Content-Type: multipart/mixed; boundary="-------------1102040938702" This is a multi-part message in MIME format. ---------------1102040938702 Content-Type: text/plain; name="11-14.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="11-14.keywords" Quantization, Birkhoff normal form, Quantum normal form, quantum KAM iteration, homological equation ---------------1102040938702 Content-Type: application/x-tex; name="STAMS.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="STAMS.tex" % \documentclass[11pt]{article} %\documentclass[a4paper,12pt,reqno]{amsart} %\documentclass[a4paper,draft,reqno]{amsart} %master.tex \documentclass[11pt,reqno]{amsart} \usepackage{amsmath} \usepackage{array} \topmargin .06in \oddsidemargin 0.7in \evensidemargin 0.7in \textheight 8.5 in \textwidth 6.4 in \hoffset= -.65in \addtolength{\baselineskip}{4pt} %\setlength{\extrarowheight}{0.3cm} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \input{amssym.def} \input{amssym} %\usepackage[notref]{showkeys} %\setlength{\textwidth}{16.0cm} %%DB margin change%% %\setlength{\textheight}{23.0cm} %\hoffset=-1.4cm %\voffset=-2.0cm %\newtheorem{theorem}{Theorem} %\newtheorem{proposition}[theorem]{Proposition} %\newtheorem{lemma}[theorem]{Lemma} %\newtheorem{corollary}[theorem]{Corollary} %\newtheorem{definition}[theorem]{Definition} %\newtheorem{remark}[theorem]{Remark} %\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}} %\renewcommand{\theremark}{\thesection.\arabic{remark}} %\def\C{{\mathcal C}} \def\L{{\mathcal L}} \def\N{{\mathcal N}} \def\B{{\mathcal B}} \def\D{{\mathcal D}} \def\H{{\mathcal H}} \def\P{{\mathcal P}} \def\M{{\mathcal M}} %\def\R{{\mathcal R}} \def\V{{\mathcal V}} \def\W{{\mathcal W}} \def\G{{\mathcal G}} \def\F{{\mathcal F}} \def\F{{\mathcal F}} \def\X{{\mathcal X}} \def\K{{\mathcal K}} \def\Y{{\mathcal Y}} \def\J{{\mathcal J}} \def\FA{{\mathcal FA}} \def\Ar{{\rm A}} \def\Br{{\rm B}} \def\v{{\mathcal \nu}} \def\vf{\varphi} \def\ve{epsilon} \def\R{\Bbb R} \def\Z{\Bbb Z} \def\Na{\Bbb N} \def\T{\Bbb T} \def\C{\Bbb C} \def\Q{\Bbb Q} \def\He{\Bbb H} \def\A{{\mathcal A}} \def\res{{\mathcal R}} \def\ha{Ha\-mil\-to\-nian} \def\Sc{Schr\"o\-din\-ger} \def\la{\langle} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\bea{\begin{eqnarray}} \def\eea{\end{eqnarray}} \def\ra{\rangle} \def\ds{\displaystyle} \def\om{\omega} \def\Om{\Omega} \def\ep{\varepsilon} \def\RR{\mathcal R} \def\imma{{\rm Im}} \def\pd{pseudo\-dif\-ferential operator} \def\pp{p^\prime} \def\qp{{q^\prime}} \def\up{u^\prime} \def\vp{{v^\prime}} \def\rp{{r^\prime}} \newcommand{\hau}{{\scr H}} \newcommand{\leb}{{\scr L}} \newcommand{\esssup}{\mathop{\rm ess\,sup}} \newcommand{\essinf}{\mathop{\rm ess\,inf}} \newcommand{\e}{\epsilon} \overfullrule=5pt %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% %%%%%%%% begin %%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \baselineskip=18pt \title{Convergence of a quantum normal form and an exact quantization for\-mu\-la} \author{Sandro Graffi} \address {Dipartimento di Matematica, Universit\`{a} di Bologna, 40127 Bologna, Italy} \email{graffi@dm.unibo.it} %thanks{First author supported by NSF grant DMS 890771} \author{Thierry Paul} \address{Centre de Math\'ematiques Laurent Schwartz, \'Ecole Polytechnique, 91128 Palaiseau Cedex, Fran\-ce} \email{paul@math.polytechnique.fr} %\setcounter{page}{0} \date{} %\date{\today} %\maketitle \begin{abstract} {Let the quantization of the linear flow of diophantine frequencies $\om$ over the torus $\T^l$, $l>1$, namely the Schr\"odinger operator $-i\hbar\omega\cdot\nabla$ on $L^2(\T^l)$, be perturbed by the quantization of a function $\V_\om: \R^l\times\T^l\to\R$ of the form \vskip 5pt\noindent $$ \V_\om(\xi,x)=\V(z\circ \L_\om(\xi),x),\quad \L_\om(\xi):= \om_1\xi_1+\ldots+\om_l\xi_l $$ \vskip 4pt\noindent where $z\mapsto \V(z,x): \R\times\T^l \to\R$ is real-holomorphic. We prove that the corresponding quantum normal form converges uniformly with respect to $\hbar\in [0,1]$. Since the quantum normal form reduces to the classical one for $\hbar=0$, this result simultaneously yields an exact quantization formula for the quantum spectrum, as well as a convergence criterion for the Birkhoff normal form, valid for a class of perturbations holomorphic away from the origin. The main technical aspect concerns the quantum homological equation $\ds {[F(-i\hbar\om\cdot\nabla),W]}/{i\hbar}+V=N$, $F:\R\to\R$ being a smooth function $\ep-$close to the identity. Its solution is constructed, and estimated uniformly with respect to $\hbar\in [0,1]$, by solving the equation $\{F(\L_\om),\W\}_M+\V=\N$ for the corresponding symbols. Here $\{\cdot,\cdot\}_M$ stands for the Moyal bracket. As a consequence, the KAM iteration for the symbols of the quantum operators can be implemented, and its convergence proved, uniformly with respect to $(\xi,\hbar,\ep)\in \R^l\times [0,1]\times \{\ep\in\C\,|\;|\ep|<\ep^\ast\}$, where $\ep^\ast>0$ is explicitly estimated in terms only of the diophantine constants. This in turn entails the uniform convergence of the quantum normal form. } \end{abstract} %\date{} \maketitle %\subjclass{???} \keywords{????} % \tableofcontents %vskip 1cm\noindent %%%%%%%%%%%%%%%%%%%%%%%%%ajout Weyl%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Approximate solutions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \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}} \renewcommand{\theremark}{\thesection.\arabic{remark}} \setcounter{equation}{0}% \setcounter{theorem}{0}% %\setcounter{lemma}{0}% %\setcounter{proposition}{0}% %\setcounter{corollary}{0}% \noindent \subsection{Quantization formulae} The establishment of a quantization formula (QF) for the eigenvalues of the \Sc\ operators is a classical mathematical problem of quantum mechanics (see e.g.\cite{FM}). To review the notion of QF, consider first a semiclassical pseudodifferential operator $H$ (for this notion, see e.g.\cite{Ro}) acting on $L^2(\R^l)$, $l\geq 1$, of order $m$, self-adjoint with pure-point spectrum, with (Weyl) symbol $\sigma_H(\xi,x)\in C^\infty(\R^l\times\R^l;\R)$. %%%%%%%% \begin{definition}\label{quant} {\it We say that $H$ admits an $M$-smooth {\rm exact} QF, $M\geq 2$, if there exists a function $\mu:$ $(A,\hbar)\mapsto \mu(A,\hbar)\in C^M(\R^l\times [0,1]; \R)$ such that: \begin{enumerate} \item $\mu(A,\hbar)$ admits an asymptotic expansion up to order $M$ in $\hbar$ uniformly on compacts with respect to $A\in\R^l$; \item %\be\label{qf} $\forall\hbar\in]0,1]$, there is a sequence $n_k:=(n_{k_1},\ldots,n_{k_l})\subset \Z^l$ such that all eigenvalues $\lambda_{k}(\hbar)$ of $H$ admit the representation: \be \label{FQ1} \lambda_{k}(\hbar)=\mu(n_k\hbar,\hbar). \ee \end{enumerate}} \end{definition} % \noindent \begin{remark} (Link with the Maslov index) \label{maslov} Consider any function $f:\ \R^l\to\R^l$ with the property $\la f(A),\nabla\mu(A,0)\ra $ $=\partial_\hbar\mu(A,0)$. Then we can rewrite the asymptotic expansion of $\mu$ at second order as : \be \mu(n_k\hbar,\hbar)=\mu(n_k\hbar+\hbar f(n_k\hbar))+O(\hbar^2). \ee When $f(m\hbar)=\nu, \;\nu\in\Q^l$, the Maslov index \cite{Ma} is recovered. Moreover, when \be \label{QF2} |\lambda_{k}(\hbar)-\mu(n_k\hbar,\hbar)|=O(\hbar^M), \quad \hbar\to 0, \quad M\geq 2 \ee then we speak of {\it approximate} QF of order $M$. \end{remark} \begin{example} (Bohr-Som\-mer\-feld-Ein\-stein for\-mu\-la). Let $\sigma_H$ fulfill the conditions of the Liouville-Arnold theorem (see e.g.