Content-Type: multipart/mixed; boundary="-------------0406140934753" This is a multi-part message in MIME format. ---------------0406140934753 Content-Type: text/plain; name="04-185.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-185.keywords" quantization, Kolmogorov's theorem, uniform semiclassical estimates ---------------0406140934753 Content-Type: application/x-tex; name="BoGrsub.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BoGrsub.tex" \documentclass[11pt]{article} %\documentclass[a4paper,11pt,reqno]{amsart} %\documentclass[a4paper,draft,reqno]{amsart} %\input{amssym.def} %\input{amssym} %\usepackage[notref]{showkeys} \setlength{\textwidth}{15.0cm} %%DB margin change%% \setlength{\textheight}{23.0cm} \hoffset=-1.0cm \voffset=-1.0cm \newtheorem{theorem}{Theorem} \newtheorem{proposition}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{corollary}{Corollary} \newtheorem{definition}{Definition} 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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% %%%%%%%% begin %%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \baselineskip=19pt \begin{center} {\large\bf A LOCAL QUANTUM VERSION OF THE \\ KOLMOGOROV THEOREM} \end{center} \vskip 13pt \begin{center} David Borthwick\footnote{Supported in part by NSF grant DMS-0204985. Department of Mathematics and Computer Science, Emory University, Atlanta 30322 (U.S.A.). (davidb@mathcs.emory.edu)}, Sandro Graffi\footnote{ Dipartimento di Matematica, Universit\`{a} di Bologna, 40127 Bologna (Italy). (graffi@dm.unibo.it)} \end{center} \begin{abstract} \noindent Consider in $L^2 (\R^l)$ the operator family $H(\epsilon):=P_0(\hbar,\omega)+\epsilon Q_0$. $P_0$ is the quantum harmonic oscillator with diophantine frequency vector $\om$, $Q_0$ a bounded pseudodifferential operator with symbol holomorphic and decreasing to zero at infinity, and $\ep\in\R$. Then there exists $\ep^\ast >0$ with the property that if $|\ep|<\ep^\ast$ there is a diophantine frequency $\om(\ep)$ such that all eigenvalues $E_n(\hbar,\ep)$ of $H(\ep)$ near $0$ are given by the quantization formula $E_\alpha(\hbar,\ep)={\cal E}(\hbar,\ep)+\la\om(\ep),\alpha\ra\hbar +|\om(\ep)|\hbar/2 + \ep O(\alpha\hbar)^2$, where $\alpha$ is an $l$-multi-index. \end{abstract} \vskip 1cm % %\date{\today} %\subjclass{???} \keywords{????} % %\maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Approximate solutions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begsection{Introduction and statement of the results} \setcounter{equation}{0}% \setcounter{theorem}{0}% %\setcounter{proposition}{0}% %\setcounter{lemma}{0}% \setcounter{corollary}{0}% %\setcounter{definition}{0}% Denote by ${\cal F}_{\rho,\sigma}$ the set of all functions $f(x,\xi):\R^{2l}\to\C$ with finite $\|f\|_{\rho,\sigma}$ norm for some $\rho>0$, $\sigma >0$ (see Section 2 for the definition and examples). Any $f\in {\cal F}_{\rho,\sigma}$ is analytic on $\R^{2l}$ and extends to a complex analytic function in the region $|\Im{z_i}|\leq a_i|\Re {z_i}|$ for suitable $a_i>0$; moreover $|f(z)|\to 0$ as $|z|\to+\infty$. Here $z:=(x,\xi)$. Let $\Phi_{\rho,\sigma}$ denote the class of semiclassical Weyl pseudodifferential operators $F$ in $L^2(\R^l)$ with symbol $f(x,\xi)$ in ${\cal F}_{\rho,\sigma}$; namely, (notation as in \cite{Ro}) \begin{eqnarray} \label{Weyl} (Fu)(x)&:=&Op^W_h(f(x,\xi))u(x) \\ \nonumber &=&\frac{1}{h^l}\int\!\!\!\int_{\R^l\times\R^l}e^{i\la (x-y),\xi\ra/\hbar} f((x+y)/2,\xi)u(y)\,dyd\xi,\; u\in{\cal S}(\R^l). \end{eqnarray} It follows directly from the definition of $\|f\|_{\rho,\sigma}$ in (\ref{norma}) that $F\in \Phi_{\rho,\sigma}$ extends to a conti\-nuous operator in $L^2(\R^l)$, with \be \|F\|_{L^2\to L^2}\leq \|f\|_{\rho,\sigma}. \ee Consider in $L^2(\R^l)$ the operator family $H(\ep)=P_0(\hbar,\om)+\ep Q_0$ and assume: \begin{itemize} \item[(A1)] $P_0(\hbar,\om)$ is the harmonic-oscillator \Sc\ operator with frequencies $\om\in [0,1]^l$: \be \label{HO} P_0(\hbar,\om)u=-\frac12\hbar^2\Delta u+[\om_1^2x_1^2+\ldots+\om_l^2x_1^2]u, \;\; D(P_0)=H^2(\R^l)\cap L_2^2(\R^l). \ee \item[(A2)] $Q_0\in \Phi_{\rho,\sigma}$; its symbol $q_0(x,\xi)=q_0(z)$ is real-valued for $z=(x,\xi)\in\R^\times\R^l$, and $q_0(z)=O(z^2)$ as $z\to 0$. \item[(A3)] There exist $ \tau>l-1,\gamma >0$ such that \begin{equation} \label{Diofanto} \la{ \om}, { k}\ra \geq {\gamma}{|{k} |^{-\tau}}, \quad \forall { k}\in \Z^l\setminus\{0\}, \quad |k|:=|k_1|+\ldots+|k_l|,\; \om:=(\om_1,\ldots,\om_l). \end{equation} Denote $\Omega_0$ the set of all $\om\in[0,1]^l$ fulfilling (\ref{Diofanto}), and $|\Omega_0|$ its measure. It is well known that $|\Omega_0|=1$. \end{itemize} Under the above assumptions the operator family $H(\ep)$ defined on $D(P_0)$ is self-adjoint with pure-point spectrum $\forall\,\ep\in\R$: ${\rm Spec}\,(H(\ep))={\rm Spec}_p\,(H(\ep))$. Moreover (\ref{Diofanto}) entails in particular the rational independence of the components of $\om$ and hence the simplicity of ${\rm Spec}(P_0)$ and its density in $\overline{\R}_+:=\R_+\cup\{0\}$. Clearly, $P_0$ is a semiclassical pseudodifferential operator of order $2$ with symbol \begin{eqnarray} \label{azioni} p_0(x,\xi)=\frac12(|\xi|^2+|\om x|^2) =\frac12\sum_{k=1}^l\om_kI_k(x,\xi), \; I_k(x,\xi):=\frac{1}{2\om_k}[\xi_k^2+\om^2_kx_k^2], \; k=1,\ldots,. \end{eqnarray} \vskip 0.3cm\noindent \begin{theorem} \label{mainth} Let (A1-A3) be verified; let $h^\ast>0$. Then given $\eta>0$ there exist $\ep^\ast >0$ and, for all $\ep\in [-\ep^\ast,\ep^\ast]$, $\Omega^\ep \subset\Omega_0$ independent of $(\hbar\in[0,\hbar^\ast]$, $\eta)$ and $\om(\hbar,\ep)\in\Omega^\ep$, such that if $|\alpha\hbar|<\eta$ the spectrum of $H(\ep)$ is given by the quantization formula \be \label{quantiz} E_\alpha(\hbar,\ep)={\cal E}(\hbar;\ep)+\la \om(\hbar,\ep),\alpha\ra\hbar+\frac12|\om(\hbar,\ep)|\hbar+\ep {\cal R}(\alpha\hbar,\hbar;\ep). \ee Here: \par\noindent 1. ${\cal E}(x;\ep):[0,h^\ast]\times [-\ep^\ast,\ep^\ast]\to\R$ is continuous in $x$ and analytic in $\ep$, with ${\cal E}(x,0)=0$, ${\cal E}(0;\ep)=0$; \par\noindent 2. $\om(x;\ep): [0,h^\ast]\times [-\ep^\ast,\ep^\ast]\to\R$ is continuous in $x$ and analytic in $\ep$ with $\om(x;0) = \om$. \par\noindent 3. ${\cal R}(x,y,\ep): \overline{\R}_+^l\times [0,h^\ast]\times [-\ep^\ast,\ep^\ast]\to\R$ is continuous in $(x,y;\ep)$ and such that \be \label{resto} |{\cal R} (x,y;\ep)|=O(|x|^2), \ee uniformly with respect to $(y,\epsilon)$. \par\noindent 4. $|\Omega^\ep-\Omega_0|\to 0$ as $\ep\to 0$. \end{theorem} \par\noindent The uniformity in $\hbar$ of the estimates needed to prove Theorem 1.1 yields in this particular setting a formulation of Kolmogorov's theorem equivalent to that of \cite{BGGS}: \begin{corollary} \label{mainc} Let $\ep^\ast$, $\Omega^\ep$, ${\cal E}(x;\ep)$, $\om(x;\ep)$ be as above. Then $\forall\,\ep$ there is an analytic canonical transformation $(x,\xi)=\psi_{\ep} (I,\phi)$ of $\R^{2l}$ onto $\overline{\R}_+^l\times\T^l$ such that \be \label{classico} (p_\ep \circ\psi)(I,\phi)={\cal E}(\ep)+\langle \om(\ep),I \rangle +\ep \tilde{\cal R}(I,\phi;\ep) \ee Here ${\cal E}(\ep):={\cal E}(0;\ep)$, $\om(\ep):=\om(0;\ep)\in\Om^\ep$; $\tilde{\cal R}(I,\phi;\ep)=O(I^2)$ as $I\to 0$ uniformly in $\phi$. \end{corollary} {\bf Remarks} \begin{enumerate} \item The form (\ref{classico}) of the Hamiltonian entails that a quasi periodic-motion with diophantine perturbed frequency $\om(\ep)\in\Om^\ep$ exists on the perturbed torus $I=0$; equivalently, a quasi periodic motion with frequency $\omega(\ep)\in\Omega^\ep$ exists on the unperturbed torus with parametric equations $(x,\xi)=\psi_{\ep}(0,\phi)$. Making $I=\alpha\hbar$ (\ref{quantiz}) represents the quantization of the r.h.s. of (\ref{classico}). In the formulation of \cite{BGGS} a quasi periodic motion with the unperturbed frequency $\omega\in\Omega$ exists on an unperturbed torus with parametric equations $(x,\xi)=\psi_{\ep}(0,\phi)$. The selection of the diophantine frequency within $\Omega$ depends here on $\ep$ because of the isochrony of the Hamiltonian flow gene\-ra\-ted by $p_0$. \item KAM theory (see e.g. {Ko}, \cite{AA}, \cite{Mo}) was first introduced in quantum mechanics in \cite{DS} to deal with quasi-periodic Schr\"odinger operators. For its applications to the Floquet spectrum of non-autonomous Schr\"odinger operators see \cite{BG} and re\-fe\-rences therein. Its first application to generate quantization formulas for $\hbar$ fixed goes back to \cite{Be} for operators in $L^2(\T^l)$ and to \cite{Co} for non-autonomous perturbations of the harmonic oscillators. A uniform quantum version of the Arnold version has been obtained by Popov\cite{Po2}, within a quantization different from the canonical one. The related method of the quantum normal forms also yields (much less explicit) quantization formulas with remainders of order $O(\hbar^\infty)$, $O(e^{-1/\hbar^a}), 0dz. $$ Given $\rho>0, \sigma>0$, define the norm \be \label{norma} \|f\|_{\rho,\sigma}:=\sum_{k\in\Z^l}e^{\rho|k|}\int_{\R^{2l}}| \widehat{\tilde{f}_k}(s)| e^{\sigma |s|}\,ds. \ee \begin{definition} Let $\rho>0, \sigma>0$. Then ${\cal