_{2}^{3\tilde{m}}}\ D_{2}^{|\alpha|+|l|+4n}\ \frac{(\alpha+l)!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p'}}{p'!}\ \frac{{k}^{k}}{\dist(t)^{k}}\\[5pt] \nonumber &&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\quad \sqrt{\frac{(J+n+2(p'-|l|-k))!}{J!}}. \eea We check that the constraints on the parameters and components of the vectors are satisfied \bea\nonumber p'&\leq&n-\tilde{m}+3\quad\leq\quad n\\[3pt] \nonumber |l|+k&\leq&p+(n-\tilde{m}+2)/2\quad\leq\quad p'+n/2\\[3pt] \nonumber |j|&\leq&J+n+2+2(p-|l|-k)\quad =\quad J+n+2(p'-|l|-k). \eea Hence we see that each contribution from $\Delta_{2}(w,t)$ satisfies the required bound, provided the following two conditions are fulfilled \bea\nonumber D_{2}^{3}&\geq&d_{0}\ \|A\| \\[3pt] \nonumber D_{2}&\geq&(d_{0}\,\|A\|)^{3}\,c_{6}. \eea There are $\ds \sum_{\tilde{m}=3}^{n}\ \sum_{|m|=\tilde{m}}\ 1\ \leq\ \sigma_{0}\,\E^{\sigma n} $ such contributions, where $\sigma >0$ can be chosen arbitrarily small. \vspace{.5cm} \noindent {\bf The Term $\int_{0}^{t}\,\Gamma_{7}$}\\ This terms depends on the $d$'s. Recall the estimates are a little different for them. \bea\nonumber &&\left\|\,D_{w}^{\alpha}\ \int_{0}^{t}\ \sum_{i=1}^{d}\ \frac 12\ \bra\Phi(w,s),\,(D^{2}_{w_{i}}{d}_{n-2,p,l,k})(w,s)\ket\,ds\, \right\| \\[5pt] \nonumber &\leq&\sum_{i=1}^{d}\ \frac{c_{2}\,\nu\,D_{1}}{2\,D_{2}^{12}}\ D_{2}^{|\alpha|+|l_{i,2}|+4n}\ \frac{(\alpha+l_{i,2})!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p'}}{p'!}\ \frac{{k}^{k}}{\dist(t)^{k}}\ \sqrt{\frac{(J+n+2(p'-|l_{i,2}|-k))!}{J!}}, \eea with all constraints on $|j|,p',|l_{i,2}|,k$ satisfied. Thus each of the $d$ contributions stemming from $\Gamma_{7}(w,t)$ satisfies the required estimate, provided $$ D_{2}^{12}\ \geq\ c_{2}\,\nu/2. $$ \vspace{.5cm} \noindent {\bf The Term $\int_{0}^{t}\,\Gamma_{8}$}\\ Similarly, \bea\nonumber &&\left\|\,D_{w}^{\alpha}\ \int_{0}^{t}\ \sum_{i=1}^{d}\ \bra\ffi,\,D_{y_{i}}\ffi\ket\ \bra\Phi(w,s),\,(D_{w_{i}}{d}_{n-1,p,l,k})(w,s)\ket\,ds\, \right\| \\[5pt] \nonumber &&\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\! \leq\ \sum_{i=1}^{d}\ \frac{c_{2}\,\nu\,d_{0}\,\|B\|\,D_{1}}{D_{2}^{8}}\ D_{2}^{|\alpha|+|l_{i,1}|+4n}\ \frac{(\alpha+l_{i,1})!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p'}}{p'!}\ \frac{{k}^{k}}{\dist(t)^{k}}\ \sqrt{\frac{(J+n+2(p'-|l_{i, 1}|-k))!}{J!}}, \eea with all constraints on $|j|,p',|l_{i,1}|,k$ satisfied. Thus, each of the $d$ contributions stemming from $\Gamma_{8}(w,t)$ satisfies the required estimate provided $$ D_{2}^{8}\ \geq\ c_{2}\,\nu\,d_{0}\,\|B\|. $$ \vspace{.5cm} \noindent {\bf The Term $\int_{0}^{t}\,\Gamma_{9}$}\\ Finally, \bea\nonumber &&\left\|\,D_{w}^{\alpha}\ \int_{0}^{t}\ \bra\dot{\Phi}(w,s),\,d_{n,p,l,k}(w,s)\ket\,ds\,\right\|\\[5pt] \nonumber &\leq&\frac{c_{3}\,\nu\,D_{1}}{D_{2}^{4}}\ D_{2}^{|\alpha |+|l|+4n}\ \frac{(\alpha+l)!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^{p'}}{p'!}\ \frac{{k}^{k}}{\dist(t)^{k}}\ \sqrt{\frac{(J+n+2(p'-|l|-k))!}{J!}}, \eea with all constraints on $|j|,p',|l|,k$ satisfied. Thus, the contribution stemming from $\Gamma_{9}(w,t)$ satisfies the required estimate, provided $$ D_{2}^{4}\ \geq\ c_{3}\,\nu . $$ By choosing $D_2$ large enough, all conditions are satisfied. This completes the induction for part iii) of Proposition \ref{mainprop}. The integration required to construct the $c$'s shows that we obtain non-zero results for $c_{n,p,l,k,\beta}$ for $n>0$ only when $p\geq 1$. This proves part i) of Proposition \ref{mainprop}. \vskip .3cm We now turn to the proof of part ii) of Proposition \ref{mainprop}. \vskip .5cm \subsection{Counting the Number of Terms that Occur in Our Expansion} In our Born--Oppenheimer expansion, the $n^{\mbox{\scriptsize th}}$ order term has the form\\ $\phi_n(w,y,t)\,=\,g_n(w,y,t)\Phi(w,t)\,+\,\phi_n^{\perp}(w,y,t)$. The