\cite{Ar1}, \S 50). Denote $A=(A_1,\ldots,A_l)\in \R^l$ the action variables, and $E(A_1,\ldots,A_l)$ the symbol $\sigma_H$ expressed as a function of the action variables. Then the Bohr-Som\-mer\-feld-Ein\-stein for\-mu\-la (BSE) QF is \be \label{QF3} \lambda_{n,\hbar}=E((n_1+\nu/4)\hbar,\ldots,(n_l+\nu/4)\hbar)+O(\hbar^2) \ee where $\nu=\nu(l)\in\Na\cup\{0\}$ is the Maslov index \cite{Ma}. When $H$ is the \Sc\ operator, and $\sigma_H$ the corresponding classical Hamiltonian, (\ref{QF3}) yields the approximate eigenvalues, i.e. the approximate quantum energy levels. In the particular case of a quadratic, positive definite Hamiltonian, which can always be reduced to the harmonic oscillator with frequencies $\om_1>0,\ldots,\om_l>0$, the BSE is an exact quantization formula in the sense of Definition 1.1 with $\nu=2$, namely: $$ \mu(A,\hbar)=E(A_1+\hbar/2,\ldots,A_l+\hbar/2) =\sum_{k=1}^l\om_k(A_k+\hbar/2) $$ \end{example} \vskip 10pt To our knowledge, if $l>1$ the only known examples of exact QF in the sense of Definition 1.1 correspond to classical systems integrable by separation of variables, such that each separated system admits in turn an exact QF, as in the case of the Coulomb potential (for exact QFs for general one-dimensional \Sc\ operators see \cite{Vo}). For general integrable systems, only the approximate BSE formula is valid. Non-integrable systems admit a formal approximate QF, the so-called Einstein-Brillouin-Keller (EBK), recalled below, provided they possess a normal form to all orders. In this paper we consider a perturbation of a linear Hamiltonian on $T^\ast\T^l=\R^l\times\T^l$, and prove that the corresponding quantized operator can be unitarily conjugated to a function of the differentiation operators via the construction of a quantum normal form which converges uniformly with respect to $\hbar\in [0,1]$. This yields immediately an exact, $\infty$-smooth QF. The uniformity with respect to $\hbar$ yields also an explicit family of classical Hamiltonians admitting a convergent normal form, thus making the system integrable. \subsection{Statement of the results} Consider the Hamiltonian family $\H_\ep: \R^l\times \T^l\rightarrow \R, (\xi,x)\mapsto \H_\ep(\xi,x)$, indexed by $\ep\in\R$, defined as follows: \be \H_\ep(\xi,x):=\L_\om(\xi)+\ep \V(x,\xi);\quad \L_\om(\xi):=\la\om,\xi\ra, \quad\om\in\R^l,\quad \V\in C^\infty(\R^l\times\T^l;\R). \ee Here $\xi\in\R^l, x\in\T^l$ are canonical coordinates on the phase space $\R^l\times\T^l$, the $2l-$cylinder. $\L_\om(\xi)$ generates the linear Hamiltonian flow $\xi_i\mapsto \xi_i, x_i\mapsto x_i+\om_it$ on $\R^l\times\T^l$. For $l>1$ the dependence of $\V$ on $\xi$ makes non-trivial the integrability of the flow of $\H_\ep$ when $\ep\neq 0$, provided the {\it frequencies} $\om:=(\om_1,\ldots, \om_l)$ are independent over $\Q$ and fulfill a diophantine condition such as (\ref{DC}) below. Under this assumption it is well known that $\H_\ep$ admits a {\it normal form} at any order (for this notion, see e.g. \cite{Ar2}, \cite{SM}). Namely, $\forall\,N\in\Na$ a canonical bijection ${\mathcal C}_{\ep,N}:\R^l\times\T^l\leftrightarrow \R^l\times\T^l$ close to the identity can be constructed in such a way that: \be \label{CNF} (\H_\ep\circ {\mathcal C}_{\ep,N})(\xi,x)=\L_\om(\xi)+\sum_{k=1}^N \B_k(\xi;\om)\ep^k+\ep^{N+1}{\mathcal R}_{N+1,\ep}(\xi,x) \ee This makes the flow of $\H_\ep(\xi,x)$ integrable up to an error of order $\ep^{N+1}$. In turn, ${\mathcal C}_{\ep,N}$ is the Hamiltonian flow at time $1$ generated by \be \label{FGen} \W^N_\ep(\xi,x):=\la\xi,x\ra+\sum_{k=1}^N\W_k(\xi,x)\ep^k, \ee where the functions $\W_k(\xi,x): \R^l\times \T^l\to\R$ are recursively computed by canonical perturbation theory via the standard Lie transform method of Deprit\cite{De} and Hori\cite{Ho} (see also e.g \cite{Ca}). To describe the quantum counterpart, let $H_\ep=L_\om+\ep V$ be the operator in $L^2(\T^l)$ of symbol $\H_\ep$, with domain $D(H_\ep)= H^1(\T^l)$ and action specified as follows: \bea \forall u\in D(H_\ep), \quad H_\ep u= L_\om u+Vu, \quad L_\om u=\sum_{k=1}^l\om_kD_ku, \;\; D_k u:=-i\hbar\partial_{x_k}u, \eea and $V$ is the Weyl quantization of $\V$ (formula (\ref{1erweyl}) below). Since {\it uniform} quantum normal forms (see e.g. \cite{Sj},\cite{BGP},\cite{Po1}, \cite{Po2}) are not so well known as the classical ones, let us recall here their definition. The construction is reviewed in Appendix. \begin{definition} \label{QuNF}[Quantum normal form (QNF)] {\it We say that a family of operators $H_\ep$ $\ep$-close (in the norm resolvent topology) to $H_0=L_\omega$ admits a uniform quantum normal form (QNF) at any order if \begin{itemize} \item[(i)] There exists a sequence of continuous self-adjoint operators $W_k(\hbar)$ in $L^2(\T^l)$, $k=1,\ldots$ and a sequence of functions $B_k(\xi_1,\ldots,\xi_l,\hbar)\in C^\infty(\R^l\times [0,1];\R)$, such that, defining $\forall\,N\in\Na$ the family of unitary operators: \bea \label{QNF} U_{N,\ep}(\hbar)=e^{iW_{N,\ep}(\hbar)/\hbar}, \quad W_{N,\ep}(\hbar)=\sum_{k=1}^N W_k(\hbar)\ep^k \eea we have: \bea \label{AQNF} && U_{N,\ep}(\hbar)H_\ep U_{N,\ep}^\ast(\hbar)=L_\om+\sum_{k=1}^N B_k(D_1,\ldots,D_l,\hbar)\ep^k+\ep^{N+1}R_{N+1,\ep}(\hbar). \eea \item [(ii)] The operators $B_k(D,\hbar): k=1,2\ldots$, $R_{N+1}$ are continuous in $L^2(\T^l)$; the corresponding symbols $\W_k, \B_k, \RR_{N+1}(\ep)$ belong to $ C^\infty(\R^l\times\T^l\times [0,1])$, and reduce to the classical normal form construction (\ref{CNF}) and (\ref{FGen}) as $\hbar\to 0$: \be \label{princip} \B_k(\xi;0)=\B_k(\xi);\quad \W_k(\xi,x,0)=\W_k(\xi,x),\quad \RR_{N+1,\ep}(x,\xi;0)=\RR_{N+1,\ep}(x,\xi) \ee \end{itemize}} \end{definition} (\ref{AQNF}) entails that $H_\ep$ commutes with $H_0$ up to an error of order $\ep^{N+1}$; hence the following approximate QF formula holds for the eigenvalues of $H_\ep$: \be \label{AQF} \lambda_{n,\ep}(\hbar)=\hbar\la n,\om\ra+\sum_{k=1}^N \B_k(n_1\hbar,\ldots,n_l\hbar,\hbar)\ep^k+O(\ep^{N+1}). \ee %\vskip 5pt \vskip 6pt\noindent \begin{definition} \label{QNFConv}{(Uniformly convergent quantum normal forms)} {\it We say that the QNF converges $M$-smoothly, $M > 2l$, {\rm uniformly with respect to the Planck constant $\hbar$}, if there is $\ep^\ast>0$ such that \vskip 5pt\noindent \bea && \label{convunifQ1} \sum_{k=1}^\infty\,\sup_{\R^l\times\T^l\times [0,1]}\sum_{|\alpha|\leq M}|D^\alpha\W_k(\xi,x;\hbar)\ep^k|<+\infty \\ && \label{convunifQ2} \sum_{k=1}^\infty\,\sup_{\R^l\times [0,1]}\sum_{|\alpha|\leq M}|D^\alpha\B_k(\xi,\hbar)\ep^k|<+\infty , \quad |\ep|<\ep^\ast. \eea \vskip 5pt\noindent Here $\ds D^\alpha=\partial ^{\alpha_1}_\xi \partial ^{\alpha_2}_x \partial ^{\alpha_3}_\hbar$, $|\alpha|=|\alpha_1|+|\alpha_2|+\alpha_3$. } \end{definition} \noindent (\ref{convunifQ1},\ref{convunifQ2}) entail that, if $|\ep|<\ep^\ast$, we can define the symbols \vskip 3pt\noindent \bea \label{somma} && \W_{\infty}(\xi,x;\ep,\hbar):=\la \xi,x\ra+\sum_{k=1}^\infty\W_k(\xi,x;\hbar)\ep^k\in C^M(\R^l\times\T^l\times [0,\ep^\ast] \times[0,1];\C), \\ \label{somma1} && \B_{\infty}(\xi;\ep,\hbar):=\L_\om(\xi)+\sum_{k=1}^\infty\B_k(\xi;\hbar)\ep^k \in C^M(\R^l\times [0,\ep^\ast] \times[0,1];\C) \eea \vskip 3pt\noindent By the Calderon-Vaillancourt theorem (see \S 3 below) their Weyl quantizations $W_{\infty}(\ep,\hbar)$, $B_{\infty}(\ep,\hbar)$ are continuous operator in $L^2(\T^l)$. Then: \be e^{iW_{\infty}(\ep,\hbar)/\hbar}H_\ep e^{-iW_{\infty}(\ep,\hbar)/\hbar}=B_{\infty}(D_1,\ldots,D_l;\ep,\hbar). \ee Therefore the uniform convergence of the QNF has the following straightforward consequences: \begin{itemize} \item[(A1)] {\it The eigenvalues of $H_\ep$ are given by the {\rm exact} quantization formula:} \be \label{QF} \lambda_{n}(\hbar,\ep)=\B_{\infty}(n\hbar,\hbar,\ep), \qquad n\in\Z^l, \quad \ep\in {\frak D}^\ast:=\{\ep\in \R\,|\,|\ep|<\ep^\ast\} \ee \item [(A2)] {\it The classical normal form is convergent, uniformly on compacts with respect to $\xi\in\R^l$, and therefore if $\ep\in {\frak D}^\ast$ the Hamiltonian $\H_\ep(\xi,x)$ is integrable.