\documentclass[12pt]{article} \input{amssym.def} \input{amssym} \setlength{\textwidth}{15.5cm} \setlength{\textheight}{22.7cm} %\setlength{\textwidth}{15.7cm} %\setlength{\textheight}{22.7cm} \setlength{\topmargin}{-1.0cm} \setlength{\oddsidemargin}{-1mm} \setlength{\evensidemargin}{-1mm} \pagestyle{myheadings} \newtheorem{theorem}{Theorem} \newtheorem{proposition}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{corollary}{Corollary} \newtheorem{definition}{Definition} %\renewcommand{\thesection} {\arabic{section}} %\renewcommand{\thetheorem} {\thesection.\arabic{theorem}} %\renewcommand{\theproposition} {\thesection.\arabic{proposition}} %\renewcommand{\thelemma} {\thesection.\arabic{lemma}} %\renewcommand{\thedefinition} {\thesection.\arabic{definition}} %\renewcommand{\thecorollary} {\thesection.\arabic{corollary}} %\renewcommand{\theequation} {\thesection.\arabic{equation}} \def\non#1{\check{#1}} \def\quadratino{ \hfill\vbox{\hrule\hbox{\vrule\vbox to 7 pt {\vfill\hbox to 7 pt {\hfill\hfill}\vfill}\vrule}\hrule}\par} \def\R{\Bbb R} \def\Z{\Bbb Z} \def\N{\Bbb N} \def\T{\Bbb T} \def\C{\Bbb C} \def\A{{\cal A}} \def\res{{\cal R}} \def\etab{\overline{\eta}} \def\ha{Ha\-mil\-to\-nian} \def\Sc{Schr\"o\-din\-ger} \def\hp{{\hbar^{\prime}\null}} \def\hpp{{h^{\prime}}} \def\zbar{\overline{z}} \def\wbar{\overline{w}} \def\la{\langle} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\ra{\rangle} \def\ds{\displaystyle} \def\chit{\tilde{\chi}} \def\om{\omega} \def\Om{\Omega} \def\ep{\epsilon} \def\gk{\tilde{g}_k} \def\op#1{{\rm Op}^{W}\left(#1\right)} \def\norma#1{\left\Vert#1\right\Vert} \def\snorma#1{\left\langle\left\Vert#1\right\Vert\right\rangle} \def\imma{{\rm Im}} \def\quadratino{ \hfill\vbox{\hrule\hbox{\vrule\vbox to 7 pt {\vfill\hbox to 7 pt {\hfill\hfill}\vfill}\vrule}\hrule}\par} \def\fou{{\cal F}_{\hp}} \begin{document} \baselineskip=18pt \date{} \begin{center} %{\Large\bf\sc Normal forms and quantization formulae} {\large\bf ADDENDUM TO: NORMAL FORMS AND QUANTIZATION FORMULAE}\vskip 13pt Dario Bambusi\footnote{ Dipartimento di Matematica, Universit\`{a} di Milano, 20133 Milano (Italy). (bambusi@mat.unimi.it)}, Sandro Graffi\footnote{ Dipartimento di Matematica, Universit\`{a} di Bologna, 40127 Bologna (Italy). (graffi@dm.unibo.it)} and Thierry Paul\footnote{DMI, \`Ecole Normale Superieure, 75776 Paris (France). (paulth@dmi.ens.fr)} \end{center} \begin{abstract} \noindent {\noindent The main result of \cite{BGP} is reformulated in terms of the standard quantization formula constructed quantizing the Lie transformation.} \end{abstract} %\vskip 1cm % % %\section{Introduction and statement of the results} Theorem 1.1 of \cite{BGP} yields a quantization formula modulo exponentially small terms in $\hbar$ for the semi-excited eigenvalues of the \Sc\ operator $Q(\hbar):=-\hbar^2\Delta/2 +V$ on $L^2(\R^n), n\geq 1$ whose symbol $q(x,\xi):=\xi^2/2+V(x)$ is a non-integrable classical Hamiltonian. It extends a formula obtained by Sj\"ostrand \cite{Sj} quantizing the classical normal form of $q$ modulo terms of order $\hbar^{\infty}$; however it has an {\it a priori} different form since it is generatated by an {\it a priori} different normal form for the symbol $q$. Here we solve the question left open