way we compute $g_n(w,y,t)$ and $\phi_n^{\perp}(w,y,t)$, they decompose naturally as sums over the parameter $\beta$. We define $u_n$ to be the number of such terms in $g_n(w,y,t)$ and $v_n$ to be the number of terms in $\phi_n^{\perp}(w,y,t)$. An examination of our construction shows that $u_n$ and $v_n$ satisfy the recursive estimates \bea\label{uinequality} u_{n+1}&\le&\sum_{j=0}^3\,a_j\,u_{n-j}\,+\, \sum_{j=0}^3\,b_j\,v_{n-j}\,+\,\sum_{j=0}^n\,c_1\,\gamma_1^j\,u_{n-j} \,+\,\sum_{j=0}^n\,c_2\,\gamma_2^j\,v_{n-j}\,+\,v_{n+1}\\ v_{n+1}&\le&\sum_{j=0}^3\,d_j\,u_{n-j}\,+\, \sum_{j=0}^3\,e_j\,v_{n-j}\,+\,\sum_{j=0}^n\,c_3\,\gamma_3^j\,u_{n-j} \,+\,\sum_{j=0}^n\,c_4\,\gamma_4^j\,v_{n-j},\label{vinequality} \eea where $a_i$, $b_i$, $c_i$, $d_i$, $e_i$ and $\gamma_i$ are fixed numbers. The exponentials $\gamma_i^j$ arise from an estimate (proven in the proof of Lemma 5.2 of \cite{hagjoy5}) for the number of Taylor series terms of any given order in the expansion of $E(a(t)+\epsilon y)$. We substitute (\ref{vinequality}) for the last term in (\ref{uinequality}) and add the result to (\ref{vinequality}). By some simple estimates this leads to a recursive estimate for the single quantity $z_n=u_n+v_n$ of the form $$z_{n+1}\ \le\ \sum_{j=0}^3\ \widetilde{a}_j\,z_{n-j}\ +\ \sum_{j=0}^n\ \widetilde{c}\ \widetilde{\gamma}^j\,z_{n-j}.$$ An easy induction on $n$ shows that this implies that $z_n$ grows at most like $e^{kn}$ for a sufficiently large value of $k$. The quantity $z_n$ is the number of terms in $\phi_n(w,y,t)$, so this proves the assertion.\ep \vskip .3cm Proposition \ref{mainprop} now follows easily. \ep \vskip .5cm \section{Exponential Error Bounds} \setcounter{equation}{0} In this section, we prove Theorem \ref{main}. \subsection{The Explicit Error Term} We use the following abstract lemma, whose proof is an easy application of Duhamel's formula (see e.g. \cite{raise}). \vskip .25cm \noindent \begin{lem}\label{magic} Suppose $H(\hbar)$ is a family of self-adjoint operators for $\hbar>0$. Suppose $\psi(t,\,\hbar)$ belongs to the domain of $H(\hbar)$, is continuously differentiable in $t$, and approximately solves the Schr\"odinger equation in the sense that %\be\label{xidef} $$ i\,\hbar\,\frac{\partial\psi}{\partial t}(t,\,\hbar)\ =\ H(\hbar)\,\psi(t,\,\hbar)\ +\ \xi(t,\,\hbar), $$ where $\xi(t,\,\hbar)$ satisfies %\be\label{xiest} $$ \|\,\xi(t,\,\hbar)\,\|\ \le \,\mu(t,\,\hbar). $$ Then, for $t>0$, %\be\label{ultimate} $$ \|\,\E^{-itH(\hbar)/\hbar}\,\psi (0,\,\hbar)\ -\ \psi(t,\,\hbar)\,\|\ \le\ \hbar^{-1}\ \int_0^{t}\,\mu(s,\,\hbar)\,ds. $$ The analogous statement holds for $t<0$. \end{lem} \vskip .4cm We substitute our approximate solution (\ref{approx})\\ $\ds F\ \E^{iS/\eps^2}\ \E^{i\eta\cdot y/\eps}\ \left(\,\sum_{n=0}^N\,\epsilon^n\,\phi_n\,+\,\eps^{N+1}\,\phi_{N+1}^\perp\,+\, \eps^{N+2}\,\phi_{N+2}^\perp\,\right)$ into the Schr\"odinger equation and compute the residual term $\xi_N$. It is more convenient to write this term in the multiple scales notation. We also use the notation $\ds \eps^m\,\frac{E^{(m)}(a(t))}{m!}y^m$ to denote the Taylor series term $\ds \sum_{|j|=m}\,\eps^{|j|}\,\frac{(D^jE)(a(t))}{j!}\,y^j$. In this notation, the residual $\xi_N(w,y,t)$ is given, up to a phase factor, by two sums of terms. The first one contains all terms that do not involve derivatives of the cut-off. The second contains all terms that do involve derivatives of the cut-off.