} \end{itemize} Let us now state explicit conditions on $V$ ensuring the uniform convergence of the QNF. \newline Given $\F(t,x)\in C^\infty(\R\times\T^l;\R)$, consider its Fourier expansion \be \label{FFE} \F(t,x)=\sum_{q\in\Z^l}\F_q(t)e^{i\la q,x\ra}. \ee and define furthermore $ \F_\om: \R^l\times\T^l\to \R; \F_\om \in C^\infty(\R^l\times\T^l;\R)$ in the following way: \vskip 4pt\noindent \bea && \label{Fouom} \F_\om(\xi,x):=\F(\L_\om(\xi),x)=\sum_{q\in\Z^l}\F_{\om,q}(\xi)e^{i\la q,x\ra}, \\ && \F_{\om,q}(\xi):=(\F_q\circ \L_\om)(\xi)=\frac1{(2\pi)^{l/2}}\int_\R\widehat{\F}_q(p)e^{-ip\L_\om(\xi)}\,dp= \\ && = \frac1{(2\pi)^{l/2}}\int_\R\widehat{\F}_q(p)e^{-i\la p\om,\xi\ra}\,dp, \quad p\om :=(p\om_1,\ldots,p\om_l ). \eea \vskip 4pt\noindent Here, as above, $\L_\om(\xi)=\la\om,\xi\ra$. \vskip 4pt\noindent Given $\rho>0$, introduce the weighted norms: \bea && \|\F_{\om,q}(\xi)\|_\rho:=\int_\R|\widehat{\F}_q(p)|e^{\rho |p|}|\,dp \\ && \|\F_\om(x,\xi)\|_{\rho}:=\sum_{q\in\Z^l}\,e^{\rho |q|}\|\F_{\om,q}\|_\rho \eea \vskip 4pt\noindent We can now formulate the main result of this paper. Assume: \vskip 4pt\noindent \begin{itemize} \item[(H1)] There exist $\gamma >1, \tau >l-1$ such that the frequencies $\om$ fulfill the diophantine condition \be \label{DC} |\la\om,q\ra|^{-1}\leq \gamma |q|^{\tau}, \quad q \in\Z^l, \; q\neq 0. \ee \item[(H2)] $V_\om$ is the Weyl quantization of $\V_\om(\xi,x)$ (see Sect.3 below), that is: \vskip 8pt\noindent \be\label{1erweyl} V_\om f(x)=\int_{\R}\sum_{q\in\Z^l}\widehat{\V}_q(p) e^{i\la q,x\ra+\hbar p\la \om,q\ra/2}f(x+\hbar p\om)\,dp, \quad f\in L^2(\T^l). \ee \vskip 5pt\noindent with $\V(\xi,x;\hbar)=\V(\la\omega,\xi\ra,x)=\V_\om(\xi,x)$ for some function $\V(t;x): \R\times\T^l\to \R$. \vskip 4pt\noindent \item[(H3)] $$ \|\V_\om\|_{\rho}<+\infty, \qquad \rho >1+16\gamma\tau^\tau. $$ \end{itemize} \vskip 4pt\noindent Clearly under these conditions the operator family $ H_\ep:=L_\om+\ep V_\om$, $D(H_\ep) =H^1(\T^l)$, $\ep\in\R$, is self-adjoint in $L^2(\T^l)$ and has pure point spectrum. We can then state the main results. \vskip 4pt\noindent %%%%%%%%%%% \begin{theorem} %%%%%%%%%% \label{mainth} Under conditions (H1-H3), $ H_\ep$ admits a uniformly convergent quantum normal form $\B_{\infty,\om}(\xi,\ep,\hbar)$ in the sense of Definition 1.5, with radius of convergence no smaller than: \vskip 6pt\noindent \be \label{rast} \ep^\ast(\gamma,\tau):=\frac{1}{e^{24(3+2\tau)}2^{2\tau}\|\V\|_{\rho}}. \ee \vskip 6pt\noindent \end{theorem} If in addition to (H1-H2) we assume, for any fixed $r\in\Na$: \begin{itemize} \item[(H4)] \be \label{condizrho} \rho> \lambda(\gamma,\tau,r):=1+8\gamma\tau[(2(r+1)^2] \ee \vskip 6pt\noindent \end{itemize} we can sharpen the above result proving smoothness with respect to $\hbar$: \begin{theorem} \label{regolarita} Let conditions (H1-H2-H4) be fulfilled. For $r\in {\Bbb N}$ define ${\frak D}^\ast_r:=\{\ep\in\C\,|\,|\ep|<\ep^\ast(\gamma,\tau,r)\}$, where: \vskip 8pt\noindent \bea \label{epastr} \ep^\ast(\gamma,\tau,r):=\frac{1}{e^{24(3+2\tau)}(r+2)^{2\tau}\|\V\|_{\rho}}\eea \vskip 8pt\noindent Then $\ds \hbar\mapsto \B_\infty(t,\ep,\hbar)\in C^\infty ([0,1]; C^\om( \{t\in\C\,|\,|\Im t|<{\rho}/{2}\times{\frak D}_r^\ast(\rho) \})$; i.e. there exist $C_r(\ep^\ast)>0$ such that, for $\ep\in {\frak D}_r^\ast$: \vskip 4pt\noindent \bea \label{stimaG1} \sum_{\gamma=0}^r\max_{\hbar\in [0,1]} \|\partial^\gamma_\hbar \B_{\infty,\om}(\xi;\ep,\hbar) \|_{\rho/2}\leq C_r, \;\;r=0,1,\ldots \eea \end{theorem} \vskip 4pt In view of Definition \ref{quant}, the following statement is a straightforward consequence of the above Theorems: \begin{corollary}[Quantization formula]\label{QFE} $\H_\ep$ admits an $\infty$-smooth quantization formula in the sense of Definition 1.1. That is, $\forall\,r\in\Na$, $\forall \,|\epsilon|<\ep^\ast (\gamma,\tau,r)$ given by (\ref{epastr}), the eigenvalues of $H_\ep$ are expressed by the formula: \be \label{EQF} \lambda(n,\hbar,\ep)=\B_{\infty,\om}(n\hbar,\ep, \hbar) =\L_\om(n\hbar)+\sum_{s=1}^\infty\B_s(\L_\om(n\hbar),\hbar)\ep^s \ee where $\B_{\infty,\om}(\xi,\ep, \hbar)$ belongs to $C^r(\R^l\times [0,\ep^\ast(\cdot,r)]\times [0,1])$, and admits an asymptotic expansion at order $r$ in $\hbar$, uniformly on compacts with respect to $(\xi,\ep)\in\R^l\times [0,\ep^\ast(\cdot,r)]$. \end{corollary} %\newpage {\bf Remarks} \begin{itemize} \item[(i)] (\ref{stimaG1}) and (\ref{EQF}) entail also that the Einstein-Brillouin-Keller (EBK) quantization formula: \be \label{EBK} \lambda_{n,\ep}^{EBK}(\hbar):=\L_\om(n\hbar)+\sum_{s=1}^\infty \B_s(\L_\om(n\hbar))\ep^s=\B_{\infty,\om}(n\hbar,\ep),\quad n\in\Z^l \ee reproduces here ${\rm Spec}(H_\ep)$ up to order $\hbar$. \item[(ii)] Apart the classical Cherry theorem yielding convergence of the Birkhoff normal form for smooth perturbations of the harmonic flow with {\it complex} frequencies when $l=2$ (see e.g. \cite{SM}, \S 30; the uniform convergence of the QNF under these conditions is proved in \cite{GV}), no simple convergence criterion seems to be known for the QNF nor for the classical NF as well. (See e.g.\cite{PM}, \cite{Zu}, \cite{St} for reviews on convergence of normal forms). Assumptions (1) and (2) of Theorem \ref{mainth} entail Assertion (A2) above. Hence they represent, to our knowledge, a first explicit convergence criterion for the NF. \end{itemize} Remark that $\L_\om(\xi)$ is also the form taken by harmonic-oscillator Hamiltonian in $\R^{2l}$, $$ \P_0(\eta,y;\om):= \sum_{s=1}^l\om_s(\eta^2_s+y_s^2), \quad (\eta_s,y_s)\in\R^2,\quad s=1,\ldots,l $$ if expressed in terms of the action variables $\xi_s>0, \,s=1,\ldots,l$, where $$ \xi_s:=\eta^2_s+y_s^2=z_s\overline{z}_s, \quad z_s:=y_s+i\eta_s. $$ {Assuming} (\ref{DC}) {\it and} the property \be \label{Rk1} \B_k(\xi)=(\F_k\circ\L_\om(\xi))=\F_{k}(\sum_{s=1}^l \om_s z_s\overline{z}_s), \quad k=0,1,\ldots \ee R\"ussmann \cite{Ru} (see also \cite{Ga}) proved convergence of the Birkhoff NF if the perturbation $\V$, expressed as a function of $(z,\overline{z})$, is {\it in addition} holomorphic at the origin in $\C^{2l}$. No explicit condition on $\V$ seems to be known ensuring {\it both} (\ref{Rk1}) and the holomorphy. In this case instead we {\it prove} that the assumption $\V(\xi,x)=\V(\L_\om(\xi),x)$ entails (\ref{Rk1}), uniformly in $\hbar\in [0,1]$; namely, we construct $\F_s(t;\hbar):\R\times [0,1]\to\R$ such that: \be \label{Rk} \B_s(\xi;\hbar)=\F_s(\L_\om(\xi);\hbar):=\F_{\om,s}(\xi;\hbar), \quad s=0,1,\ldots \ee The conditions of Theorem \ref{mainth} cannot however be transported to R\"ussmann's case: the map \vskip 6pt\noindent $$ {\mathcal T}(\xi,x)=(\eta,y):= \begin{cases} \eta_i=-\sqrt{\xi_i}\sin x_i, \\ y_i=\sqrt{\xi_i}\cos x_i, \end{cases}\quad i=1,\ldots,l, $$ \vskip 6pt\noindent namely, the inverse transformation into action-angle variable, is defined only on $\R_+^l\times\T^l$ and does not preserve the analyticity at the origin. On the other hand, ${\mathcal T}$ is an analytic, canonical map