F}_{\rho,\sigma}:=\{ f:\R^{2l}\to\C\,|\,\|f\|_{\rho,\sigma}<+\infty\}$. \end{definition} \noindent {\bf Remarks}. \begin{enumerate} \item If $f\in {\cal F}_{\rho,\sigma}$ then $f$ is analytic on $\R^{2l}$, and extends to a complex analytic function on a region ${\cal B}_{\rho,\sigma}\subset\C^{2l}$ of the form ${\cal B}_{\rho,\sigma}:=|\Im z_i|\leq a_i|\Re z_i|$, with suitable $a_i$. \item $F:=Op^W_hW(f)$ is a trace-class, self-adjoint $\hbar$-pseudodifferential operator in $L^2(\R^l)$ if $f\in {\cal F}_{\rho,\sigma}$. Let $\widehat{f}(s)$ be the Fourier transform of $f$. Since $\ds \|\widehat{f}\|_{L^1}\leq \|f\|_{\rho,\sigma}$, we have \be \label{normaL^2} \|F\|_{L^2\to L^2}\leq \int_{\R^{2l}}|\widehat{f}(s)|\,ds\equiv \|\widehat{f}\|_{L^1}, \qquad \|F\|_{L^2\to L^2}\leq \|f\|_{\rho,\sigma}. \ee \item v We introduce also the space ${\cal F}_{\sigma}$ of all functions $f:\R^{2l}\to \C$ such that $$ \|g\|_{\sigma}:=\int_{\R^{2l}}|\widehat{g}(s)|e^{\sigma |s|}\,ds < +\infty. $$ Obviously if $f\in {\cal F}_{\sigma}$ then $f$ is analytic on $\R^{2l}$, and extends to a complex analytic function in the multi-strip ${\cal S}:=\{z\in C^{2l}|\,|\Im z_i|<\sigma\}$. \vskip 0.2cm\noindent \item Example of $f\in {\cal F}_{\rho,\sigma}$: $\ds f(x,\xi)=P(x,\xi)e^{-(|x|^2+|\xi|^2)}$, $P(x,\xi)$ any polynomial. \end{enumerate} The starting point of the proof is represented by the first step of the Kolmogorov iteration, and is summarized in the following \begin{proposition} \label{prop1} Let $\om\in\Omega_0$. Then, for any $00$ such that, for any eigenvector $\psi_\alpha$ of $P_0(\om)$: \be \label{resto1bis} |\la\psi_\alpha, R_1(\ep)\psi_\alpha\ra|\leq D_1(|\alpha|\hbar)^2. \ee 3. $\forall\,K>0$ with $\ds (1+K^\tau)<\frac{\gamma}{\ep\|q_0\|_{\rho,\sigma}}$ $\exists$ $\Omega_1\subset\Omega_0$ closed and $d_1>1$ independent of $K$ such that \be \label{Omega1} |\Omega_0-\Omega_1|\leq \gamma (1+1/K^{d^1}). \ee Moreover if $\om_1\in\Omega_1$ then (\ref{Diofanto}) holds with $\gamma$ replaced by \be \label{g1} \gamma_1:=\gamma-\ep\|q_0\|_{\rho,\sigma}(1+K^\tau). \ee \end{proposition} {\bf Proof} To prove Assertion 1 we first recall some relevant results of \cite{BGP}. \begin{lemma}[Lemma 3.6 of \cite{BGP}] \label{omologico} Let $g\in\F$. Then the homological equation, \be \label{qhom} \{p_0,w\}+{\cal N}=g,\qquad \{p_0,{\cal N}\}=0 \ee admits the analytic solutions \be \label{soluzioni} \label{Zg} {\cal N}:=\tilde{g}_0; \qquad w:=\sum_{k\neq 0}\frac{\tilde{g}_k}{i\la\om,k\ra}, \ee with the property ${\cal N}\circ\Psi_{\phi}={\cal N}$. Equivalently, ${\cal N}$ depends only on $I_1,\ldots,I_l$. Moreover, for any $d<\rho$: \be \label{stimaom} \|{\cal N}\|_{\rho,\sigma}\leq \|g\|_{\rho,\sigma}; \quad \|w\|_{\rho-d,\sigma} \leq c_{\Psi}\frac{\|g\|_{\rho,\sigma}}{d^{\tau}}; \qquad c_{\Psi}:=\left(\frac{\tau}{e}\right)^{\tau}\frac{1}{\gamma}. \ee \end{lemma} Given $(g,g^{\prime})\in\F$, let $\{g,g^{\prime}\}_M$ be their Moyal bracket, defined as $$ \{g,g^{\prime}\}_M=g\# g^{\prime}-g^{\prime}\#g, $$ where $\#$ is the composition of $g, g^{\prime}$ considered as Weyl symbols. We recall that in Fourier transform representation, used throughout the paper, the Moyal bracket is (see e.g. \cite{Fo}, $3.4$): \be \label{twisted} (\{g,g^{\prime}\}_M)^{\wedge}(s)= \frac{2}{\hbar'}\int_{\R^{2n}}\widehat{g}(s^1) \widehat{g^{\prime}}(s-s^1) \sin{\left[{\hp}(s-s^1)\wedge s^1/{2}\right]}\,ds^1, \ee where, given two vectors $s=(v,w)$ and $s^1=(v^1,w^1)$, $s\wedge s^1:=\la w,v_1\ra-\la v,w_1\ra$.\par\noindent We also recall that $\{g,g^{\prime}\}_M=\{g,g^{\prime}\}$ if either $g$ or $g^{\prime}$ is quadratic in $(x,\xi)$. \begin{lemma}[Lemmas 3.1 and 3.3 of \cite{BGP}] \label{stimeM} Let $g\in{\cal F}_{\sigma}$, $g^{\prime}\in{\cal F}_{\sigma-\delta}$. Then: \par\noindent 1. $\forall\,0<\delta^{\prime}<\sigma-\delta$: \be \label{stimaM} \|\{g,g^{\prime}\}_M\|_{\sigma-\delta-\delta^{\prime}} \leq \frac{1}{e^2\delta^{\prime}(\delta+\delta^{\prime})}\|g\|_{\sigma} \|g^{\prime}\|_{\sigma-\delta}. \ee 2. Let $g\in\F$ and $g^{\prime}\in{\cal F}_{\rho,\sigma-\delta}$. Then, for any positive $\delta^{\prime}<\sigma-\delta$: \be \label{stimaM1} \|\{g,g^{\prime}\}_M\|_{\rho,\sigma-\delta-\delta^{\prime}}\leq \frac{1}{e^2\delta^{\prime}(\delta+\delta^{\prime})} \|g\|_{\rho,\sigma}\, \|g^{\prime}\|_{\rho,\sigma-\delta}. \ee \end{lemma} \vskip 0.2cm As a simple corollary of Lemmas \ref{omologico} and \ref{stimeM}, we find: \begin{lemma}[Lemma 3.4 of \cite{BGP}] \par\noindent Let $g\in{\cal F}_{\rho,\sigma}$, $w\in{\cal F}_{\rho,\sigma}$. \par\noindent 