in \cite{BGP} of the relation between these two normal forms by proving their coincidence. The main result of [BGP] can thus be reformulated, within the same assumptions and notations (in particular, $\ell$ is the Gevrey order of $V\in C^{\infty}(\R^n)$, $\tau$ the power in the diophantine condition fulfilled by the frequencies $\ds \omega_i:=\partial^2_iV(0)$) as follows: \begin{theorem} \label{maintheorem} Assume (A1-A3), and let once more $\ds b:=[(\tau+2)(2\ell+3)]^{-1}$. Then: \vskip 0.2cm\noindent There exist $h_*>0, \ep_*>0, A>0, B>0$ (independent of $\hbar$ and $\ep$) and a smooth real valued function ${\cal W}(I_1,\ldots,I_n;\hbar,\ep)$ with asymptotic expansion in $\hbar$: \be \label{FF} {\mathcal W}(I_1,\ldots,I_n;\hbar,\ep)\sim {\cal W}_0(I_1,\ldots,I_n;\ep)+\hbar {\cal W}_1(I_1,\ldots,I_n;\ep)+\ldots \ee such that $\forall \ep\leq \ep_*, \hbar\leq \hbar_*\epsilon^2$ the eigenvalues $\lambda(k,\hbar,\ep)$ of $Q(\hbar)$ in $[0,\ep^2]$ have the representation \be \label{qfe} \lambda(k,\hbar,\ep)=\la({ k}+1/2),\om\ra\hbar+{\cal W}\left(({ k} +1/2){\hbar}, {\hbar},\ep\right)+\ep^2 O\left(e^{-({B\epsilon^2}/{\hbar})^{1/(\ell+1)}} \right)+O\left(\ep^3 e^{-{A}/{\ep^b}}\right) \ee Here ${\cal W}_0$ is the classical Birkhoff normal form of $q$ truncated at a suitable order. \end{theorem} {\bf Remark}. \vskip 0.1cm\noindent The main difference with the original formulation of [BGP], Theorem 1.1, is that the quantization formula (\ref{qfe}) has the same form as that of [Sj], Theorem 0.1, even in the most general case. We will indeed explicitly reabsorb the rescaling of $\hbar$ introduced in \cite{BGP} which is responsible for the possible noncoincidence of the two normal forms for $q$. Thus (\ref{qfe}) extends the validity range of the quantization formula of [Sj] to a larger class of eigenvalues (Corollary 1.1), providing in addition an explicit error estimate. Choosing $\ep^2=\varphi(\hbar)$ as in \cite{BGP} one actually obtains that the eigenvalues of $Q(\hbar)$ in $[0,\varphi(\hbar)]$ are given by \be \lambda(k,\hbar)=\la({ k}+1/2),\om\ra\hbar+{\cal W}\left(({ k} +1/2){\hbar}, {\hbar},\varphi(\hbar)\right)+O\left(e^{-A/\varphi(\hbar)^{b/2}}\right) \ee and the choice of $\varphi(\hbar)$ can be optimized as in Corollary 1.1. \vskip 0.1cm\noindent {\bf Proof}. We outline the additions and modifications with respect to the arguments of \cite{BGP}, whose formulas and statements are directly referred to here. \par\noindent 1. Consider once more the (Weyl) pseudodifferential operator $P_R(\hp,\ep)$ of Gevrey symbol $p_{\ep,R}(x,\xi):=p_0(x,\xi)+a_{\ep,R}(x,\xi)$, $\hp=\hbar/\ep^2$. Perform the inverse unitary rescaling (1.3). The result is the pseudodifferential operator $P'(\hbar,\ep)$ of Weyl symbol $p_\epsilon'(x,\xi):=p_0(x,\xi)+\ep^3a_{\ep,R}(x/\ep, \xi/\ep)$. We omit the easy verification. \par\noindent 2. As in Section 3, we approximate $p'_\epsilon$ with an analytic symbol fulfilling the conditions of Proposition 3.1. The modifications of the argument leading from Proposition 3.1 to Theorem 3.1 are as follows: define $a'(x,\xi):=\epsilon^3a_{\epsilon,R}(x/\epsilon,\xi/\epsilon)$, and recalling formulas (3.7-11) define also the analogous of the norm $\snorma{a_{\epsilon}}$ of (3.33) \be \label{pp} \snorma{a'}_{\mu,\mu\epsilon^\beta}:=\sum_{k\in\Z^n}e^{\mu|k|^\beta} \int_{\R^{2n}} |\widehat{\tilde a\null'}_k(s)|e^{\mu(|s|\epsilon)^\beta}ds\ , \ee Performing the change of variables $\epsilon s=s^{\prime}$ in (3.33) we immediately get $\ds \snorma{a'}_{\mu,\mu\epsilon^\beta}= \epsilon^3\snorma{a_{\epsilon}}_{\mu,\mu}$ whence $$ \snorma{a'}_{\mu,\mu\epsilon^\beta}=\epsilon^3\snorma{a_{\epsilon,R}}\leq C_{11}\epsilon^3\ , $$ where $C_{11}$ is the constant whose existence is ensured by Corollary 3.1. \par\noindent 3. Define now $a'_{\Lambda,K}(z)$ by (3.38) (up to the obvious modifications induced by the replacement of $a_{\ep}$ by $a^{\prime}$). Then we repeat the argument in the proof of Theorem 3.1 to get $$ \norma{\op{a'}-\op{a'_{\Lambda,K}}}_{L^2\to L^2}\leq \epsilon^3 C_{11}\left[e^{-\mu K^\beta}+e^{-\mu(\Lambda\epsilon)^{\beta}} \right]\ . $$ 4. Now we approximate $p'_{\epsilon}$ by $p_0+a'_{\Lambda,K}$, and we apply to it Proposition 3.1. Before doing that we reformulate Proposition 3.1 in a more convenient way (the proof does not change): \par\noindent {\bf Proposition 3.1} {\it Consider the symbol $p_0+f$, with $f\in\A_{\rho,\sigma}$ for some $\rho>0,\sigma>0$. Denote $P_f:=P_0(\hbar)+F,\,F:=Op^W(f)$ the corresponding operator. Let (A3) hold. Set: $$ E:=\|f\|_{\rho,\sigma}\ , \epsilon_0:=\frac{2^{\tau+5}e^2\tau^\tau}\gamma\frac{E}{\rho^\tau\sigma^2} $$ and assume $ \ep_0 <1\ .$ Then for $0\leq \epsilon \leq \epsilon_0$ there exists a unitary transformation $T_{\epsilon} :L^2(\R^n)\to L^2(\R^n)$ such that $S_f(\hbar):=T_{\epsilon}P_f(\hbar)T^{-1}_{\epsilon}$ is an $\hbar -$ pseudodifferential operator family with symbol \be \label{nuovosimbolo1} \sigma_f(x,\xi,\hbar,\ep) = p_0(x,\xi)+ {\cal Z}({ I}(x,\xi);\hbar,\ep)+{\cal R}(x,\xi,\hbar,\ep). \ee Here ${\cal Z}\in A_{\rho/2,\sigma/2}$ and fulfills the estimate \be \label{Z} \|{\cal Z}\|_{\rho/2,\sigma/2}\leq 2E, \ee The principal symbol of ${\cal Z}$ is the normal form of the \ha\ $p_0(x,\xi)+f(x,\xi)$ computed up to order $\ds (\ep_0)^{-\frac{1}{\tau+2}}$, and ${\cal R}\in \A_{\rho/2,\sigma/2}$ is exponentially small, namely \be \label{stimaresto1} \|{\cal R}\|_{\rho/2,\sigma/2}\leq e E\exp{\left[-(\tau+2) (1/{\ep_0})^{\frac{1}{\tau+2}}\right]} \ee } \par\noindent 5. Apply now Propostion 3.1 to the regularized symbol. A simple computation gives $$ E=e^{\rho K+\sigma \Lambda}\snorma{a'_{\Lambda,K}}_{0,0} \leq e^{\rho K+\sigma \Lambda} C_{11}\epsilon^3\ . $$ Choosing $ K=\epsilon^{-\alpha} $, $\Lambda=\epsilon^{-(\alpha+1)}$, $\rho=\epsilon^\alpha$, $\sigma=\epsilon^{\alpha+1}$ we obtain the normal form result for the regularized system. Finally, choose $\alpha:=[(2\ell+1)/2(\tau+2)(\ell+1)]$ (exactly as in the proof of Theorem 3.1). This yields the result for the symbol $p'_\epsilon$ and therefore for the corresponding operator. %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% \begin{thebibliography}{DGH} {\small \bibitem[BGP]{BGP} {\sc D.Bambusi, S.Graffi,T.Paul}, {\it Normal forms and quantization formulae}. Commun. Math. Phys., {\bf 207}, 173-195, (1999). \bibitem[Sj]{Sj} {\sc J.Si\"ostrand}, {\it Semi-excited levels in non-degenerate potential wells}, Asymptotic Analysis {\bf 6} (1992), 29-43 } \end{thebibliography} \end{document}