\\ The first sum is $F(w)$ times the following: \bea\label{errterm} &&\frac{\eps^{N+3}}{2}\,\left(\Delta_wg_{N-1}\right)\,\Phi \\[7pt] %\nonumber &+&\frac{\eps^{N+4}}{2}\,\left(\Delta_wg_{N}\right)\,\Phi \\[7pt] %\nonumber &+&\eps^{N+3}\,\left(\nabla_wg_{N-1}\right)\cdot\left(\nabla_w\Phi\right) \\[7pt] %\nonumber &+&\eps^{N+4}\,\left(\nabla_wg_{N}\right)\cdot\left(\nabla_w\Phi\right) \\[7pt] %\nonumber &+&\frac{\eps^{N+3}}{2}\,g_{N-1}\,\left(\Delta_w\Phi\right) \\[7pt] %\nonumber &+&\frac{\eps^{N+4}}{2}\,g_{N}\,\left(\Delta_w\Phi\right) \\[7pt] %\nonumber &+&\frac{\eps^{N+3}}{2}\,\left(\Delta_w\phi_{N-1}^\perp\right) \\[7pt] %\nonumber &+&\frac{\eps^{N+4}}{2}\,\left(\Delta_w\phi_{N}^\perp\right) \\[7pt] %\nonumber &+&\eps^{N+3}\,\left(\nabla_w\cdot\nabla_yg_{N}\right)\Phi \\[7pt] %\nonumber &+&\eps^{N+3}\,\left(\nabla_yg_{N}\right)\cdot\left(\nabla_w\Phi\right) \\[7pt] %\nonumber &+&\eps^{N+3}\,\left(\nabla_w\cdot\nabla_y\phi_{N}^\perp\right) \\[7pt] %\nonumber \label{dot1} &+&i\,\eps^{N+3}\,\dot{\phi}_{N+1}^\perp \\[7pt] %\nonumber \label{dot2} &+&i\,\eps^{N+4}\,\dot{\phi}_{N+2}^\perp \\[7pt] %\nonumber &+&\frac{\eps^{N+5}}{2}\,\left(\Delta_w\phi_{N+1}^\perp\right) \\[7pt] %\nonumber &+&\frac{\eps^{N+6}}{2}\,\left(\Delta_w\phi_{N+2}^\perp\right) \\[7pt] %\nonumber &+&\eps^{N+4}\,\left(\nabla_w\cdot\nabla_y\phi_{N+1}^\perp\right) \\[7pt] %\nonumber &+&\eps^{N+5}\,\left(\nabla_w\cdot\nabla_y\phi_{N+2}^\perp\right) \\[7pt] %\nonumber &+&\frac{\eps^{N+3}}{2}\,\left(\Delta_y\phi_{N+1}^\perp\right) \\[7pt] %\nonumber &+&\frac{\eps^{N+4}}{2}\,\left(\Delta_y\phi_{N+2}^\perp\right) \\[7pt] %\nonumber &-&\frac{\eps^{N+3}}{2}\,E^{(2)}(a(t))\,y^2\,\phi_{N+1}^\perp \\[7pt] %\nonumber &-&\frac{\eps^{N+4}}{2}\,E^{(2)}(a(t))\,y^2\,\phi_{N+2}^\perp \\[7pt] %\nonumber \label{sumerr1} &-&\sum_{n=0}^N\,\eps^{N-n}\,\left(E(a(t)+\eps y)\,-\, \sum_{m\le 2+n}\,\eps^m\,\frac{E^{(m)}(a(t))}{m!}\,y^m\, %TS_{2+n} \right)\,g_{N-n}\,\Phi \\[7pt] %\nonumber \label{sumerr2} &-&\sum_{n=0}^N\,\eps^{N-n}\,\left(E(a(t)+\eps y)\,-\, \sum_{m\le 2+n}\,\eps^m\,\frac{E^{(m)}(a(t))}{m!}\,y^m\, %TS_{2+n} \right)\,\phi_{N-n}^\perp \\[7pt] %\nonumber &-&\eps^{N+1}\,\left(E(a(t)+\eps y)\,-\, \sum_{m\le 2}\,\eps^m\,\frac{E^{(m)}(a(t))}{m!}\,y^m\, %TS_2 \right)\,\phi_{N+1}^\perp \\[7pt]\label{moose} &-&\eps^{N+2}\,\left(E(a(t)+\eps y)\,-\, \sum_{m\le 2}\,\eps^m\,\frac{E^{(m)}(a(t))}{m!}\,y^m\, %TS_2 \right)\,\phi_{N+2}^\perp . \eea The second sum arises from terms in which the cut-off $F(w)$ is differentiated. It is %The presence of the cut-off $F(w)$ causes all of the %above terms to be multiplied by $F(w)$ and introduces some new error %terms. The additional terms in $\xi(w,y,t)$ are \bea\label{cutoff1} &&\sum_{n=0}^N\ \frac{\eps^{n+4}}2\ (\Delta_wF)\,g_n\,\Phi \\[7pt] %\nonumber &+&\sum_{n=0}^{N+2}\ \frac{\eps^{n+4}}2\ (\Delta_wF)\,\phi_n^\perp \\[7pt] %\nonumber &+&\sum_{n=0}^N\ \eps^{n+4}\ (\nabla_wF)\cdot(\nabla_wg_n)\,\Phi \\[7pt] %\nonumber &+&\sum_{n=0}^N\ \eps^{n+4}\ g_n\,(\nabla_wF)\cdot(\nabla_w\Phi) \\[7pt] %\nonumber &+&\sum_{n=0}^{N+2}\ \eps^{n+4}\ (\nabla_wF)\cdot(\nabla_w\phi_n^\perp) \\[7pt] %\nonumber &+&\sum_{n=0}^N\ \eps^{n+3}\ (\nabla_wF)\cdot(\nabla_yg_n)\,\Phi \\[7pt] \label{cutoff2}%\nonumber &+&\sum_{n=0}^{N+2}\ \eps^{n+3}\ (\nabla_wF)\cdot(\nabla_y\phi_n^\perp) \eea \vskip .5cm \subsection{Optimal Truncation}\label{optimal} Each error term in the first sum (\ref{errterm})--(\ref{moose}) can be written as a uniformly bounded function times one of the following two forms: \bea\nonumber {\cal A}&=&\Psi(w,t)\ \sum_{r}\ \sum_{|j|\leq\rho(r)}\ c_{r,j}(w,t)\ \ffi_{j}(y,t)\\[4pt]\nonumber {\cal B}&=&\sum_{r'}\ \sum_{|j|\leq\rho'(r')}\ d_{r',j}(w,t)\ \ffi_{j}(y,t), \eea where $\Psi(w,t)\in {\cal H}_{\mbox{\scriptsize el}}$, $\ffi_j(y,t)=\eps^{-d/2}\ffi_j(A(t), B(t), 1, 0, 0, y)$, $r,\,r'$ denote a collective set of indices that belong to some finite set, and $\rho(r)$ and $\rho'(r')$ limit the number of multi-indices $j$ allowed in the second sum. The error term $\xi(w,y,t)\in{\cal H}_{\mbox{\scriptsize el}}$ needs to be estimated for $t\in\R$, in the following norm \bea\nonumber \|\xi(t)\|&=&\left\{\,\int_{\smallR^d}\ \|\xi\left(x-a(t),\,(x-a(t))/\eps,\,t\right)\, \|^2_{{\cal