1. Define $$ g_r:=\frac{1}{r}\{w,g_{r-1}\}_{M}, \qquad r\geq 1; \;\;g_0:=g. $$ Then $g_r\in{\cal F}_{\rho,\sigma-\delta}$ for any $0<\delta<\sigma$, and the following estimate holds \be \label{stimaindiv} \|g_r\|_{\rho,\sigma-\delta}\leq \left(\delta^{-2}\|w\|_{\rho,\sigma}\right)^r \|g\|_{\rho,\sigma}. \ee 2. Let $g\in{\cal F}_{\rho,\sigma}$, and $w$ be the solution of the homological equation (\ref{qhom}). Define the sequence $p_{r0}: r=0,1,\ldots$ as follows: $$ p_{00}:=p_0; \qquad p_{r0}:=\frac{1}{r}\{w,p_{r-10}\}_M, \;r\geq 1. $$ Then, for any $0l-1$ we can write $$ \left|\bigcup_{|k|\geq K}{\cal T}_k\left(\frac{\gamma_1}{|k|^\tau}\right)\right|\leq \sum_{|k|\geq K}\frac{\gamma_1}{|k|^{\tau+1}} <\frac{\gamma_1}{K^{d_1}}. $$ Since $\ds |\la\om_1(\ep),k\ra|\geq \gamma_1/|k|^\tau$ by construction when $|k|\leq K$, the proposition is proved. \vskip 1.0cm\noindent \section{Iteration} \setcounter{equation}{0}% \setcounter{theorem}{0}% \setcounter{proposition}{0}% \setcounter{lemma}{0}% \setcounter{corollary}{0}% \setcounter{definition}{0}% The above result represents the starting point for the iteration. To ensure convergence, we first preassign the values of the parameters involved in the iterative estimates. Keeping $\ep$, $K$, $\gamma$, $\rho$ and $\sigma$ fixed define, for $p\geq 1$: \begin{eqnarray} \label{iter1} \sigma_p &:=&\frac{\sigma}{4p^2},\quad s_p:=s_{p-1}-\sigma_p,\quad \rho_p:=\frac{\rho}{4p^2},\quad r_p:=r_{p-1}-\rho_p, \\ \label{iter2} \gamma_p&:=&\gamma_{p-1}-\frac{4\ep_p}{1+K_p^{\tau}}, \quad K_p:=pK. \end{eqnarray} where $\ep_p$ is defined in (\ref{epsp}) below. The initial values of the parameter sequences are chosen as follows: \be \label{iter3} \gamma_0:=\gamma;\quad s_0:=\sigma;\quad r_0:=\rho, \quad \ep_0=0. \ee We then have: \begin{proposition} \label{prop2} let $\om\in\Omega_0$. There exist $\ep^\ast(\gamma)>0$ and, $\forall\,p\geq 1$, a closed set $\Omega_p^\gamma\subset \Omega_0$ such that, if $|\ep|<\ep^\ast(\gamma)>0$ and $\om_p(\hbar;\ep)\in\Omega_p^\gamma$: \par\noindent 1. One can construct two sequences of unitary transformations $\{X_p\}$, $\{Y_p\}$ in $L^2(\R^l)$ with the property \begin{eqnarray} \label{equivk} X_p(P_0(\om)+\ep Q_0)X_p^{-1}=\qquad\qquad\qquad\qquad\qquad\qquad \\ \nonumber P_0(\om_p(\hbar;\ep))+\ep{\cal E}_p(\hbar;\ep)I+e^{2^p}Q_p+ \\ \nonumber \ep^{2^p}R_p(\hbar;\ep)+ \ep\sum_{s=2}^{p}Y_sR_{s-1}(\hbar)Y_s^{-1}\ep^{2^{s-2}}. \end{eqnarray} \par\noindent 2. $X_p$ and $Y_p$ have the form \begin{eqnarray} \label{unitarie} X_p=U_1U_2\cdots U_p;\\ Y_s=U_pU_{p-1}\cdots U_s. \end{eqnarray} Here $\ds U_p(\om,\ep,\hbar)=\exp{[i\ep^{2^{p-1}} W_p/\hbar}]: L^2\leftrightarrow L^2$, $W_p=W_p^\ast$ \begin{eqnarray} \label{wp} W_p=Op^W_h(w_p)\in \Phi_{r_p,s_p}, \quad Q_p(\ep,\hbar)=Op^W(q_p)\in\Phi_{r_p,s_p}, \\ \label{stimewp} \|w_p\|_{r_p,s_p}\leq \rho_p^{-2\tau}\|q_{p-1}\|_{r_{p-1},s_{p-1}}\;\quad \|q_p\|_{r_p,s_p}\leq \rho_p^{-2\tau}\sigma_p^{-2}\|q_{p-1}\|^2_{r_{p-1},s_{p-1}}, \\ {\cal E}_p(\hbar;\ep)=\sum_{s=0}^{p}{\cal N}_s(\hbar)\ep^{2^{s}}, \quad {\cal N}_s(\hbar)=(\tilde{q}_s)_0(\hbar).\qquad \end{eqnarray} \par\noindent 3. $R_s(\ep)$ is a self-adjoint semiclassical pseudodifferential operator of order $4$; $[R_s(\ep),P_0]=0$; there exist $D_{p}>0, \overline{D}_{p}>0$ such that, for any eigenvector $\psi_\alpha$ of $P_0(\om)$: \begin{eqnarray} \label{resto1ter} |\la\psi_\alpha, R_p(\ep)\psi_\alpha\ra|\leq \ D_{p}(|\alpha|\hbar)^2,\qquad\quad \\ \label{resto1ter2} |\la\psi_\alpha, \sum_{s=2}^{p}Y_sR_{s-1}Y_s^{-1}\ep^{2^{s-2}}\psi_\alpha\ra|\leq \overline{D}_{p}(|\alpha|\hbar)^2. \end{eqnarray} 4. $\forall\,K_{p-1}>0$ such that \be \label{Kp} (1+K_{p-1}^\tau)<\frac{\gamma_{p-1}}{\ep\|q_{p-1}\|_{r_{p-1},s_{p-1}}}, \ee $\exists$ $\Omega_p\subset\Omega_{p-1}$ closed and $d_p>1$ independent of $K_p$ such that \be \label{Omegap} |\Omega_p-\Omega_{p-1}|\leq \frac{\gamma_{p-1}} {1+1/(K_{p-1})^{d_p}}. \ee Moreover if $\om_p(\ep)\in\Omega_p$ then (\ref{Diofanto}) holds with $\gamma$ replaced by \begin{eqnarray} \label{gp} \gamma_p&:=&\gamma_{p-1}-\ep_p(1+K_{p-1}^\tau) \\ \label{epsp} \ep_p&:=&\ep^{2^{p-1}}\|q_{p-1}\|_{r_{p-1},s_{p-1}} \end{eqnarray} \end{proposition} {\bf Proof} \par\noindent We proceed by induction. For $p=1$ the assertion is true because we can take $W_1$, $Q_1$, $R_1$, $\om_1$, $\Om_1^\ep$, $K_1$ as in Proposition \ref{prop1}. To go from step $p-1$ to step $p$ we consider the operator \begin{eqnarray*} X_{p-1}(P_0(\om)+\ep Q_0)X_{p-1}^{-1}:=\qquad\qquad\qquad\qquad \\ P_0(\om_{p-1}(\hbar;\ep))+\ep{\cal