between $\R_+^l\times\T^l$ and $\R^{2l}\setminus\{0,0\}$. Assuming for the sake of simplicity $\V_0=0$ the image of $\H_\ep$ under ${\mathcal T}$ is: \bea \label{H0} (\H_\ep \circ {\mathcal T})(\eta,y)= \sum_{s=1}^l\om_s(\eta^2_s+y_s^2)+\ep (\V\circ {\mathcal T})(\eta,y):=\P_0(\eta,y)+\ep \P_1(\eta,y) \eea where \bea && \label{H1} \P_1(\eta,y)=(\V\circ {\mathcal T})(\eta,y)=\P_{1,R}(\eta,y)+\P_{1,I}(\eta,y), \;(\eta,y)\in\R^{2l}\setminus\{0,0\}. \eea \bea && \nonumber \P_{1,R}(\eta,y)=\frac12\sum_{k\in\Z^l}(\Re{\V}_k\circ\H_0)(\eta,y)\prod_{s=1}^l \left(\frac{\eta_s-iy_s}{\sqrt{\eta^2_s+y_s^2}}\right)^{k_s} \\ \nonumber && \P_{1,I}(\eta,y)=\frac12\sum_{k\in\Z^l} (\Im{\V}_k\circ\H_0)(\eta,y)\prod_{s=1}^l \left(\frac{\eta_s-iy_s}{\sqrt{\eta^2_s+y_s^2}}\right)^{k_s} \end{eqnarray} \vskip 4pt\noindent If $\V$ fulfills Assumption (H3) of Theorem \ref{mainth}, both these series converge uniformly in any compact of $\R^{2l}$ away from the origin and $\P_1$ is holomorphic on $\R^{2l}\setminus\{0,0\}$. Therefore Theorem \ref{mainth} immediately entails a convergence criterion for the Birkhoff normal form generated by perturbations holomorphic away from the origin. We state it under the form of a corollary: \begin{corollary} \label{mainc} {\rm (A convergence criterion for the Birkhoff normal form)} Under the assumptions of Theorem \ref{mainth} on $\om$ and $\V$, consider on $\R^{2l}\setminus\{0,0\}$ the holomorphic Hamiltonian family $P_\ep(\eta,y):=\P_0(\eta,y)+\ep\P_1(\eta,y)$, $\ep\in\R$, where $\P_0$ and $\P_1$ are defined by (\ref{H0},\ref{H1}). Then the Birkhoff normal form of $H_\ep$ is uniformly convergent on any compact of $\R^{2l}\setminus\{0,0\}$ if $|\ep|<\ep^\ast (\gamma,\tau)$. \end{corollary} \vskip 0.5cm\noindent %%%%%%%%%%%%%% \subsection{Strategy of the paper} The proof of Theorem \ref{mainth} rests on an implementation in the quantum context of R\"ussmann's argument\cite{Ru} yielding convergence of the KAM iteration when the complex variables $(z,\overline{z})$ belong to an open neighbourhood of the origin in $\C^{2l}$. Conditions (\ref{DC}, \ref{Rk}) prevent the occurrence of accidental degeneracies among eigenvalues at any step of the quantum KAM iteration, in the same way as they prevent the formation of resonances at the same step in the classical case. However, the global nature of quantum mechanics prevents phase-space localization; therefore, and this is the main difference, at each step the coefficients of the homological equation for the operator symbols not only have an additional dependence on $\hbar$ but also have to be controlled up to infinity. These difficulties are overcome by exploiting the closeness to the identity of the whole procedure, introducing adapted spaces of symbols i(Section \ref{not}), which account also for the properties of differentiability with respect to the Planck constant. The link between quantum and classical settings is provided by a sharp (i.e. without $\hbar^\infty$ approximation) Egorov Theorem established in section \ref{sectionegorov}. Estimates for the solution of the quantum homological equation and their recursive properties are obtained in sections \ref{hom} (Theorem \ref{homo}) and \ref{towkam} (Theorem \ref{resto}) respectively. Recursive estimates are established in Section \ref{recesti} (Theorem \ref{final}) and the proof of our main result is completed in section \ref{iteration}. The link with the usual construction of the quantum normal form described in Appendix. \vskip 1cm\noindent \section{Norms and first estimates} \label{not} \setcounter{equation}{0}% \setcounter{theorem}{0}% Let $m,l=1,2,\dots$. For $\F\in C^\infty(\R^m\times\T^l\times [0,1]; \C),\ (\xi,x,\hbar)\to\F(\xi,x;\hbar)$, and $\G\in C^\infty(\R^m\times [0,1]; \C),\, (\xi,\hbar)\to\G(\xi;\hbar)$, consider the Fourier transforms \be \widehat{\G}(p;\hbar)=\frac1{(2\pi)^{m/2}}\int_{\R^m}\G(\xi;\hbar)e^{-i\la p,\xi\ra}\,dx \ee \be \F(\xi,q;\hbar):=\frac1{(2\pi)^{m/2}}\int_{\T^l}\F(\xi,x;\hbar)e^{-i\la q,x\ra}\,dx . \quad \ee \be \label{FE1} \F(\xi,x;\hbar)=\sum_{q\in\Z^l}\F(\xi,q;\hbar)e^{-i\la q,x\ra} \ee \be \label{FE2} \widehat{\F}(p,q;\hbar)=\frac1{(2\pi)^{m/2}}\int_{\R^m}\F(\xi,q;\hbar)e^{-i\la p,\xi\ra}\,dx \ee It is convenient to rewrite the Fourier representations (\ref{FE1}, \ref{FE2}) under the form a single Lebesgue-Stieltjes integral. Consider the product measure on $\R^m\times \R^l$: \bea \label{pm1} && d\lambda (t):=dp\,d\nu(s), \quad t:=(p,s)\in\R^m\times \R^l; \\ \label{pm2} && dp:=\prod_{k=1}^m\,dp_k;\quad d\nu(s):=\prod_{h=1}^l \sum_{q_h\leq s_h} \delta (s_h-q_h), \;q_h\in\Z, h=1,\ldots,l \eea Then: \be \label{IFT} \F(\xi,x;\hbar)=\int_{\R^m\times\R^l}\,\widehat{\F}(p,s;\hbar)e^{i\la p,\xi\ra +i\la s,x\ra}\,d\lambda(p,s) \ee \begin{definition} {\it For $\rho\geq 0$, $\sigma\geq 0$, we introduce the weighted norms } \vskip 3pt\noindent \bea \label{norma1} |\G|^\dagger_{\sigma}&:=&\max_{\hbar\in [0,1]}\|\widehat{\G}(.;\hbar)\|_{L^1(\R^m,e^{\sigma |p|}dp)}=\max_{\hbar\in [0,1]}\int_{\R^l}\|\widehat{\G}(.;\hbar)\|\,e^{\sigma |p|}\,dp. \\ \label{norma1k} |\G|^\dagger_{\sigma,k}&:=&\max_{\hbar\in [0,1]}\sum_{j=0}^k\|(1+|p|^2)^{\frac{k-j}{2}}\partial^j_\hbar\widehat{\G}(.;\hbar)\|_{L^1(\R^m,e^{\sigma |p|}dp)};\quad |\G|^\dagger_{\sigma;0}:=|\G|^\dagger_{\sigma}. \eea \end{definition} \begin{remark} By noticing that $\vert p\vert\leq\vert p^\prime-p\vert+\vert p^\prime\vert$ and that, for $x\geq 0$, $\ds x^je^{-\delta x}\leq \frac 1 e(\frac j{\delta})^j$, we immediately get the inequalities \be\label{plus} \vert \F\G\vert^\dagger_{\sigma}\leq\vert \F\vert_{\sigma}\vert \G\vert_{\sigma}, \ee \be \label{diff} \vert (I-\Delta^{j/2})\F\vert_{\sigma-\delta}\leq \frac1 e\left(\frac j\delta\right)^j\vert \F\vert_\sigma, \quad k\geq 0. \ee \end{remark} Set now for $ k\in\Na\cup\{0\} $: \be\label{muk} \mu_{k}(t):=(1+|t|^{2})^{\frac k 2}=(1+|p|^{2}+|s|^{2})^{\frac k 2}. \ee and note that \be \mu_k(t-t^\prime)\leq 2^{\frac k 2} \mu_k(t)\mu_k( t ^\prime). \ee because $|x-x^\prime|^2\leq 2(|x|^2+|x^\prime|^2)$. \begin{definition} {\it Consider $\F(\xi,x;\hbar)\in C^\infty(\R^m\times \T^l\times[0,1];\C)$, with Fourier expansion \be \label{FF} \F(\xi,x;\hbar)=\sum_{q\in\Z^l}\,\F(\xi,q;\hbar)e^{i\la q,x\ra} \ee \begin{itemize} \item [(1)] Set: \bea \label{sigmak} \Vert \F\Vert^\dagger_{\rho,k}:=\max_{\hbar\in [0,1]}\sum_{\gamma=0}^k \int_{\R^m\times \R^l}\vert \mu_{k-\gamma}(p,s)\partial^\gamma_\hbar\widehat{\F}(p,s;\hbar)\vert e^{\rho(\vert s\vert+\vert p\vert)}\,d\lambda(p,s). \eea \item [(2)] Let ${\mathcal O}_\omega$ be the set of functions ${\Phi}:\R^l\times\T^l\times[0,1]$ such that $\Phi(\xi,x;\hbar)=\F(\L_\omega(\xi),x;\hbar)$ for some $\F:\ \R\times\T^l\times [0,1]\to \C$. Define, for $\Phi\in {\mathcal O}_\omega$: \bea\label{sigom} \Vert \Phi\Vert_{\rho,k}:=\max_{\hbar\in [0,1]}\sum_{\gamma=0}^k \int_{\R}\vert \mu_{k-\gamma}( p\omega,q) \partial^\gamma_\hbar\widehat{\F}(p,s;\hbar)\vert e^{\rho(\vert s\vert+\vert p\vert}\,d\lambda(p,s). \eea \item [(3)] Finally we denote $Op^W(\F)$ the Weyl quantization of $\F$ recalled in Section \ref{sectionweyl} and \bea \label{normsymb'} \J^\dagger_k(\rho)&=&\{\F \,|\,\Vert \F\Vert^\dagger_{\rho,k}<\infty\}, \\ \label{normop'} J^\dagger_k(\rho)&=&\{Op^W(\F)\,|\,\F\in\J^\dagger(\rho,k)\}, \\ \label{normsymb} \J_k(\rho)&=&\{\F\in {\mathcal O}_\omega\,|\,\Vert \F\Vert_{\rho,k}<\infty\}, \\ \label{normop} J_k(\rho)&=&\{\F \,|\,\Vert \F\Vert_{\rho,k}<\infty\}, \eea \end{itemize}} \end{definition} Finally we denote: $L^1_\sigma(\R^m):=L^1(\R^m,e^{\sigma |p|}dp)$. %\end{definition} \begin{remark} Note that, if $\F(\xi,q,\hbar)$ is independent of $q$, i.e. $\F(\xi,q,\hbar)=\F(\xi,\hbar)\delta_{q,0}$, then: \be \label{normeid} \|\F\|^\dagger_{\rho,k}=|\F|^\dagger_{\rho,k}; \quad \|\F\|_{\rho,k}=|\F|_{\rho,k} \ee while in general \bea && \|\F\|_{\rho,k}\leq \|\F\|_{\rho^\prime,k^\prime} \quad {\rm whenever}\; k\geq k^\prime,\,\rho\leq \rho^\prime; \eea \end{remark} \begin{remark} (Regularity properties) Let $\F\in \J_k^\dagger(\rho), k\geq 0$. Then: \begin{enumerate} \item There exists $K(\alpha,\rho,k)$ such that \be \label{maggC} \max_{\hbar\in [0,1]}\|\F(\xi,x;\hbar)\|_{C^\alpha(\R^m\times\T^l)}\leq K \|\F\|^\dagger_{\rho,k}, \quad \alpha\in\Na \ee and analogous statement for the norm $\|\cdot\|_{\rho,k}$. \item Let $\rho>0$, $k\geq 0$. Then $\F(\xi,x;\hbar)\in C^k([0,1];C^\om(\{|\Im \xi|<\rho\}\times \{|\Im x|<\rho\})$ and \be \label{supc} \sup_{\{|\Im \xi|0. \eea In what follows we will often use the notation $\F$ also to denote the function $\F(\L_\om(\xi))$, because the indication of the belonging to $J$ or $J^\dagger$, respectively, is already sufficient to mark the distinction of the two cases. \begin{remark} Without loss of generality we may assume: \be |\om |:=|\om_1|+\ldots+|\om_l |\leq 1 \ee Indeed, the general case $|\om|=\alpha |\om^\prime|$, $|\om^\prime|\leq 1$, $\alpha>0$ arbitrary reduces to the former one just by the rescaling $\ep\to \alpha\ep$. \end{remark} \vskip 1.0cm\noindent %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%% \section{Weyl quantization, matrix elements, commutator estimates}\label{sectionweyl} \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}} \renewcommand{\theremark}{\thesection.\arabic{remark}} \setcounter{equation}{0}% \setcounter{theorem}{0}% \subsection{Weyl quantization: action and matrix elements} We sum up here the canonical (Weyl) quantization procedure for functions (classical observables) defined on the phase space $\R^l\times\T^l$. In the present case it seems more convenient to consider the representation (unique up to unitary equivalences) of the natural Heisenberg group on $\R^l\times\T^l$. Of course this procedure yields the same quantization as the standard one via the Br\'ezin-Weil-Zak transform (see e.g. \cite{Fo}, \S 1.10) and has already been employed in \cite{CdV}, \cite{Po1},\cite{Po2}). \par Let $\He_l(\R^l\times\R^l\times\R)$ be the Heisenberg group over $\ds \R^{2l+1}$ (see e.g.\cite{Fo}, Chapt.1). Since the dual space of $\R^l\times\T^l$ under the Fourier transformation is $\R^l\times\Z^l$, the relevant Heisenberg group here is the subgroup of $\He_l(\R^l\times\R^l\times\R)$, denoted by $\He_l(\R^l\times\Z^l\times\R)$, defined as follows: \begin{definition} \label{HSG} {\it Let $u:=(p,q), p\in\R^l, q\in\Z^l$, and let $ t\in\R$. Then $\He_l(\R^l\times\Z^l\times\R)$ is the subgroup of $\He_l(\R^l\times\R^l\times\R)$ topologically equivalent to $\R^l\times\Z^l\times\R$ with group law \be \label{HGL} (u,t)\cdot (v,s)= (u+v, t+s+\frac12\Omega(u,v)) \ee Here $\Omega(u,v)$ is the canonical $2-$form on $\R^l\times\Z^l$:} \be \label{2forma} \Omega(u,v):=\la u_1,v_2\ra-\la v_1,u_2\ra \ee \end{definition} $\He_l(\R^l\times\Z^l\times\R)$ is the Lie group generated via the exponential map from the Heisenberg Lie algebra ${\mathcal H}{\mathcal L}_l(\Z^l\times\R^l\times\R)$ defined as the vector space $\R^l\times\Z^l\times\R$ with Lie bracket \be \label{LA} [(u,t)\cdot (v,s)]= (0, 0,\Omega(u,v)) \ee The unitary representations of $\He_l(\R^l\times\Z^l\times\R)$ in $L^2(\T^l)$ are defined as follows \be \label{UR} (U_\hbar(p,q,t)f)(x):=e^{i\hbar t +i\la q,x\ra+\hbar\la p.q\ra/2}f(x+\hbar p) \ee $\forall\,\hbar\neq 0$, $\forall\,(p,q,t)\in{\mathcal H}_l$, $\forall\,f\in L^2(\T^l)$. These representations fulfill the Weyl commutation relations \be \label{Weyl} U_\hbar(u)^\ast =U_\hbar(-u), \qquad U_\hbar(u)U_\hbar(v)=e^{i\hbar\Omega(u,v)}U(u+v) \ee For any fixed $\hbar>0$ $U_\hbar$ defines the \Sc\ representation of the Weyl commutation relations, which also in this case is unique up to unitary equivalences (see e.g. \cite{Fo}, \S 1.10). Consider now a family of smooth phase-space functions indexed by $\hbar$, $\A(\xi,x,\hbar):\R^l\times\T^l\times [0,1]\to\C$, written under its Fourier representation \be \label{FFR} \A(\xi,x,\hbar)=\int_{\R^l}\sum_{q\in\Z^l}\widehat{\A}(p,q;\hbar)e^{i(\la p.\xi\ra +\la q,x\ra)}\,dp=\int_{\R^l\times \R^l}\widehat{\A}(p,s;\hbar)e^{i(\la p.\xi\ra +\la s,x\ra)}\,d\lambda(p,s) \ee \begin{definition} \label{Qdef} {\it The (Weyl) quantization of $\A(\xi,x;\hbar)$ is the operator $A(\hbar)$ definde as} \bea \label{Wop} && (A(\hbar)f)(x):=\int_{\R^l}\sum_{q\in\Z^l}\widehat{\A}(p,q;\hbar)U_\hbar(p,q)f(x)\,dp \\ && \nonumber = \int_{\R^l\times\R^l}\widehat{\A}(p,s;\hbar)U_\hbar(p,s)f(x)\,d\lambda(p,s) \quad f\in L^2(\T^l) \eea \end{definition} \noindent \begin{remark} Formula \eqref{Wop} can be also be written as \be \label{Wopq} (A(\hbar)f)(x)=\sum_{q\in\Z^l}\,A(q,\hbar)f, \quad (A(q,\hbar)f)(x)=\int_{\R^l}\,\widehat{\A}(p,q;\hbar)U_\hbar(p,q)f(x)\,dp \ee \end{remark} \noindent From this we compute the action of $A(\hbar)$ on the canonical basis in $L^2(\T^l)$: $$ e_m(x):=(2\pi)^{-l/2}e^{i\la m, x\ra}, \quad x\in\T^l, \;m\in\Z^l . $$ \begin{lemma} \label{azione} \be \label{azioneAop} A(\hbar)e_m(x)= \sum_{q\in\Z^l}e^{i\la (m+q),x\ra}\A(\hbar (m+q/2),q,\hbar) \ee \end{lemma} \begin{proof} By \eqref{Wopq}, it is enough to prove that the action of $A(q,\hbar)$ is \be \label{azioneAq} A(q,\hbar)e_m(x)= e^{i\la (m+q),x\ra}\A(\hbar(m+q/2),q,\hbar) \ee Applying Definition \ref{Qdef} we can indeed write: \begin{eqnarray*} && (A(q,\hbar)e_m)(x)=(2\pi)^{-l/2}\int_{\R^l}\widehat{A}(p,q;\hbar)e^{i\la q,x\ra+i\hbar \la p,q\ra/2}e^{i\la m,(x+\hbar p)\ra}\,dp \\ && =(2\pi)^{-l/2}e^{i\la (m+q),x\ra}\,\int_{\R^l}\widehat{A}(p;q,\hbar)e^{i\hbar \la p,(m+q/2)\ra}\,dp =e^{i\la (m+q),x\ra}\A(\hbar(m+q/2),q,\hbar). \end{eqnarray*}. \end{proof} We note for further reference an obvious consequence of \eqref{azioneAq}: \be \label{ortogq} \la A(q,\hbar)e_m,A(q,\hbar)e_n\ra_{L^2(\T^l)}=0,\;m\neq n;\quad \la A(r,\hbar)e_m,A(q,\hbar)e_n\ra_{L^2(\T^l)}=0,\;r\neq q. \ee As in the case of the usual Weyl quantization, formula (\ref{Wop}) makes sense for tempered distributions $\A(\xi,x;\hbar)$ \cite{Fo}. Indeed we prove in this context, for the sake of completeness, a simpler, but less general, version of the standard Calderon-Vaillancourt criterion: \begin{proposition} Let $A(\hbar)$ by defined by (\ref{Wop}). Then \vskip 8pt\noindent \be \label{CV} \Vert A(\hbar)\Vert_{L^2\to L^2}\leq \frac{2^{l+1}}{l+2}\cdot \frac{\pi^{(3l-1)/2}}{\Gamma(\frac{l+1}{2})}\,\sum_{|\alpha|\leq 2k}\,\Vert \partial_x^k\A(\xi,x;\hbar)\Vert_{L^\infty(\R^l\times\T^l)}. \ee where $$ k=\begin{cases} \frac{l}{2}+1,\quad l\;{\rm even} \\ {} \\ \frac{l+1}{2}+1,\quad l\;{\rm odd}. \end{cases} $$ \end{proposition} \begin{proof} Consider the Fourier expansion $$ u(x)=\sum_{m\in\Z^l}\,\widehat{u}_me_m(x),\quad u\in L^2(\T^l). $$ Since: $$ \|A(q,\hbar)\widehat{u}_me_m\|^2=|\A(\hbar(m+q/2),q,\hbar) |^2\cdot |\widehat{u}_m|^2 $$ by Lemma \ref{azione} and \eqref{ortogq} we get: \begin{eqnarray*} \|A (\hbar)u\|^2&\leq & \sum_{(q,m)\in\Z^l\times\Z^l}\|A(q,\hbar)\widehat{u}_m e_m\|^2 = \sum_{(q,m)\in\Z^l\times\Z^l}|\A(\hbar (m+q/2),q,\hbar)|^2\cdot |\widehat{u}_m|^2 \\ &\leq& \sum_{q\in\Z^l}\,\sup_{\xi\in\R^l}|\A(\xi,q,\hbar) |^2\sum_{m\in\Z^l}|\widehat{u}_m|^2 = \sum_{q\in\Z^l}\,\sup_{\xi\in\R^l}|\A(\xi,q,\hbar) |^2\|u\|^2 \\ &\leq& \big[ \sum_{q\in\Z^l}\,\sup_{\xi\in\R^l}|\A(\xi,q,\hbar) |\big]^2\|u\|^2 \end{eqnarray*} Therefore: \[ \Vert A(\hbar)\Vert_{L^2\to L^2} \leq \sum_{q\in\Z^l}\,\sup_{\xi\in\R^l}\vert\A(\xi,q,\hbar)\vert. \] Integration by parts entails that, for $k\in \Na$, and $\forall \,g\in C^\infty(\T^l)$: \vskip 8pt\noindent \begin{eqnarray*} && \left |\int_{\T^l}e^{i\la q,x\ra}g(x)dx\right |=\frac 1 {1+|q|^{2k}}\left |\int_{\T^l} e^{i\la q,x\ra}(1+(-\triangle_x)^k) g(x)dx\right | \\ && \leq \frac 1 {1+\vert q\vert^{2k}}(2\pi)^l\sup_{\T^l}\sum_{|\alpha|\leq 2k}\vert \partial_x^\alpha g(x)\vert . \end{eqnarray*} \vskip 8pt\noindent Let us now take: \be \label{kappa} k=\begin{cases} \frac{l}{2}+1,\quad l\;{\rm even} \\ {} \\ \frac{l+1}{2}+1,\quad l\;{\rm odd} \end{cases} \Longrightarrow \begin{cases} 2k-l+1=3,\quad l\;{\rm even} \\ 2k-l+1=2,\quad l\;{\rm odd} \end{cases} \ee \vskip 4pt\noindent Then $2k-l+1\geq 2$, and hence: $$ \sum_{q\in\Z^l}\,\frac1{1+\vert q\vert^{2k}}\leq 2\int_{\R^l}\,\frac{du_1\cdots du_l}{1+\|u\|^{2k}}\leq 2\frac{\pi^{(l-1)/2}}{\Gamma(\frac{l+1}{2})}\int_0^\infty\frac{\rho^{l-1}}{1+\rho^{2k}}\,d\rho. $$ Now: \begin{eqnarray*} && \int_0^\infty\frac{\rho^{l-1}}{1+\rho^{2k}}\,d\rho =\frac 1 {2k}\int_0^\infty\,\frac{u^{l/2k -1}}{1+u}\,du %=\int_0^1\,\frac{u^{l/2k -1}}{1+u}\,du+\int_1^\infty\,\frac{u^{l/2k -1}}{1+u}\,du \\ && \leq \frac 1 {2k}\left(\int_0^1\, u^{l/2k -1}\,du+\int_1^\infty\,{u^{l/2k -2}}\,du\right)=\frac{1}{(4k-l)(2k-l)} \end{eqnarray*} This allows us to conclude: \begin{eqnarray*} \sum_{q\in\Z^l}\,\sup_\xi\vert\A(\xi,q,\hbar)\vert &\leq &(2\pi)^l \sum_{|\alpha|\leq 2k}\Vert \partial_x^{\alpha}\A(\xi,x;\hbar)\Vert_{L^\infty(\R^l\times\T^l)}\cdot \sum_{q\in\Z^l}\,\frac1{1+\vert q\vert^{2k}} \\ &\leq & 2^{l+1}\cdot \frac{\pi^{(3l-1)/2}}{\Gamma(\frac{l+1}{2})}\frac{1}{l+2}\sum_{|\alpha|\leq 2k}\,\Vert \partial_x^k\A(\xi,x;\hbar)\Vert_{L^\infty(\R^l\times\T^l)}. \end{eqnarray*} with $k$ given by (\ref{kappa}). This proves the assertion. \end{proof} \begin{remark} Thanks to Lemma \ref{azione} we immediately see that, when $\A(\xi, x,\hbar)=\F(\L_\omega(\xi),x;\hbar)$, \vskip 8pt\noindent \bea \label{quant2} && \A(\hbar)f=\int_{\R}\sum_{q\in\Z^l}\widehat{\F}(p,q;\hbar)U_h(p\om ,q)f\,dp \\ \nonumber && = \int_{\R}\sum_{q\in\Z^l}\widehat{\F}(p,q;\hbar)e^{i\la q,x\ra+i\hbar p\la\om,q\ra/2}f(x+\hbar p\om)\,dp\quad f\in L^2(\T^l) \eea \vskip 5pt\noindent where, again, $p\om:=(p\om_1,\dots,p\om_l)$. Explicitly, \eqref{azioneAq} and \eqref{azioneAop} become: \begin{eqnarray} \label{azioneAom} && A(\hbar)e_m(x)= \sum_{q\in\Z^l}e^{i\la (m+q),x\ra}\A(\hbar \la\om,(m+q/2)\ra,q,\hbar) \\ && \label{azioneAqom} A(q,\hbar)e_m(x)= e^{i\la (m+q),x\ra}\A(\hbar\la\om,(m+q/2)\ra,q,\hbar) \end{eqnarray} \end{remark} \begin{remark} If $\A$ does not depend on $x$, then $\A(\xi,q,\hbar)=0, q\neq 0$, and (\ref{azioneAop}) reduces to the standard (pseudo) differential action \bea (A(\hbar) u)(x)=\sum_{m\in\Z^l}\overline{\A}(m\hbar ,\hbar) \widehat{u}_m e^{i\la m,x\ra}=\sum_{m\in\Z^l}\overline{\A}(-i\hbar\nabla,\hbar) \widehat{u}_m e^{i\la m,x\ra} \eea because $-i\hbar\nabla e_m=m\hbar e_m$. On the other hand, if $\F$ does not depend on $\xi$ (\ref{azioneAop}) reduces to the standard multiplicative action \be (A (\hbar)u)(x)=\sum_{q\in\Z^l}\A(q,\hbar)e^{i\la q,x\ra}\sum_{m\in\Z^l}\widehat{u}_m e^{i\la m,x\ra}=\A(x,\hbar)u(x) \ee \end{remark} \noindent \begin{corollary} \label{corA} Let $A(\hbar): L^2(\T^l)\to L^2(\T^l)$ be defined as in \ref{Qdef}. Then: \begin{enumerate} \item $\forall\rho\geq 0, \forall\,k\geq 0$ we have: \be\label{stimz} \Vert A(\hbar)\Vert_{L^2\to L^2}\leq\Vert\A\Vert^\dagger_{\rho,k} \ee and, if $\A(\xi, x,\hbar)=\A(\L_\omega(\xi),x;\hbar)$ \be\label{stimg} \Vert A(\hbar)\Vert_{L^2\to L^2}\leq\Vert\A\Vert_{\rho,k}. \ee \item \bea \label{elm44} && \la e_{m+s}, A(q,\hbar)e_m\ra =\delta_{q,s}\A( (m+q/2)\ra\hbar,q,\hbar) \\ && \label{elm55} \la e_{m+s},A(\hbar)e_m\ra =\A((m+s/2)\hbar,s,\hbar) \eea and, if $\A(\xi, x,\hbar)=\F(\L_\omega(\xi),x;\hbar)$ \bea \label{elm4} && \la e_{m+s}, F(q,\hbar)e_m\ra =\delta_{q,s}\F(\la\om, (m+q/2)\ra\hbar,q,\hbar) =\delta_{q,s}\F(\L_\om (m+s/2)\hbar,q,\hbar) \\ && \label{elm5} \la e_{m+s},F(\hbar)e_m\ra =\F(\la\om,(m\hbar+s\hbar/2)\ra,s,\hbar) =\F(\L_\om(m\hbar+s\hbar/2),s,\hbar) \eea Equivalently: \be \la e_m,A(\hbar) e_n\ra=\A((m+n)\hbar/2,m-n,\hbar) \ee \item $A(\hbar)$ is an operator of order $-\infty$, namely there exists $C(k,s)>0$ such that \be \|A(\hbar)u\|_{H^k(\T^l)}\leq C(k,s)\|u\|_{H^s(\T^l)}, \quad (k,s)\in\R,\; k\geq s \ee \end{enumerate} \end{corollary} \begin{proof} (1) Formulae \eqref{stimz} and \eqref{stimg} are straighforward consequences of Formula (\ref{maggC}). \vskip 5pt\noindent (2) (\ref{elm4}) immediately yields (\ref{elm5}). In turn, (\ref{elm4}) follows at once by \eqref{azioneAq}. \vskip 5pt\noindent (3) The condition $\A\in\J(\rho)$ entails: \begin{eqnarray} \label{stimaexp} \sup_{(\xi;\hbar)\in\R^l\times [0,1]}|\A(\xi;q,\hbar)|e^{\rho |q|}\leq e^{\rho |q|}\max_{\hbar\in [0,1]}\|\widehat{\A}(p;q,\hbar)\|_1\to 0, \;|q|\to \infty. \end{eqnarray} Therefore: \begin{eqnarray*} \|A(\hbar)u\|^2_{H^k}&\leq& \sum_{(q,m)\in\Z^l\times\Z^l}(1+|q|^2)^k\A((m+q/2)\hbar,q,\hbar) |^2\cdot |\widehat{u}_m|^2 \\ &\leq& \sum_{q\in\Z^l}\,\sup_{q,m}(1+|q|^2)^k|\A((m+q/2)\hbar,q,\hbar) |^2\sum_{m\in\Z^l}\,(1+|m|^2)^{s}|\widehat{u}_m|^2 \\ &=& C(k,s)\|u\|^2_{H^s} \\ C(k,s)&:=&\sum_{q\in\Z^l}\,\sup_{q,m}(1+|q|^2)^k|\A((m+q/2)\hbar,q,\hbar)|^2 \end{eqnarray*} where $00$, $k=0,1,\ldots$. Then $\{\F,\L_\om\}_M\in\J_k(\rho-d)$ $\forall\,00$. \end{proof} \subsubsection{ Assertion {\mbox{\bf ($1^\dagger$)}}}\label{1'} By definition \begin{eqnarray} && \|\F(\hbar)\sharp\G(\hbar)\|_{\rho,k}^\dagger= \nonumber \sum_{\gamma=0}^k\int_{\R^{2l}\times\R^{2l}}| \partial^\gamma_\hbar [\widehat{\F}(t^\prime-t,\hbar) \widehat{\G}(t^\prime,\hbar)e^{i\hbar\Om_\om(t^\prime,t^\prime-t)}] |\mu_{k-\gamma}(t)e^{\rho |t|}\,d\lambda(t^\prime) d\lambda(t) \nonumber \end{eqnarray} whence \begin{eqnarray} && \|\F(\hbar)\sharp\G(\hbar)\|^\dagger_{\rho,k}= \nonumber \\ && \sum_{\gamma=0}^k\sum_{j=0}^\gamma\binom {\gamma}{j}\int_{\R^{2l}\times \R^{2l} }\vert\partial_\hbar^{\gamma-j} [\widehat{\F}(t^\prime-t,\hbar) \widehat{\G}(t^\prime,\hbar)]\vert\Omega_\om(t^\prime-t,t^\prime)\vert^j \mu_{k-\gamma}(t)e^{\rho |t|}\,d\lambda(t^\prime) d\lambda(t)= \nonumber \\ && \nonumber \sum_{\gamma=0}^k\sum_{j=0}^\gamma\sum_{i=0}^{\gamma-j}\binom {\gamma}{j}\binom{j}{i}\int_{\R^{2l}\times\R^{2l}} \vert\partial_\hbar^{\gamma-j-i}\widehat{\F}(t^\prime-t,\hbar) \partial_\hbar^{i}\widehat{\G}(t^\prime,\hbar) \vert\vert\Omega_\om(t^\prime-t,t^\prime)\vert^j\mu_{k-\gamma}(t)e^{\rho|t|}\,d\lambda(t^\prime) d\lambda(t) \end{eqnarray} By Lemma \ref{symp} and the inequality $\ds \mu_k(t^\prime-t)\leq 2^{k/2}\mu_k(t^\prime)\mu_k(t)$ we get, with $t=(p,s): t^\prime=(p^\prime,s^\prime)$ \begin{eqnarray*} && \vert\Omega_\om(t^\prime-t,t^\prime)\vert^j\mu_{k-\gamma}(t)\leq 2^j\mu_j(t^\prime-t)\mu_j(t^\prime)\mu_{k-\gamma}(t) \\ && \leq 2^j\mu_jt^\prime-t)\mu_j(t^\prime)\mu_{k-\gamma}(t)2^{(k-\gamma)/2}\mu_{k-\gamma}(t^\prime -t)\mu_{k-\gamma}(t) \\ && \leq 2^{j+(k-\gamma)/2}\mu_{k-\gamma+j}(t^\prime -t)\mu_{k-\gamma+j}(t) \end{eqnarray*} Denote now $\gamma-j-i=k-\gamma^\prime$, $i=k-\gamma^{\prime\prime}$ and remark that $j\leq\gamma^\prime$, $i\leq\gamma-j$. Then: \begin{eqnarray*} 2^{j+(k-\gamma)/2}\mu_{k-\gamma+j}(t^\prime -t)\mu_{k-\gamma+j}(t) \leq 2^k\mu_{\gamma^\prime}(t^\prime)\mu_{\gamma^{\prime\prime}}(t) \end{eqnarray*} Since $\ds \binom {\gamma}{j}\binom{j}{i}\leq 4^k$ and the sum over $k$ has $(k+1)$ terms we get: \begin{eqnarray*} && \|\F(\hbar)\sharp\G(\hbar)\|^\dagger_{\rho,k} \leq \\ && (k+1)4^k\,\sum_{\gamma^\prime,\gamma^{\prime\prime}=0}^k\int_{\R^{2l}\times\R^{2l}} |\partial^{k-\gamma^\prime}_\hbar\widehat{\F}(t^\prime -t,\hbar) |\partial^{k-\gamma^{\prime\prime}}_\hbar\widehat{\G}(t^\prime,\hbar)| \mu_{\gamma^\prime}(t^\prime -t)\mu_{\gamma^{\prime\prime}}(t)e^{\rho |t|}\,d\lambda(t^\prime) d\lambda(t) \end{eqnarray*} Now we can repeat the argument of Lemma \ref{MoyalS} to conclude: \begin{eqnarray*} \|\F(\hbar)\sharp\G(\hbar)\|_{\rho,k}^\dagger \leq (k+1)4^k \|\F\|^\dagger_{\rho,k} \cdot \|\G\|^\dagger_{\rho,k} \end{eqnarray*} which is (\ref{2conv'}). Assertion {\mbox{\bf ($3^\dagger$)}}, formula (\ref{simple'}) is the particular case of (\ref{2conv'}) obtained for $\Om_\om=0$, and Assertion ${\bf (3)}$, formula (\ref{simple}), is in turn particular case of (\ref{simple'}) . \subsubsection{ Assertion{\mbox{\bf ($2^\dagger$)}}}\label{2'} By definition: \begin{eqnarray*} \|\{\F(\hbar),\G(\hbar)\}_M\|^\dagger_{\rho,k}= \sum_{\gamma=0}^k\int_{\R^{2l}\times\R^{2l}}| \partial^\gamma_\hbar [\widehat{\F}(t^\prime -t,\hbar) \widehat{\G}(t^\prime,\hbar)\sin\hbar\Omega(t^\prime-t,t^\prime)/\hbar] |\mu_{k-\gamma}(t)e^{\rho |t|}\,d\lambda(t^\prime) d\lambda(t). \end{eqnarray*} Lemma \ref{sin} entails: $$ \vert\partial_\hbar^j \sin\hbar\Omega(t^\prime-t,t^\prime)/\hbar\vert\leq \vert \Omega(t^\prime-t,t^\prime)\vert^{j+1} $$ and therefore: \begin{eqnarray} && \|\{\F(\hbar),\G(\hbar)\}_M\|_{\rho,k}\leq \nonumber \\ && \sum_{\gamma=0}^k\sum_{j=0}^\gamma\binom {\gamma}{j}\int_{\R^{2l}\times \R^{2l} }\vert\partial_\hbar^{\gamma-j} [\widehat{\F}(t^\prime -t,\hbar) \widehat{\G}(t^\prime,\hbar)]\vert\Omega_\om(t^\prime-t,t^\prime)\vert^{j+1} \mu_{k-\gamma}(t)e^{\rho(|t|}\,d\lambda(t^\prime) d\lambda(t)= \nonumber \\ && \nonumber \sum_{\gamma=0}^k\sum_{j=0}^\gamma\sum_{i=0}^{\gamma-j}\binom {\gamma}{j}\binom{j}{i}\int_{\R^{2l}\times\R^{2l}} \vert\partial_\hbar^{\gamma-j-i}\widehat{\F}(t^\prime -t,\hbar) \partial_\hbar^{i}\widehat{\G}(t^\prime,\hbar) \vert\vert\Omega_\om(t^\prime-t,t^\prime)\vert^{j+1}\mu_{k-\gamma}(t)e^{\rho |t|}\,d\lambda(t^\prime) d\lambda(t) \end{eqnarray} Let us now absorb a factor $\vert\Omega_\om(t^\prime-t,t^\prime)\vert^{j}$ in exactly the same way as above, and recall that $\vert\Omega_\om(t^\prime-t,t^\prime)\vert\leq \vert (t^\prime-t)t^\prime\vert$. We end up with the inequality: \begin{eqnarray*} && \|\{\F(\hbar),\G(\hbar)\}_M\|^\dagger_{\rho,k} \leq \\ && (k+1)4^k\,\sum_{\gamma^\prime,\gamma^{\prime\prime}=0}^k\int_{\R^{2l}\times\R^{2l}} |\partial^{k-\gamma'}_\hbar\widehat{\F}(t^\prime -t,\hbar) |\partial^{k-\gamma"}_\hbar\widehat{\G}(t^\prime,\hbar)| |t^\prime -t||t^\prime| \mu_{\gamma^\prime}(t^\prime -t)\mu_{\gamma^{\prime\prime}}(t^\prime)e^{\rho( |t|}\,d\lambda(t^\prime) d\lambda(t) \end{eqnarray*} Repeating once again the argument of Lemma \ref{MoyalS} we finally get: \begin{eqnarray*} \|\{\F(\hbar),\G(\hbar)\}_M\|^\dagger_{\rho-d-d_1,k} \leq \frac{(k+1)4^k}{e^2d_1(d+d_1)} \|\F\|^\dagger_{\rho,k} \cdot \|\G\|^\dagger_{\rho-d,k} \end{eqnarray*} which is (\ref{normaM2'}). Once more, Assertion ${\bf (2)}$ is a particular case of (\ref{normaM2'}) and Assertion ${\bf (1)}$ a particular case of (\ref{2conv'}). This completes the proof of Proposition 3.10. \vskip 1cm \section{A sharper version of the semiclassical Egorov theorem}\label{sectionegorov} Let us state and prove in this section a particular variant of the semiclassical Egorov theorem (see e.g.\cite{Ro}) which establishes the relation between the unitary transformation $\ds e^{i\ep W/i\hbar}$ and the canoni\-cal transformation $\phi^\ep_{\W_0}$ generated by the flow of the symbol $\W(\xi,x;\hbar)|_{\hbar=0}:=\W_0(\xi,x)$ (principal symbol) of $W$ at time $1$. The present version is sharper in the sense that the usual one allows for a $O(\hbar^\infty)$ error term. \begin{theorem} Let $\rho>0, k=0,1,\ldots$ and let $A,W\in J^\dagger_k(\rho)$ with symbols $\mathcal A,\ \mathcal W$. Then: \be \nonumber S_\ep:=e^{i\frac {\ep W}\hbar}(L_\omega+A)e^{-i\frac {\ep W}\hbar}=L_\omega+B \ee where: \begin{enumerate} \item $\forall\,00$ we can always find $00; \quad \lim_{|u|\to \infty}\frac{|\F(u,\hbar)|}{|u|}=C>0 $$ {\it uniformly with respect to $\hbar\in [0,1]$.} \vskip 5pt\noindent {\textbf{Condition (3)}} {\it Set: \be \label{Kappa} \K_\F(u,\eta,\hbar)=\frac{\eta}{\F(u+\eta,\hbar)-\F(u,\hbar)} \ee Then there is $0<\Lambda(\F)<+\infty$ such that} \be \label{KB} \sup_{u\in\R,\eta\in\R,\hbar\in [0,1]}\vert\K_\F(u,\eta,\hbar)\vert<\Lambda. \ee \vskip 5pt\noindent The first result deals with the identification of the operators $W$ and $M$ through the determination of their matrix elements and corresponding symbols $\W$ and $\M$. \begin{proposition} \label{WN} Let $V\in J(\rho)$, $\rho>0$, and let $W$ and $M$ be the minimal closed operators in $L^2(\T^n)$ generated by the infinite matrices \be \label{sheq1} \la e_m,We_{m+q}\ra =\frac{i\hbar\la e_m,Ve_{m+q}\ra}{\F(\la \om,m\ra\hbar,\hbar)-\F(\la\om,(m+q)\ra\hbar,\hbar)},\quad q\neq 0, \quad \la e_m,We_m\ra=0 \ee \be \la e_m,Me_m\ra=\la e_m,Ve_m\ra,\qquad \la e_m,Me_{m+q}\ra=0, \quad q\neq 0 \label{sheq2} \ee on the eigenvector basis $e_m: m\in\Z^l$ of $L_\om$. Then: \begin{enumerate} \item $W$ and $M$ are continuous and solve the homological equation (\ref{heq}); \item The symbols $\W(x,\xi;\hbar)$ and $\M(\xi,\hbar)$ have the expression: \bea \label{defW} && \M(\xi;\hbar)=\overline{\V}(\L_\om(\xi);\hbar);\quad \W(\L_\om(\xi),x;\hbar)=\sum_{q\in\Z^l,q\neq 0}\W(\L_\om(\xi),q;\hbar)e^{i\la q,x\ra} \\ && \W(\L_\om(\xi),q;\hbar):=\frac{i\hbar\V(\L_\om(\xi);q;\hbar)}{\F(\L_\om(\xi);\hbar)-\F(\L_\om(\xi+q),\hbar)}, \;q\neq 0; \quad \overline{\W}(\L_\om(\xi);\hbar)=0. \eea \vskip 4pt\noindent Here the series in (\ref{defW}) is $\|\cdot\|_\rho$ convergent; $\overline{\V}(\L_\om(\xi);\hbar)$ is the $0$-th coefficients in the Fourier expansion of $\V(\L_\om(\xi),x,\hbar)$. \end{enumerate} \end{proposition} \begin{proof} Writing the homological equation in the eigenvector basis $e_m: m\in\Z^l$ we get \vskip 7pt\noindent \be \label{mheq} \la e_m,\frac{[\F(L_\om),W]}{i\hbar}e_n\ra+\la e_m,Ve_n\ra=\la e_m,M(L_\om)e_n\ra\delta_{m,n} \ee \vskip 5pt\noindent which immediately yields (\ref{sheq1},\ref{sheq2}) setting $n=m+q$. As far the continuity is concerned, we have: \vskip 6pt\noindent $$ \frac{i\hbar}{\F(\la \om,m\ra\hbar,\hbar)-\F(\la\om,(m+q)\ra\hbar,\hbar)}=\la\om,q\ra^{-1}\frac{\eta} {\F(\la \om,m\ra\hbar,\hbar)-\F(\la\om,m\ra\hbar+\eta,\hbar)},\quad \eta:=\la q,\om\ra\hbar. $$ \vskip 7pt\noindent and therefore, by (\ref{KB}) and the diophantine condition: $$ |\la e_m,We_{m+q}\ra|\leq \gamma |q|^\tau\Lambda |\la e_m,Ve_{m+q}\ra|. $$ The assertion now follows by Corollary \ref{corA}, which also entails the $\|\cdot\|_\rho$ convergence of the series (\ref{defW}) because $\V\in \J_\rho$. Finally, again by Corollary \ref{corA}, formulae \eqref{elm4}, \eqref{elm5}, we can write $$ \la e_m,We_{m+q}\ra= \W(\la \om,(m+q/2)\ra\hbar,q,\hbar); \quad \la e_m,Me_m\ra=\M(\om,m\ra\hbar,\hbar)=\V(\L_\om(\om,m\ra\hbar,0,\hbar) $$ and this concludes the proof of the Proposition. \end{proof} The basic example of $\F$ is the following one. Let: \bea \label{FNl} && \bullet \qquad \F_{\ell}(u,\ep;\hbar)=u+\Phi_{\ell}(u,\ep,\hbar),\qquad \ell=0,1,2,\ldots \\ && \bullet \qquad \Phi_{\ell}(\ep,\hbar):=\ep\N_{0}(u;\ep,\hbar)+\ep^2\N_{1}(u;\ep,\hbar)+\ldots+\ep_{\ell}\N_{\ell}(u,\ep,\hbar), \quad \ep_{j}:=\ep^{2^{j}}. \eea where we assume holomorphy of $\ep\mapsto \N_s(u,\ep,\hbar)$ in the unit disk and the existence of $\rho_0>\rho_1>\ldots>\rho_{\ell}>0$ such that: \begin{itemize} \item[($N_s$)] $\ds\qquad\qquad\qquad\qquad\quad \max_{|\ep|\leq 1} \vert\N\vert_{\rho_s}<\infty, \qquad .