H}_{\mbox{\scriptsize el}}}\,dx\,\right\}^{1/2}\\[4pt]\nonumber &=&\left\{\,\int_{\smallR^d}\ \|\xi(w,w/\eps,t)\|^2_{{\cal H}_{\mbox{\scriptsize el}}}\,dw\,\right\}^{1/2}. \eea With that norm, using the Cauchy--Schwarz inequality and the $L^2(\R^d)$ orthonormality of the $\ffi_j(y,t)$, we obtain the following estimate for the norm of ${\cal A}$ in terms of the norm of vector $c_r(w,t)\in l^2(\N^d,\C)$, \be\label{norma} \|{\cal A}\|\ \leq\ \sum_{r}\,\sup_{w\in\mbox{\scriptsize supp}\,F}\ \|\Psi(w,t)\|_{{\cal H}_{\mbox{\scriptsize el}}} \,\sup_{w\in\mbox{\scriptsize supp}\,F}\ \|c_r(w,t)\|\,\left(\,\sum_{|j|\leq\rho(r)}\,1\,\right)^{1/2}. \ee By similar arguments we get the following estimate for the norm of ${\cal B}$ in terms of the norm of the vector $d_{r'}(w,t)\in l^2(\N^d,{\cal H}_{\mbox{\scriptsize el}})$, \be\label{normb} \|{\cal B}\|\ \leq\ \sum_{r'}\,\sup_{w\in \mbox{\scriptsize supp}\,F} \|d_{r'}(w,t)\|\,\left(\,\sum_{|j|\leq\rho'(r')}\,1\,\right)^{1/2}. \ee Note also that \be\label{numbindic} \sum_{|j|\leq\rho'(r')}\,1\ \leq\ \pmatrix{\rho'(r')+d\cr d}, \ee which grows at most polynomially with $\rho'(r')$. \begin{lem}\label{nbehave} For $t\in [0,\,T]$, and for any $\alpha\in\N^d$ and $\gamma\in\N^d$, there exist $C_0>0$ and $\tau_0>0$, such that \bea\nonumber %\!\!\!\!\!\!\!\!\!\!\!\!\!\! &&\sum_{\beta\in{\cal B}_{n,1}}\ \sum_{p\leq n}\ \sum_{k+|l|\leq p+\frac n2}\ \sum_{|j|\leq J+n +2(p-|l|-k)}\left\| \Psi(w, t) D_w^\alpha D_y^\gamma c_{n,p,l,k,\beta,j}(w,t) \ffi_j(y,t)\right\|\\[5pt]\label{goat} &\leq&C_0\left\{n^{1/2}\tau_0\right\}^{n} \eea and \bea\nonumber %\nonumber \\[5pt] %&&\!\!\!\!\!\!\!\!\!\!\!\!\!\! &&\sum_{\beta\in{\cal B}_{n,2}}\ \sum_{p\leq n-1}\ \sum_{k+|l|\leq p+\frac{n-1}{2}}\ \sum_{|j|\leq J+n-1+2(p-|l|-k)} \left\|D_w^\alpha D_y^\gamma d_{n, p, l, k, \beta, j}(w,t) \ffi_j(y,t)\right\|\\[5pt]\label{sheep} &\leq&C_0\left\{n^{1/2}\tau_0\right\}^{n} \eea If the operator $D^\gamma_y$ is replaced by the operator $y^\gamma$, the same bounds are valid. \end{lem} \noindent {\bf Proof:}\quad We begin with (\ref{goat}). We have \bea &&\sum_{|j|\leq J+n+2(p-|l|-k)}D_w^\alpha c_{n, p, l, k, \beta, j}(w,t)\ D_y^\gamma \ffi_j(y,t)\nonumber\\[5pt] &=&\sum_{|\tilde{k}|\leq J+|\gamma|+n+2(p-|l|-k)} \left(\,\bra\ffi,\,D_y^\gamma\ffi\ket\ D_w^\alpha c_{n, p, l, k, \beta}(w,t)\right)_{\tilde{k}}\, \ffi_{\tilde{k}}(y,t).\nonumber \eea We know that the vector $\bra\ffi,\,D_y^\gamma\ffi\ket\ D_w^\alpha c_{n,p,l,k,\beta}(w,t)$ satisfies the estimate \bea\label{replace} &&\left\|\,\bra\ffi,\,D_y^\gamma\ffi\ket\, D_w^\alpha c_{n,p,l,k,\beta}(w,t)\,\right\|\ \leq\ D_1\,D_2^{|\alpha|+|l|+4n}\ \frac{(\alpha+l)!}{(1+|\alpha|)^{d+1}}\ \frac{|t|^p}{p!}\ \frac{k^k}{\delta^k}\ (\|B\|\,d_0)^{|\gamma|}\nonumber\\[5pt] &&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\times\quad \sqrt{\frac{(J+|\gamma|+n+2(p-|l|-k))!}{J!}}. \eea Here $\delta>0$ is the distance in the complex plane from $[0,\,T]$ to the complement of $\Omega$. Since the number of indices in ${\cal B}_{n,1}$ is bounded by $\E^{K_0 n}$,\ \, $D_2\geq 1$,\ \, and $(\alpha+l)!\leq (|\alpha|+|l|)!$, we can estimate the sum (\ref{goat}) by \bea &&\frac{D_1\,D_2^{|\alpha|}\,(\|B\|d_0)^{|\gamma|}} {\sqrt{J!}\,(1+|\alpha|)^{d+1}}\ \E^{K_0 n}\ D_2^{11 n/2}\ \sum_{p\leq n}\ \frac{|t|^p}{p!}\ \sum_{|l|+k\leq p+n/2}\left(\frac{k}{\delta}\right)^k\nonumber\\[5pt] &&\qquad\qquad\qquad\quad\quad\times\quad(|\alpha|+|l|)!\ \sqrt{(J+|\gamma|+n+2(p-|l|-k))!}. \label{above} \eea Then, using $a!b!\leq (a+b)! $, the fact that $(a+2p)!/(p!)^2$ is increasing in $p$, and $p\leq n$, we have $$ \frac{(J+|\gamma|+n+2(p-|l|-k))!\ ((|\alpha|+|l|)!)^2} {(p!)^2}\ \leq\ \frac{(J+|\gamma|+2|\alpha|+3n-2k)!}{(n!)^2}, $$ so that (\ref{above}) is bounded by \bea\nonumber &&\frac{D_1\,D_2^{|\alpha|}\,(\|B\|d_0)^{|\gamma|}} {n!