E}_{p-1}(\hbar;\ep)I+e^{2^{p-1}}Q_{p-1} \\ +\ep^{2^{p-1}}R_{p-1}(\hbar;\ep)+ \ep\sum_{s=2}^{{p-1}}Y_sR_{s-1}(\hbar)Y_s^{-1}\ep^{2^{s-2}}. \end{eqnarray*} We have to determine and estimate the unitary map $U_p$ transforming it into the form (\ref{equivk}) via the definitions (\ref{unitarie}). With $\ds U_p=e^{i\ep W_p/\hbar}$, $W_p$ continuous and self-adjoint, we have at the $p$-th iteration step \begin{eqnarray*} U_p(P_0(\om_{p-1}+\ep^{2^{p-1}} Q_{p-1})U_p^{-1}=P_0(\om_p)+\ep^{2^{p-1}}P_p+\ep^{2^p} Q_p, \qquad\qquad \\ P_p:= Q_{p-1}+[W_p,P_0]/i\hbar, \qquad\qquad \qquad\qquad \\ Q_p:=\ep^{-2}\left( U_p(P_0(\om_{p-1})+\ep Q_0)U_1^{-1}-P_0(\om_{p-1})-\ep (Q_{p-1}+[W_p,P_0]/i\hbar)\right). \end{eqnarray*} (the explicit dependence of the frequencies on $(\hbar,\ep)$ has been omitted). We will look therefore for $W_p\in\Phi_{r_p,s_p}$ and an operator $N_p\in \Phi_{r_p,s_p}$ such that \be \label{qhomp} Q_p+[W_p,P_0]/i\hbar =N_p,\quad [N_p,P_0]=0. \ee Denoting $w_p$, ${\cal N}_p$ the (Weyl) semiclassical symbols of $W_p$, $N_p$, respectively, eq.(\ref{qhomp}) is again equivalent to the classical homological equation in $\F$ $$ \{p_0,w_p\}_M+{\cal N}_p=q_p, \qquad \{p_0,{\cal N}_p\}_M=0 $$ which once more becomes $$ \{p_0,w_p\}+{\cal N}_p=q_p, \qquad \{p_0,{\cal N}_p\}=0. $$ The existence of $w_p\in{\cal F}_{r_p,s_p}$, ${\cal N}_p\in{\cal F}_{r_p,s_p}$ with the stated properties now follows by direct application of Lemma \ref{omologico}. Expanding ${\cal N}_p$ as in the proof of Proposition \ref{prop1} and taking into account the definitions (\ref{unitarie}) we immediately check that $\ds X_pX_{p-1}(P_0(\om)+\ep Q_0)X_{p-1}^{-1}X_p$ has the form (\ref{equivk}). The estimate of $Q_p$ and the small denominator estimates follow by exactly the same argument of Proposition 2.1. The estimate (\ref{resto1ter}) is proved exactly as (\ref{resto1bis}). It remains to prove the estimate (\ref{resto1ter2}). By the inductive assumption, it is enough to prove the existence of $D^\prime_p>0$ such that $$ |\la \psi_\alpha,U_pR_{p-1}U_{p}^{-1}\psi_\alpha\ra|\leq D^\prime_p(|\alpha|\hbar)^2. $$ We only have to prove that the operator $U_pR_{p-1}U_{p}^{-1}$ is an $\hbar$-pseudo\-differential operator of order $4$ fulfilling the hypotheses of Proposition A.1, assuming by the inductive argument the validity of these properties for $R_{p-1}$. On the other hand, $\ds U_p=\exp{(i\ep^{2^{p-1}}W_p/\hbar)}$, and $W_p$ is an $\hbar$-pseudo\-differential operator of order $0$. We can therefore apply the semiclassical Egorov theorem (see e.g. \cite{Ro}, Chapter 4) to assert that $U_pR_{p-1}U_{p}^{-1}$ is again an $\hbar$-pseudo\-differential operator. Denote $\sigma (x,\xi;\ep;\hbar)$ the Weyl symbol of $U_pR_{p-1}U_{p}^{-1}$, and consider its expansion $$ \sigma(x,\xi;\ep;\hbar)=\sigma_0(x,\xi;\ep)+\sum_{j=2}^M\hbar^j\sigma_j(x,\xi;\ep )+O(h^{M+1}). $$ It is clearly enough to prove that the principal symbol $\sigma_0(x,\xi;\ep)$ has order $4$. Denote by $$ \phi(x,\xi;\ep):=\exp{[\ep^{2^p}{\cal L}_{w_p}]}(x,\xi) $$ the Hamiltonian flow on $\R^{2l}$ generated by the Hamiltonian vector field $\ds {\cal L}_{w_p}$ at time $\ds \ep^{2^p}$; here $w_p^0(x,\xi)$ is the principal symbol of $W_p$. Then $\sigma_0(x,\xi;\ep)={\cal R}_{p-1}^0(\phi(x,\xi;\ep))$ where ${\cal R}_{p-1}^0(x,\xi)$ is in turn the principal symbol of $R_{p-1}$. Now $$ \phi(x,\xi;\ep)=(x+\int_0^{\ep^{2^p}}\nabla_\xi w_p(x,\xi;\eta)\,d\eta, \xi-\int_0^{\ep^{2^p}}\nabla_x w_p(x,\xi;\eta)\,d\eta). $$ By Assumption A2 and the inductive hypothesis we know that $w_p(z)=O(|z|^2)$ as $|z|\to 0$. Hence we can write $\phi(z)=z+\epsilon r(z)$ where $r(z)=O(z), z\to 0$. This concludes the proof of Proposition 3.1. \vskip 0.3cm \noindent {\bf Proof of Theorem \ref{mainth}} \vskip 0.2cm\noindent Applying the estimates on $q_p$ in Propositions \ref{prop1} and \ref{prop2} iteratively, we have \begin{equation} \label{qp} \|q_p\|_{r_p,s_p}\leq \left(\frac{4p^2}{\rho}\right)^{2\tau p}\cdot \left(\frac{4p^2}{\sigma}\right)^{2p}\|q_0\|^{2^p}, \end{equation} whence \begin{equation} |\ep|^{2^p}\|Q_p\|_{L^2\to L^2}\leq |\ep|^{2^p}(4p^2)^{2p(\tau+1)}\rho^{-2\tau p} \sigma^{-2p} \|q_0\|^{2^p} \to 0\quad \hbox{as }p\to\infty, \end{equation} for all $|\ep|\leq \ep^\ast$ provided $\ep^\ast>0$ is small enough. At the $p$-th iteration the frequency is given by \begin{equation} \label{omk} \om_p(\hbar;\ep)=\om +\sum_{s=1}^p\nabla_I{\cal N}_s(\hbar)\ep^{2^s}. \end{equation} Since $\ds \|\nabla_z f(z)\|_{\rho-d,\sigma-\delta}\leq \frac{1}{d\delta}\|f(z)\|_{\rho,\sigma}$, by (\ref{qp}) we have \begin{equation} \label{Nk} \sum_{s=1}^p|\nabla_I{\cal N}_s(\hbar)\ep^{2^s}|\leq \sum_{s=1}^p |\ep|^{2^s}(4s^2)^{2s(\tau+1)}\rho^{-2\tau s} \sigma^{-2s} \|q_0\|^{2^s}. \end{equation} Hence the series (\ref{omk}) converges as $p\to \infty$ for $|\ep|<\ep^\ast$ if $\ep^\ast $ is small enough, uniformly with respect to $\hbar)\in [0,h^\ast]$. In the same way, the estimate (\ref{qp}) entails, by the definition (\ref{gp}), the existence of $\ds \lim_{p\to\infty}\gamma_p:=\gamma_\infty$. Let $\om(\hbar;\ep) := \lim_{p\to\infty} \om_p(\hbar\;\ep)$. Then $\om(\hbar;\ep)$ is diophantine with constant $\gamma_\infty$ by Proposition \ref{prop2}. In the same way: $$ {\cal E}(\hbar;\ep)=\sum_{s=1}^\infty{\cal N}_s(\hbar)\ep^{2^s}, \quad |\ep|<\ep^\ast. $$ Finally, let ${\cal R}(\alpha\hbar,\ep)$ be an asymptotic sum of the power series $\ds \sum_{s=2}^{\infty}Y_sR_{s-1}Y_s^{-1}\ep^{2^{s-2}}$. Then the validity of (\ref{resto}) follows by its validity term by term. This concludes the proof of Theorem \ref{mainth}. \par\noindent %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\bf Proof of Corollary 1.1} \par\noindent It is enough to illustrate the specialization of the argument of Propositions 2.1 and 3.1 to the $\hbar=0$ case. Denoting by $\ds e^{\ep {\cal L}_{w_1}}$ the canonical flow at time $\ep$ generated by the Hamiltonian vector field generated by the symbol $w_1$, we have: \begin{eqnarray} \label{U1c} e^{\ep {\cal L}_{w_1}}(p_0+\ep q_0)(x,\xi)=(p_0+\ep p_1+\ep ^2 q_1^0)(x,\xi), \qquad\qquad \\ \label{P1c} p_1:= q_0+\{w_1,p_0\}, \qquad\qquad \qquad\qquad \\ \label{Q1c} q_1^0:=\ep^{-2}\left( e^{\ep {\cal L}_{w_1}}(p_0+\ep q_0)(x,\xi)- p_0-\ep (q_0+\{w_1,p_0\})\right). \end{eqnarray} Remark that $\ds e^{\ep{\cal L}_{w_1}}(p_0+\ep q_0)(x,\xi)$ is the principal symbol of $U_1(P_0+\ep Q_0)U_1^{-1}$ by the semiclassical Egorov theorem; $p_1$ is the full, and hence principal, symbol of $P_1$ because $p_0$ is quadratic. Likewise, $q_1^0$ is the principal symbol of $Q_1$. Hence the classical definitions (\ref{U1c},\ref{P1c},\ref{Q1c}) correspond to the principal symbols of the semiclassical pseudodifferential operators $U_1(P_0+\ep Q_0)U_1^{-1}$, $P_1$, $Q_1$ defined in (\ref{U1},\ref{P1},\ref{Q1}). Therefore we can take over the homological equation (\ref{chom2}) and apply Lemma \ref{omologico} once more. This yields the same $w_1$ and ${\cal N}_1$ of Proposition 2.1. To prove the estimate (\ref{q_1}) for $q_1^0$ we write $$ q_1^0=\int_0^1e^{s\ep {\cal L}_{w_1}}\{\{p_0+\ep q_0,w_1\},w_1\}\,ds $$ Now as in \cite{BGGS}, Lemma 1, note that if $|\ep |<\ep^\ast$ and $z=(x,\xi)\in {\cal B}_{\rho-d,\sigma-\delta}$ then $e^{s{\cal L}_{w_1}}z\in {\cal B}_{\rho,\sigma}$ for $0\leq s\leq 1$ because (Lemma 2.1) $\ds \ep\|\nabla w_1\|_{\rho-d,\sigma} \leq \ep (\tau/e)c_\psi d^{-\tau}\|q_0\|_{\rho,\sigma}$. Therefore we can apply Lemma 2.3, valid a fortiori for the Poisson bracket, and, as in the proof of Proposition 2.1, get the estimate corresponding to the second one of (\ref{q_1}): \begin{eqnarray} \label{stimaq10} \|q_1^0\|_{\rho-d,\sigma-\delta} \leq \|\{\{p_0+\ep q_0,w_1\},w_1\}\|_{\rho-d,\sigma-\delta}\leq \delta^{-2} d^{-2\tau}\|q_0\|_{\rho,\sigma}^2. \end{eqnarray} Now, writing: \begin{eqnarray} \psi^1_{\ep}(x,\xi)&=&e^{\ep {\cal L}_{w_1}}(x,\xi), \quad \label{N1c} {\cal E}_1:={\cal N}_1(0); \\ \label{omega1} \om_1(\ep)&=&\om+\ep(\nabla_I{\cal N}_1)(0), \\ \label{R1c} \tilde{\cal R}_1(I,\ep)&=& {\cal N}_1(0)-\la(\nabla_I{\cal N}_1)(0),I\ra-{\cal E}_1, \end{eqnarray} we can sum up the above argument by writing (compare with (\ref{passo1bis})) \be \label{passo1c} \psi^1_{\ep}\circ (p_0+\ep q_0)={\cal E}_1+\la\om_1(0;\ep),I\ra+\ep^2q_1(I,\phi)+\ep {\cal R}_1^0(I,\ep) \ee where ${\cal R}^0_1$ is the principal symbol of $R_1$. Morover, Assertion 3 of Proposition 2.1 holds without change. \newline Let us now specialize the iterative argument of Proposition 3.1. First, the parameters defined in (\ref{iter1},\ref{iter2},\ref{iter3}) remain unchanged. Then: \newline 1. The construction of the two sequences of canonical transformations \begin{eqnarray} \chi^p_{\ep}&=&\psi^1_{\ep}\circ\psi^2_{\ep}\cdots\circ \psi^p_{\ep}, \quad p=1,2,\ldots \\ \zeta^s_{\ep}&=&\psi^p_{\ep}\circ\psi^{p-1}_{\ep}\cdots\circ \psi^s_{\ep}, \quad p=1,2,\ldots \\ \psi^s_{\ep}(x,\xi)&=&e^{\ep {\cal L}_{w^0_s}}(x,\xi) \end{eqnarray} such that \begin{eqnarray} \label{equivkc} \psi^p_{\ep,I_0}\circ (p_0+\ep q_0)=\qquad\qquad\qquad\qquad\qquad \\ \nonumber \la\om_p(0,\ep),I\ra+{\cal E}_p(\ep)+e^{2^p}q^0_p+\ep^{2^p}{\cal R}^0_p+ \ep\sum_{s=2}^{p}\psi^s_{\ep}\circ {\cal R}_{s-1}^0\ep^{2^{s-2}}. \end{eqnarray} follows as in the above argument valid for $p=1$. Here $w_s^0$, $q^0_p$, ${\cal R}^0_s$ are the principal symbols of the semiclassical pseudodifferential operators $W_s$, $Q_p$ and $R_s$, once reexpressed on the $(x,\xi)$ canonical variables via, with $\om_p$ in place of $\om_1$. Morover: \begin{eqnarray} {\cal E}_p(\ep)=\sum_{s=0}^{p}{\cal N}_s(0)\ep^{2^{s}}, \quad {\cal N}_s(0)=(\tilde{q}_s^0)_0(0).