\,\sqrt{J!}\,(1+|\alpha|)^{d+1}}\ \E^{K_0 n}\ D_2^{11 n/2}\ \sum_{p\leq n}\ |t|^p\ \sum_{k=0}^{p+n/2}\ \left(\frac{k}{\delta}\right)^k\\[5pt]\nonumber % \label{below} &&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\quad \sqrt{(J+|\gamma|+2|\alpha|+3n-2k)!}\ \sum_{|l|\leq p+n/2-k}\ 1. \eea The last term is bounded by $\pmatrix{\grintl 3n/2\grintr +d\cr d}\,\leq\, \sigma_0\,\e^{3\sigma n/2}$, where $\grintl x\grintr$ denotes the integer part of $x$. Using $k^{2k}\leq (2k)^{2k}$, $a^ab^b\leq (a+b)^{a+b}$ and $a!\leq a^a$ we have $$ (J+|\gamma|+2|\alpha|+3n-2k)!\ k^{2k}\ \leq\ (J+|\gamma|+2|\alpha|+3n)^{J+|\gamma|+2|\alpha|+3n}. $$ Since we can assume without loss that $\delta<1$, this implies \bea &&\sum_{k=0}^{p+n/2}\, \left(\frac{k}{\delta}\right)^k\, \sqrt{(J+|\gamma|+2|\alpha|+3n-2k)!}\nonumber\\ &\leq&(J+|\gamma|+2|\alpha|+3n)^{\frac{J+|\gamma|+2|\alpha|+3n}{2}}\quad \sum_{k=0}^{p+n/2}\,\delta^{-k}\nonumber \\[5pt] &\leq&(J+|\gamma|+2|\alpha|+3n)^{\frac{J+|\gamma|+2 |\alpha| +3n}{2}}\ \frac{K_1}{\delta^{n/2 }}\ \delta^{-p},\label{45} \eea for some constant $K_1$ that satisfies $\delta^{-1}-1\geq K_1^{-1}\delta^{-1}$. Together with \be\label{46} \sum_{p\leq n}\ (t/\delta)^p\ \leq\ \left\{\,\matrix{K_2 (t/\delta)^n&\mbox{if}&t/\delta>1\cr K_2^n&\mbox{if}&t/\delta=1\cr K_2&\mbox{if}&t/\delta<1,}\right. \ee where $K_2$ is constant, we get (in the first case above) \bea\nonumber % \label{sumc} &&\sum_{\beta\in {\cal B}_{n,1}}\ \sum_{p\leq n}\ \sum_{k+|l|\leq p+\frac n2}\ \left\|\,\bra\ffi,\,D_y^\gamma\ffi\ket\, D_w^\alpha c_{n,p,l,k,\beta}(w,t)\,\right\| \\[4pt]\nonumber &\leq&\frac{\sigma_0K_1K_2D_1 D_2^{|\alpha|}(\|B\|d_0)^{|\gamma|}} {(1+|\alpha|)^{d+1}}\ \frac{\E^{(K_0+3\sigma/2 )n}D_2^{11 n/2}t^n}{\sqrt{J!}\,n!\,\delta^{3n/2}}\ (J+|\gamma|+2|\alpha|+3n)^{\frac{J+|\gamma|+2|\alpha|+3n}{2}}. \eea We postpone the study of the dependence of our estimates on $t$ and $J$ to Section \ref{gene}. So, using the above, $$ (J+|\gamma|+2|\alpha|+3n)^{\frac{J+|\gamma|+2|\alpha|+3n}{2}} \leq (J+|\gamma|+2|\alpha|+3n)^{\frac{J+|\gamma|+2|\alpha|}{2}} ((J+|\gamma|+2|\alpha|+3)n)^{\frac{3n}{2}}, $$ and the existence of $00$, there exist $\Gamma(g)$ and $C(g)>0$ such that the choice $N(\eps)=\grintl g^2/\eps^2\grintr$ implies that the norm of the error term $\xi_{N(\eps)}(t)$ given by (\ref{errterm}) satisfies $$\|\xi_{N(\eps)}(t)\|\ \leq\ C(g)\,\E^{-\Gamma(g)/\eps^2}.$$ \end{lem} \noindent{\bf Proof:}\quad The previous lemma, formulas (\ref{norma}), (\ref{normb}) and (\ref{numbindic}) show that all terms in the first sum defining $\xi_N$ except (\ref{dot1}), (\ref{dot2}), (\ref{sumerr1}), and (\ref{sumerr2}) are exponentially small, once we prove \be C_0\eps^{N(\eps)}\ \left\{N(\eps)^{1/2}\ \tau_0\right\}^{N(\eps)}\ \le\ C\ \E^{-\Gamma/\eps^2}. \ee Because $g^2/\eps^2-1 \leq N\leq g^2/\eps^2$, if we choose $00$ that satisfies $$ \max\ \{\,|\Delta_w F(w)|,\ \|\nabla_w F(w)\|\,\}\ \leq\ F_0, $$ uniformly in $w$, and recall that for any $i=1,\dots,d$, \be \mbox{supp}\,\partial_{w_i}F(w)\,\subseteq\,\{w\in\R^d\;:\; b_0<|w|0$, such that for all $t\in\R$ $$ \|A(t)\|\,+\,\|B(t)\|\ \leq\ L\,\E^{\lambda t}. $$ Then, there exist $\tau'$, $C'$, $T'>0$, and $0<\sigma,\,\sigma'<2$ such that the approximation defined by choosing $N(\eps)\simeq 1/\eps^{\sigma}$ is accurate up to an error whose norm is bounded by $C'\E^{-\tau'/\eps^{\sigma'}}$, uniformly for all times $0\leq t\leq T'\,\ln(1/\eps^2)$. \end{thm} {\bf Proof:}\quad It is enough to mimick the proof of the corresponding result for the semiclassical propagation of the Schr\"odinger equation in \cite{hagjoy5}, since our hypotheses imply that nothing can happen on the adiabatic side of the problem. By the conservation of energy, the exponential bound on $E(z)$ and the assumed existence of a Liapunov exponent, we easily see from the proof of Lemmas \ref{nbehave} and \ref{optrunc}, that the behavior in $t$ of all constants (independent of $N$) is at worst exponential in $t$. From the conditions $D_2\geq \E^{K T}$, with $K$ some constant, we need to take $g(T)\leq g_0\E^{-g_1 t}$ so that the optimal truncation procedure yields an error of the order $\ds\E^{K_0 T}\E^{-g_0^2\e^{-2g_1}/\eps^2}$. The choice $T(\eps)\leq T'\ln(1/\eps^2)$, with $T'>0$ sufficiently small, gives the desired result.\ep \vspace{.3cm} Similarly, we can extend our results to allow initial conditions in a wider class of vectors. Indeed, we have been careful to make explicit the $J$ dependence in all estimates so that we can control the error term as a function of $J$. Recall that $J$ is fixed arbitrarily in (\ref{2aorder}) which gives the expansion in the basis $\ffi_j(A(0),B(0),\eps^2,a(0),\eta(0),x)$ of the nuclear part of the wave function that we take as an initial condition. \vspace{.2cm} As in \cite{hagjoy5}, for $(a,\eta)\in\R^{2d}$, we introduce the operator $\Lambda_{\eps}(a,\eta)$ such that $$ (\Lambda_{\eps}(a,\eta)f)(x)\,=\, {\eps}^{-d}\,\E^{i\eta \cdot\,(x-a)/{\eps^2}}\,f((x-a)/{\eps}). $$ We define a dense set ${\cal C}$ in $L^2(\R^d)$, that is contained in the set ${\cal S}$ of Schwartz functions, by \bea {\cal C}&=&\left\{\,f(x)\,=\,\sum_{j}\,c_j\,\ffi_j(\un,\,\un,\,1,\,0,\,0,\,x)\, \in\,{\cal S},\right.\mbox{ such that }\nonumber\\ &&\qquad\qquad\quad\left.\,\mbox{there exists}\ K>0 \mbox{ with } \sum_{|j|>J}|c_j|^2\leq \E^{-K J}, \mbox{ for large } J\,\right\}.\label{woofer} \eea \vskip .2cm \noindent {\bf Remark}\quad It is easy to check that the inequality in (\ref{woofer}) is equivalent to the requirement that the coefficients of $f$ satisfy $$ |c_j|\,\leq\,\E^{-K|j|}, $$ for large $|j|$. Another equivalent definition of ${\cal C}$ is $$ {\cal C}\,=\,\cup_{t>0}\ \E^{-tH_{ho}}\,{\cal S}, $$ where $H_{ho}\,=\,-\,\Delta/2\,+\,x^2/2$ is the harmonic oscillator Hamiltonian. The set ${\cal C}$ is also called the set of analytic vectors \cite{RS} for the harmonic oscillator Hamiltonian. \vskip .4cm Let $f\in{\cal C}$. We set \bea f_J(y, t) &=& \sum_{|j|\leq J}\, c_j\,\ffi_j(A(t),B(t),\eps^2,0,0,y), \qquad\mbox{and}\nonumber\\[5pt] f(y, t) &=& \sum_{j}\,c_j\,\ffi_j(A(t),B(t),\eps^2,0,0,y)\nonumber \eea where the classical quantities $a(t)$, $\eta(t)$, $A(t)$, $B(t)$, and $S(t)$ correspond to the initial conditions $a(0)$, $\eta(0)$, $A(0)=B(0)=\un$, and $S(0)$. We consider the construction described in Section \ref{mr} corresponding to the initial condition $g_0(0,y,t)=f_J(y,t)$, making explicit the dependence on $J$ in the notation: \bea\nonumber &&\hat{\Psi}_{J, N}(w, y, t)\\ &=&F(w)\,\E^{iS(t)/\eps^2}\,\E^{i\eta(t)\cdot y/\eps}\, \left(\sum_{n=0}^N\,\eps^n\,g_{n,J}(w,y,t)\,\Phi(w,t)\,+ \sum_{n=2}^{N+2}\,\eps^n\, \phi_{n,J}^\perp(w,y,t)\right).\nonumber \eea Recall that \bea\nonumber &&\hat{\Psi}_{J,N}(w,y,0)\\ &=&F(w)\,\E^{iS(0)/\eps^2}\,\E^{i\eta(0)\cdot y/\eps}\, \left(\,f_J(y, 0)\,\Phi(w, 0)\,+\, \sum_{n=2}^{N+2}\,\eps^n\, \phi_{n, J}^\perp(w,y,0)\,\right).\nonumber \eea Let $\nu>0$, and consider $N(\eps)\,=\,\grintl g^2/\eps^2\grintr$ and $J(\eps)=\nu N(\eps)$. We define our more general initial conditions as \bea\nonumber &&\hat{\Psi}_{f}(w,y,0)\\ &=&F(w)\,\E^{iS(0)/\eps^2}\,\E^{i\eta(0)\cdot y/\eps}\, \left(f(y,0)\,\Phi(w, 0)\,+\, \sum_{n=2}^{N(\eps)+2}\,\eps^n\, \phi_{n,J(\eps)}^\perp(w,y,0)\right),\nonumber \eea which corresponds, when we get back to the variables $(X,t)$, to an initial state $\hat{\Psi}_{f}(X-a(0),(X-a(0))/\eps,0)$ whose projection along the electronic eigenvector $\tilde{\Phi}(X,0)$ yields a nuclear wave packet of the form $(\Lambda_{\eps}(a(0),\eta(0))f)(X)$. % Note that the component of the initial state perpendicular to $\tilde{\Phi}(X,0)$ necessary to achieve exponential accuracy depends on $\eps$. This component is determined by the coefficients of the function $f$. We can now state our result for such general initial conditions \begin{thm} Assume the hypotheses of Theorem \ref{main} and consider the above constructions. There exist sufficiently small $g>0$ and positive constants $C(g)$, $\Gamma(g)$, such that with the definition $$ \Psi_*(X,t,\eps)=\hat{\Psi}_{J(\eps),N(\eps)}(X-a(t),(X-a(t))/\eps,t), $$ we have $$ \left\|\,\E^{-itH(\eps)/\eps^2}\,\Psi_f(X, 0, \eps)\,-\,\Psi_*(X,t,\eps)\, \right\|_{L^2({\smallR}^d,{\cal H}_{\mbox{\scriptsize el}})} \ \leq\ C(g)\,\E^{-\Gamma(g)/\eps^2}, $$ for all $t\in [0, T]$, as $\eps\ra 0$.\\ Moreover, the result for times $T\simeq \ln(1/\eps^2)$ corresponding to Theorem \ref{ehr} is also true for these initial conditions. \end{thm} \noindent{\bf Proof:}\quad We have \bea\nonumber &&\E^{-itH(\eps)/\eps^2}\,\Psi_f(X,0,\eps)\\[7pt]\nonumber % \label{strategy} &=&\E^{-itH(\eps)/\eps^2}\, (\Psi_f(X,0,\eps)-\Psi_*(X,0,\eps))\,+\,\E^{-itH(\eps)/\eps^2}\, \Psi_*(X, 0,\eps) \\[7pt]\nonumber &=&\Psi_*(X,t,\eps)\ +\ O(\|\E^{-itH(\eps)/\eps^2}\,\Psi_*(X,0,\eps)-\Psi_*(X,t,\eps) \|_{L^2({\smallR}^d,{\cal H}_{\mbox{\scriptsize el}})})\\[7pt] \nonumber &&+\quad O\left(\,\|\Psi_f(X,0,\eps)-\Psi_*(X, 0, \eps) \|_{L^2({\smallR}^d,{\cal H}_{\mbox{\scriptsize el}})}\,\right). \eea By our choice of function $f$, the last term is exponentially small in $1/\eps^2$. The remaining norm to estimate corresponds to the situation of Theorem \ref{main} in which we let the parameter $J$ grow as $1/\eps^2$, according to our choice of $J(\eps)$. But, as in the proof of Theorem 3.6 in \cite{hagjoy5} for the corresponding result in semiclassical dynamics, we have made the dependence in $J$ of all the key estimates explicit. It is enough to go through the proof of theorem \ref{main} to check that with $J=\nu N$, all arguments can be repeated to get the same $N$ and $\eps$ behavior for the estimates on the error terms, (see \cite{hagjoy5} for details). Hence, we see that for sufficiently small $g$, we can approximate the solution corresponding to these generalized initial conditions up to an error of order $\E^{-\Gamma(g)/\eps^2} $. The Ehrenfest time regime is dealt with similarly. \ep \vskip .5cm \section{Technicalities}\label{techie} In this section we give the proofs of the auxiliary lemmas we used in the course of the main argument. \vskip .5cm \noindent {\bf Proof of Lemma \ref{cauchylem}:}\quad We first consider the case $k\ge 1$. By Cauchy's formula, we can write \be\label{cauchyformula} g'(t)\ =\ \frac 1{2\pi i}\ \int_\Gamma\,\frac{g(s)}{(t-s)^2}\,ds, \ee where $\Gamma$ is the circular contour with center $t$ and radius $\ds\frac 1{k+1}\ (\delta\,-\,|\mbox{\rm Im}\,t|)$. For $s$ on $\Gamma$, we have $\ds (\delta-|\mbox{\rm Im}\,s|)\,\ge\, \frac k{k+1}(\delta\,-\,|\mbox{\rm Im}\,t|)$. Thus, $$ \|g(s)\|\quad\le\quad C\ k^k\ (\delta\,-\,|\mbox{\rm Im}\,s|)^{-k} \quad\le\quad C\ k^k\ \left[\,\frac{k}{k+1}\, (\delta\,-\,|\mbox{\rm Im}\,t|)\,\right]^{-k} $$ So, by putting the norm inside the integral in (\ref{cauchyformula}), we have \bea\nonumber \|g'(t)\|&\le&\frac 1{2\pi}\ \frac{2\pi}{k+1}(\delta\,-\,|\mbox{\rm Im}\,t|)\ C\,k^k\,\left[\frac{k}{k+1}(\delta\,-\,|\mbox{\rm Im}\,t|)\right]^{-k}\ \left[\frac{1}{k+1}(\delta\,-\,|\mbox{\rm Im}\,t|)\right]^{-2}\\[5pt]\nonumber &=&C\ (k+1)^{k+1}\ (\delta\,-\,|\mbox{\rm Im}\,t|)^{-k-1}.\eea For $k=0$ we use the same argument with the radius of $\Gamma$ replaced by $\ds\alpha\,\,(\delta\,-\,|\mbox{\rm Im}\,t|)$ for any $\alpha<1$. This yields the bound $$\|g'(t)\|\ \le\ C\,\alpha^{-1}\,(\delta\,-\,|\mbox{\rm Im}\,t|)^{-1}.$$ The lemma follows because $\alpha<1$ is arbitrary.