\qquad \\ \om_p(\ep)=\om+\sum_{s=0}^{p}\om_s(0)\ep^{2^{s}},\quad \om_s(0)=\nabla_I{\cal N}_s(0) \end{eqnarray} 2. The estimates (\ref{stimewp}) are a fortiori valid with $w^0_p$, $q^0_p$ in place of $w_p$, $w_p$; as a consequence, (\ref{Omegap}) holds unchanged together with the definitions (\ref{Kp},\ref{gp},\ref{epsp}). Hence the uniform estimate (\ref{qp}) allows us to set $\hbar=0$ in (\ref{omk},\ref{Nk}). \par\noindent 3. Finally, remark that ${\cal R}^0_s(I)=O(I^2), s=1,\ldots,p$. Now the estimate $\psi_s^\ep{\cal R}_s(I)=O(I^2)$ as $I\to 0$ follows by exactly the same argument of Proposition 3.1 after rexpression on the canonical variables $(x,\xi)$. \vfill\eject %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% %\begsection{Appendix} \noindent {\bf\Large Appendix} \vskip 0.3cm\noindent \setcounter{equation}{0}% \setcounter{theorem}{0}% \setcounter{proposition}{0}% \setcounter{lemma}{0}% \setcounter{corollary}{0}% \setcounter{definition}{0}% To establish the remainder estimate (\ref{resto}) the key fact is that vanishing of a symbol at the origin $(x,\xi)=0$ implies bounds on harmonic oscillator matrix elements that are uniform in $\hbar$. No analyticity of the symbol is required for this result, so we will state and prove it in somewhat greater generality, using the following semiclassical symbol class defined in Shubin \cite{Sh}: $$ \Sigma^{m,\mu} = \{f\in C^\infty({\R}^{2l}\times (0,\epsilon]):\; |\partial^\gamma_z f(z,\hbar)| \le C_\gamma \langle z\rangle^{m-|\gamma|} \hbar^{\mu}\}, $$ where $z = (x,\xi)$, here considered a real variable, and $\langle z\rangle = \sqrt{1+|z|^2}$. For future reference we note that Proposition A.2.3 of \cite{Sh} gives the result: %\begin{equation} %\label{hbound} $$ \forall f \in \Sigma^{0,\mu}, \quad \Vert Op^{W}_\hbar(f) \Vert_{L^2} \le C(f) \hbar^\mu, \eqno{(A.1)} $$ %\end{equation} for all $\hbar\in(0,\epsilon]$. The matrix elements in question are most easily computed in Bargmann space, with the remainder operator written as a Toeplitz operator. Since these are anti-Wick ordered, we first must consider the translation from Weyl symbols to anti-Wick (for these notions, see e.g. \cite{BS}). Denoting by $Op^{AW}_\hbar(f)$ the anti-Wick quantization of a symbol $f\in \Sigma^{m,\mu}$, the correspondence is given by the action of the heat kernel on the symbol: $$ Op^{AW}_\hbar(f) = Op^{W}_\hbar(e^{\hbar\Delta/4} f), \eqno{(A.2)} $$ where $\Delta = \Delta_z = \partial_x\cdot\partial_x + \partial_\xi\cdot\partial_\xi$. To begin, we show that the Weyl symbol of an anti-Wick operator is given by formal expansion of the heat kernel up to a remainder. \vskip 0.2cm\noindent {\bf Lemma A1} {\it For $f,g\in \Sigma^{m,\mu}$, suppose that $Op^{AW}_\hbar(g) = Op^{W}_\hbar(f)$. Then for all $n\ge 1 $,} $$ f - \sum_{k=0}^{n-1}\frac{1}{k!} \left(\frac{\hbar}{4}\Delta\right)^k g \in \Sigma^{m-2n,\mu+n}. $$ {\bf Proof.} According to (A.2), $$ f(z,\hbar) = \frac{1}{(\pi\hbar)^l} \int e^{-|z-w|^2/\hbar} g(w) dw. $$ In this expression we will expand $g(w)$ in a Taylor series centered at $w=z$: $$ g(w,\hbar) = \sum_{|\alpha|<2n} \frac{1}{\alpha!} \partial^\alpha g(z,\hbar) (w-z)^\alpha + r(w,z,\hbar), $$ where $$ r(w,z) = \sum_{|\alpha|=2n} c'_\alpha (w-z)^\alpha \int_0^1 (1-t)^{2n-1} \partial^\alpha g(z+t(w-z)) \>dt. $$ Thus, $$ f(z,\hbar) = \sum_{|\alpha|<2n} c_\alpha \partial^\alpha g(z,\hbar) + r(z,\hbar), $$ where $$ c_\alpha = \frac{1}{(\pi\hbar)^l} \frac{1}{\alpha!} \int w^\alpha e^{-|w|^2/\hbar}\>dw, $$ and $$ r(z,\hbar) = \sum_{|\alpha|=2n} c''_\alpha \hbar^{-l} \int\int_0^1 (w-z)^\alpha e^{-|z-w|^2/\hbar} (1-t)^{2n-1} \partial^\alpha g(z+t(w-z)) \>dt\>dw. $$ Note that $c_\alpha = 0$ for $|\alpha|$ odd, and for any integer $k$ $$ \sum_{|\alpha| = 2k} c_\alpha \partial^\alpha g = \frac{1}{k!