\qquad\qquad\ep \vskip .5cm \noindent {\bf Proof of Lemma \ref{nenciuprop2}}:\quad To prove the quantity $\nu$ is finite, we estimate \bea\nonumber &&\sum_{\{\,l\,:\,0\le l_i\le\alpha_i\,\}}\ \frac{1}{(1+|l|)^{d+1}}\ \frac{1}{(1+|\alpha-l|)^{d+1}}\\[8pt] \nonumber &=&\sum_{\begin{array}{c} {\{\,l\,:\,0\le l_i\le\alpha_i\,\}}\\[2pt] |l|\le\grintl\frac{|\alpha|}2\grintr\end{array}}\ \frac{1}{(1+|l|)^{d+1}}\ \frac{1}{(1+|\alpha-l|)^{d+1}}\\[-3pt] \nonumber &&\quad\quad\qquad\qquad\qquad\qquad\ +\qquad \sum_{\begin{array}{c} {\{\,l\,:\,0\le l_i\le\alpha_i\,\}}\\[2pt] |l|>\grintl\frac{|\alpha|}2\grintr\end{array}}\ \frac{1}{(1+|l|)^{d+1}}\ \frac{1}{(1+|\alpha-l|)^{d+1}}\\[8pt] \nonumber &\le&\frac{2}{\left(\,1+\grintl\frac{|\alpha|}2\grintr\,\right)^{d+1}}\ \sum_{\begin{array}{c} {\{\,l\,:\,0\le l_i\le\alpha_i\,\}}\\[2pt] |l|\le\grintl\frac{|\alpha|}2\grintr\end{array}}\ \frac{1}{(1+|l|)^{d+1}}\\[8pt] \nonumber &\le&\frac{2^{d+2}}{\left(\,1+|\alpha|\,\right)^{d+1}}\ \sum_{\begin{array}{c} {\{\,l\,:\,0\le l_i\le\alpha_i\,\}}\\[2pt] |l|\le\grintl\frac{|\alpha|}2\grintr\end{array}}\ \frac{1}{(1+|l|)^{d+1}}\\[8pt] \nonumber &\le&\frac{2^{d+2}}{\left(\,1+|\alpha|\,\right)^{d+1}}\ \sum_{l}\ \frac{1}{(1+|l|)^{d+1}}. \eea \vskip .2cm \noindent Thus,\qquad $\ds\nu\ \le\ 2^{d+2}\ \sum_l\ (1+|l|)^{-d-1}$. \vskip .1cm \noindent To see that the right hand side of this inequality is finite, we note that the number of multi-indices $l$ with $|l|=L$ is the binomial coefficient $\ds\left(\begin{array}{c}L+d-1\\d-1\end{array}\right)^{\phantom{|}}$, with the convention that $\ds\left(\begin{array}{c}0\\0\end{array}\right)\ =\ 1$. Thus, \bea\nonumber \nu&\le&2^{d+2}\ \sum_{L=0}^\infty\ \left(\begin{array}{c}L+d-1\\d-1\end{array}\right)\ \frac{1}{(1+L)^{d+1}}\\[8pt] &=&\frac{2^{d+2}}{(d-1)!}\ \sum_{L=0}^\infty\ \frac{(L+d-1)(L+d-2)\cdots(L+1)}{(L+1)^{d+1}}.\nonumber \eea \vskip .1cm \noindent For large $L$,\quad $\ds\frac{(L+d-1)(L+d-2)\cdots(L+1)}{(L+1)^{d+1}}\ $ is asymptotic to $L^{-2}$, so $\nu$ is finite. \vskip .3cm \noindent Since $\ds D^\alpha\,(M\,N)\ =\ \sum_{\{\,l\,:\,0\le l_i\le\alpha_i\,\}}\ \left[\,\prod_{j=1}^d\, \left(\begin{array}{c}\alpha_j\\l_j\end{array}\right)\,\right]\ \left(\,D^l\,M\,\right)\ \left(\,D^{(\alpha-l)}\,N\,\right)$, we have \vskip .2cm \bea\nonumber &&\left\|\,\left(\,D^\alpha\,(M\,N)\,\right)(x)\,\right\|\\[6pt] \nonumber &\le& \sum_{\{\,l\,:\,0\le l_i\le\alpha_i\,\}}\, \left[\,\prod_{j=1}^d\, \left(\begin{array}{c}\alpha_j\\l_j\end{array}\right)\,\right]\ m(x)\ n(x)\ a(x)^{|\alpha+p+q|}\quad\frac{(l+p)!}{(1+|l|)^{d+1}}\ \frac{(\alpha-l+q)!}{(1+|\alpha-l|)^{d+1}}\\[7pt] \nonumber&=& m(x)\ n(x)\ a(x)^{|\alpha+p+q|}\quad(\alpha+p+q)!\\[6pt] \nonumber &&\ \times\quad \sum_{\{\,l\,:\,0\le l_i\le\alpha_i\,\}} \left[\,\prod_{j=1}^d\, \left(\begin{array}{c}\alpha_j\\l_j\end{array}\right)\, \left(\begin{array}{c}\alpha_j+p_j+q_j\\l_j+p_j\end{array}\right)^{-1}\,\right]\ \frac 1{(1+|l|)^{d+1}\ (1+|\alpha-l|)^{d+1}}. \eea Since $\ds\left(\begin{array}{c}\alpha_j+p_j+q_j\\l_j+p_j\end{array}\right)\ \ge\ \left(\begin{array}{c}\alpha_j+q_j\\l_j\end{array}\right)\ \ge\ \left(\begin{array}{c}\alpha_j\\l_j\end{array}\right)$,\quad we therefore have \bea\nonumber \left\|\,\left(\,D^\alpha\,(M\,N)\,\right)(x)\,\right\| &\le&m(x)\ n(x)\ a(x)^{|\alpha+p+q|}\quad (\alpha+p+q)!\\[6pt] \nonumber &&\qquad\qquad\qquad\times\qquad\sum_{\{\,l\,:\,0\le l_i\le\alpha_i\,\}}\ \frac 1{(1+|l|)^{d+1}\ (1+|\alpha-l|)^{d+1}}.\\[6pt] \nonumber &\le&m(x)\ n(x)\ \nu\ a(x)^{|\alpha+p+q|}\quad \frac{(\alpha+p+q)!}{(1+|\alpha|)^{d+1}}.\qquad\qquad\ep \eea \vskip .5cm \noindent {\bf Proof of Lemma \ref{integrate}:}\quad If $f(t)$ satisfies $\|f(t)\|\leq C\,|t|^p\,\dist(t)^{-k}$, for all $t\in\Omega$, there exists $g(t)$ analytic in $\Omega$, such that $f(t)=t^p\,g(t)$ and $\|g(t)\|\leq C\,\dist(t)^{-k}$. 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