} \left(\frac{\hbar}{4}\Delta\right)^k g. $$ The lemma is thus reduced to the claim that $r(z,\hbar) \in \Sigma^{m-2n,\mu+n}$. To see this, we change variables by $w' = (w-z)/\sqrt{\hbar}$ to write $$ r(z,\hbar) = \sum_{|\alpha|=2n} c''_\alpha \hbar^{n} \int\int_0^1 w^\alpha e^{-|w|^2} (1-t)^{2n-1} \partial^\alpha g(z+tw\sqrt{\hbar}) \>dt\>dw. $$ We must estimate the derivatives: $$ \partial^\gamma r(z,\hbar) = \sum_{|\alpha|=2n} c''_\alpha \hbar^{n} \int\int_0^1 w^\alpha e^{-|w|^2} (1-t)^{2n-1} \partial^\beta g(z+tw\sqrt{\hbar}) \>dt\>dw, $$ where $|\beta| = 2n+|\gamma|$. This integral for $\partial^\gamma r$ we then split into two pieces according to the domain of the $w$-integral, $I'_{\alpha,\beta}:|w|<|z|/2$ and $I''_{\alpha,\beta}:|w|>|z|/2$. The assumption $g \in \Sigma^{m,\mu}$ implies an estimate $$ |I'_{\alpha,\beta}| \le C \langle z\rangle^{m-2n-|\gamma|} \hbar^{n+\mu}. \eqno{(A.3)} $$ The second term is taken care of by the exponential factor in $|w|$: $$ |I''_{\alpha,\beta}| < C_l \hbar^l \langle z\rangle^{-l}, \quad\forall l. $$ Therefore $\partial^\gamma r$ satisfies an estimate of the form (A.3) for any $\gamma$, and hence $r \in \Sigma^{m-2n,\mu+n}$.\quadratino \vskip 0.2cm\noindent Our application of Lemma A.1 will be specifically to operators of order 4: \vskip 0.2cm\noindent {\bf Lemma A.2} {\it For $g \in \Sigma^{4,0}$, $$ Op^{W}_\hbar(g) = Op^{AW}_\hbar(g) - \frac{\hbar}{4} Op^{AW}_\hbar(\Delta g) + R(\hbar), $$ where $\Vert R(\hbar)\Vert_{L^2} \le C\hbar^2$ .} \vskip 0.2cm\noindent {\bf Proof.} Let $\sigma(A)$ denote the Weyl symbol of the $\hbar$-pseudodifferential operator $A$. Applying Lemma A.1 with $n=2$ gives $$ \sigma(Op^{AW}_\hbar(g)) = g + \frac{\hbar}{4} \Delta g + r_1, $$ and $$ \frac{\hbar}{4} \sigma(Op^{AW}_\hbar(\Delta g)) = \frac{\hbar}{4} \Delta g + r_2, $$ where $r_1, r_2\in \Sigma^{0,2}$. Noting that $$ Op^{W}_\hbar(g) - Op^{AW}_\hbar(g) + \frac{\hbar}{4} \sigma(Op^{AW}_\hbar(\Delta g)) = Op^{W}_\hbar(r_1-r_2), $$ the bound on $R(\hbar)$ follows from (A.1).\quadratino \vskip 0.2cm\noindent The point of introducing anti-Wick symbols is to exploit the Bargmann space representation of the harmonic oscillator. The Bargmann space is (see e.g. \cite{BS}) $$ \mathcal{H}_\hbar = L^2_{hol}({\C}^l, e^{-|z|^2/\hbar} \>dzd\bar z). $$ The Bargmann transform is an isomorphism $\mathcal{B}: L^2({\R}^l) \to \mathcal{H}_\hbar$, defined so as to intertwine anti-Wick operators with Toeplitz operators: $$ \mathcal{B}\circ Op^{AW}_\hbar(f)\circ\mathcal{B}^{-1} = T_\hbar(f). $$ The Toeplitz operator $T_\hbar(f):\mathcal{H}_\hbar\to \mathcal{H}_\hbar$ is defined for $f\in\Sigma^{m,\mu}$ by $$ T_\hbar(f) = \Pi_\hbar M(f), $$ where $M(f)$ denotes the multiplication operator on $L^2({\C}^l, e^{-|z|^2/\hbar} \,dzd\bar z )$ (identifying ${\R}^{2l} = {\C}^l$ by $z = x+i\xi$), and $\Pi_\hbar:L^2({\C}^l, e^{-|z|^2/\hbar} \>dzd\bar z) \to \mathcal{H}_\hbar$ is orthogonal projection onto the holomorphic subspace. The main result of this Appendix is the following matrix element estimate: \vskip 0.2cm\noindent {\bf Proposition A.1} {\it Let $\{\psi_\alpha\}$ be the normalized eigenstates of the standard harmonic oscillator on $L^2({\R}^{l})$. Suppose $f\in \Sigma^{4,0}$ satisfies $$ f(z,\hbar) = \sum_{|\gamma|=4} z^\gamma g_\gamma(z,\hbar), $$ where $\sup|\partial^\beta g_\gamma|\le M$ for all $|\beta|\le 2$. Then $$ |\langle \psi_\alpha, Op^W_\hbar(f) \psi_\alpha\rangle| \le C M (|\alpha| \hbar)^{2} $$ for all $\alpha, \hbar$, where $C$ depends only on the dimension.} \vskip 0.2cm\noindent {\bf Proof.} Under the Bargmann transform the harmonic oscillator eigenstates have a particularly convenient form: $$ (\mathcal{B}^{-1} \psi_\alpha) (z) = (\pi^l \hbar^{|\alpha|+l} \alpha!)^{-1/2} \cdot z^\alpha. $$ Using Lemma A.1 we write $$ Op^W(f) = Op^{AW}_\hbar(f) - \frac{\hbar}{4} Op^{AW}_\hbar(\Delta f) + R(\hbar), \eqno{(A.4)} $$ where $|\langle R(\hbar)\rangle| \le C\hbar^2$. Consider the matrix element of the first term on the right-hand side of (A.4). In Bargmann space this becomes $$ \langle \psi_\alpha, Op^{AW}_\hbar(f) \psi_\alpha\rangle = \frac{1}{\pi^l \hbar^{|\alpha|+l} \alpha!} \int \bar z^\alpha f(z,\hbar) z^\alpha e^{-|z|^2/\hbar} \>dzd\bar z. $$ Writing $f$ as a sum over $z^\gamma g_\gamma$ with $|\gamma|=4$, the estimate for a particular $\gamma$ is straightforward: \begin{eqnarray*} |\langle \psi_\alpha, Op^{AW}_\hbar(z^\gamma g_\gamma) \psi_\alpha\rangle| &\le& M \frac{1}{\pi^l \hbar^{|\alpha|+l} \alpha!} \int \left|z^{\alpha}\right|^2 |z|^4 e^{-|z|^2/\hbar} \>dzd\bar z \\ &=& M \hbar^{2} (|\alpha|+l)(|\alpha|+l+1). \end{eqnarray*} The second term on the right in (A.4) is handled in a similar way. By assumption we can write $\Delta f = \sum_{|\eta|=2} z^\eta h_\eta(z,\hbar)$, where $\sup|h_\eta| \le 12M$. 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