$ \end{itemize} Denote, for $\zeta\in\R$: \vskip 6pt\noindent \be \label{gl} g_\ell(u,\zeta;\ep,\hbar):=\frac{\Phi_{\ell-1}(u+\zeta;\ep,\hbar)-\Phi_{\ell-1}(u;\ep,\hbar)}{\zeta} \ee \vskip 6pt\noindent Let furthermore: \bea \label{ddll} && 00, \;s=0,\ldots,\ell-1 \\ && \delta_\ell:=\sum_{s=0}^{\ell-1}d_\ell <\rho \eea and set, for $j=1,2,\ldots$: \bea \label{theta} && \theta_{\ell,k}(\N,\ep):=\sum_{s=0}^{\ell-1}\frac{|\ep_s|\,|\N_s|_{\rho_s,k}}{ed_{s}}, \qquad \theta_{\ell}(\N,\ep):=\theta_{\ell,0}(\N,\ep). \eea By Remark 2.4 we have \begin{eqnarray} \label{Theta} && \theta_{\ell,k}(\N,\ep)=\sum_{s=0}^{\ell-1}\frac{|\ep_s|\,\|\N_s\|_{\rho_s,k}}{ed_{s}} \end{eqnarray} \begin{lemma} \label{propN} In the above assumptions: \begin{enumerate} \item For any $R>0$ the function $\zeta\mapsto g_\ell(u,\zeta,\ep,\hbar)$ is holomorphic in $\{\zeta\;|\,\,|\zeta|0. \ee Then: \be \label{stimaKg} \sup_{\zeta\in\R;u\in\R}|\K_\F(u,\zeta,\ep,\hbar)|_{\rho_\ell}\leq \frac{1}{|\zeta|}\cdot \frac1{1-\theta_{\ell}(\N,\ep)} \ee \item \bea && \label{stimadgu} \sup_{\zeta\in\R}\,|\partial^j_u g(u,\zeta,\ep,\hbar)|_{\rho_\ell}\leq \theta_{\ell,j}(\N,\ep) \\ && \label{stimadgeta} \sup_{\zeta\in\R}\,|\partial^j_\zeta g(u,\zeta,\ep,\hbar)|_{\rho_\ell}\leq \theta_{\ell,j}(\N,\ep) \\ && \label{stimadgh} \sup_{\zeta\in\R}\,|\partial^j_\hbar g(u,\zeta,\ep,\hbar)|_{\rho_\ell }\leq \theta_{\ell,j}(\N,\ep). \eea \end{enumerate} \end{lemma} \begin{proof} The holomorphy is obvious given the holomorphy of $\N_s(u;\ep,\hbar)$. To prove the estimate (\ref{convN}), denoting $\widehat{\N}_s(p,\ep,\hbar)$ the Fourier transform of $\N_s(\xi,\ep,\hbar)$ we write \vskip 4pt\noindent \begin{eqnarray} && \label{gF} g_\ell(u,\zeta,\ep,\hbar)=\frac{1}{\zeta}\sum_{s=0}^{\ell-1}\,\ep_s\,\int_\R\widehat{\N}_\ell(p,\ep,\hbar)(e^{i\zeta p}-1)e^{iu p}\,dp= \\ \nonumber && \frac{2}{\zeta}\sum_{s=0}^{\ell-1}\,\ep_s\,\int_\R\widehat{\N}_\ell(p,\ep,\hbar)e^{ip(u+\zeta)/2}\sin{\zeta p/2}\,dp\qquad\quad \end{eqnarray} which entails: \begin{eqnarray*} && \sup_{{\zeta\in\R}}|g_\ell(u,\zeta,\ep,\hbar)|_{\rho_\ell}=\sup_{{\zeta\in\R}}\int_\R\,|\widehat{g}_\ell (p,\zeta,\ep,\hbar)|e^{\rho_\ell |p|}\,dp \\ && \leq \max_{\hbar\in [0,1]}\sum_{s=0}^{\ell-1}|\ep_s|\, \int_\R|\widehat{\N}_s(p,\ep,\hbar) p|e^{(\rho_s-d_s) |p|}\,dp \leq \frac1{e}\sum_{s=0}^{\ell-1}\,|\ep_s|\,\frac{|\N_s|_{\rho_s}}{d_s}= \theta_\ell(\N,\ep,1)\,\qquad 0 \rho_{\ell+1}>0$, $k=0,1,\ldots$. Let $\V_\ell(\L_\om(\xi),x;\ep,\hbar)\in\J_k(\rho)$ be its symbol. Then for any $\ds \theta_{\ell}(\N,\ep)<1$ the homological equation (\ref{heq}), rewritten as \vskip 4pt\noindent \be \label{heqell} \frac{[\F_\ell(L_\om),W_\ell]}{i\hbar}+V_{\ell}=N_\ell(L_\om,\ep) \ee \vskip 6pt\noindent \be \label{Moell} \{\F_\ell(\L_\om(\xi),\ep,\hbar),\W_\ell(x,\xi;\ep,\hbar)\}_M+\V_{\ell}(x,L_\om(\xi);\ep,\hbar)={\mathcal N}_\ell(\L_\om(\xi),\ep,\hbar) \ee \vskip 4pt\noindent admits a unique solution $(W_\ell,N_\ell)$ of Weyl symbols $\W_\ell(\L_\om(\xi),x;\ep,\hbar)$, $\N_\ell(\L_\om(\xi),\ep,\hbar)$ such that \begin{enumerate} \item $W_\ell=W^\ast_\ell\in J_k(\rho_\ell)$, with: \bea && \label{Thm5.1} \|W_\ell\|_{\rho_{\ell+1},k}=\|\W\|_{\rho_{\ell+1},k}\leq A(\ell,k,\ep)\|\V_\ell\|_{\rho_{\ell},k} \\ \nonumber && {} \\ && \label{Adrk} A(\ell,k,\ep)=\gamma \frac{\tau^\tau}{(ed_\ell)^\tau}\left[1+\frac{2^{k+1}(k+1)^{2(k+1)}k^k}{(e\delta_\ell)^{k}[1-\theta_\ell(\N,\ep)]^{k+1}} \theta_{\ell,k}^{k+1}\right]. \eea \vskip 6pt\noindent \item $\N_\ell=\overline{\V}_\ell$; therefore $\N_\ell\in J_k(\rho_\ell)$ and $ \|\N \|_{\rho_\ell,k} \leq \|\V_\ell\|_{\rho_\ell,k} .$ \end{enumerate} \end{theorem} \begin{proof} The proof of (2) is obvious and follows from the definition of the norms $\Vert\cdot\Vert_\rho$ and $\Vert\cdot\Vert_{\rho,k}$. The self-adjointess property $W=W^*$ is implied by the construction itself, which makes $W$ symmetric and bounded. Consider $\W_\ell$ as defined by (\ref{defW}). Under the present assumptions, by Lemma \ref{propN} we have: \vskip 8pt\noindent $$ \W_\ell(\L_\om(\xi),q;\ep,\hbar):=\frac1{\la\om,q\ra}\frac{i\hbar\V_\ell(\L_\om(\xi);q;\ep,\hbar)}{1+ g_\ell(\L_\om(\xi);\la\om,q\ra\hbar,\ep,\hbar)}, \quad q\neq 0; \quad \W_\ell(\cdot,0;\hbar)=0. $$ \vskip 8pt\noindent By the $\|\cdot\|_{\rho_\ell}$-convergence of the series (\ref{sgg}) we can write \begin{eqnarray} && \partial^\gamma_\hbar \W_\ell(\L_\om(\xi),q;\ep,\hbar)=\sum_{n=0}^\infty\,(-\ep)^n\,\partial^\gamma_\hbar \W_{\ell,n}(\L_\om(\xi),q;\ep,\hbar), \\ && \W_{\ell,n}(\L_\om(\xi),q;\ep,\hbar)=\frac1{\la\om,q\ra}\V_\ell(\L_\om(\xi);q;\ep,\hbar)[g_\ell(\L_\om(\xi);\la\om,q\ra\hbar,\ep,\hbar)]^n \\ \label{derivateWn} && \partial^\gamma _\hbar\W_{\ell,n}(\L_\om(\xi),q;\ep,\hbar)= \\ \nonumber &&\sum_{j=0}^\gamma\,\binom{\gamma}{j}\,\partial^{\gamma-j}_\hbar \V_\ell(\L_\om(\xi);q;\ep,\hbar)D^j_\hbar [g_\ell(\L_\om(\xi);\la\om,q\ra\hbar,\ep,\hbar)]^n \end{eqnarray} \vskip 4pt\noindent where $D_\hbar$ denotes the total derivative with respect to $\hbar$. We need the following preliminary result. \begin{lemma} \label{derivateg} Let $\zeta(\hbar):=\la\om,q\ra\hbar$. Then: \begin{enumerate} \item \bea \label{stimadghh} |D^j_\hbar g_\ell(\L_\om(\xi),\zeta(\hbar),\ep,\hbar)|_{\rho_\ell} \leq (j+1) ({2|q|})^j \theta_{\ell,j}(\N,\ep)^2 \eea \item \bea \label{stimadgjn} |D^j_\hbar [g_\ell(\L_\om(\xi);\zeta(\hbar),\ep,\hbar)]^n|_{\rho_\ell}\leq 2n^j (\theta_\ell(\N,\ep))^{n-j} [2(j+1)|q|]^j\theta_{\ell,j}(\N,\ep)^{2j}. \eea \end{enumerate} \end{lemma} \begin{proof} The expression of total derivative $D_\hbar g$ is: \be \label{Dom} D_\hbar g(\cdot;\la\om,q\ra\hbar,\ep,\hbar)=(\la\om,q\ra\ \frac{\partial}{\partial\zeta}+\frac{\partial}{\partial\hbar})\left.g_\ell(\cdot;\zeta,\ep,\hbar)\right|_{\zeta=\la\om,q\ra\hbar} \ee By Leibnitz's formula we then have: \be D^j_\hbar g_\ell(\cdot;\la\om,q\ra\hbar,\ep,\hbar)=\sum_{i=0}^j\,\binom{j}{i}\la\om,q\ra^{j-i}\frac{\partial^{j-i}g_\ell}{\partial\zeta^{j-i}}\frac{\partial^i g_\ell}{\partial\hbar^{i}} \ee Apply now (\ref{simple}) with $k=0$, (\ref{stimadgu}) and (\ref{stimadgh}). We get: \vskip 5pt\noindent \begin{eqnarray*} \left\vert\frac{\partial^{j-i}g_\ell}{\partial\zeta^{j-i}}\frac{\partial^i g_\ell}{\partial\hbar^{i}}\right\vert_{\rho_\ell}\leq (j+1)2^j \theta_{\ell,j}(\N,\ep)^2 \end{eqnarray*} whence, since $|\om|\leq 1$: \begin{eqnarray} \label{stimaDjg} \left\vert\frac{D^jg_\ell}{D\hbar^j}\right\vert_{\rho_\ell} \leq (j+1)(2)^j{|q|^j}\theta_{\ell,j}(\N,\ep)^2 \end{eqnarray} This proves Assertion (1). To prove Assertion (2), let us first note that \be D^j_\hbar [g_\ell(\L_\om(\xi);\la\om,q\ra\hbar,\ep,\hbar)]^n=P_{n,j}\left(g_\ell,\frac{Dg_\ell}{D\hbar},\ldots,\frac{D^jg_\ell}{D\hbar^j}\right). \ee \vskip 5pt\noindent where $P_{n,j}(x_1,\ldots,x_j)$ is a homogeneous polynomial of degree $n$ with $n^j$ terms. Explicitly: $$ P_{n,j}\left(g_\ell,\frac{Dg_\ell}{D\hbar},\ldots,\frac{D^jg_\ell}{D\hbar^j}\right)=\sum_{j=1}^n\,{g_\ell}^{n-j}\,\prod_{{k=1}\atop {j_1+\ldots+j_k=j}}^j \frac{D^{j_k}g_\ell}{D\hbar^{j_k}}. $$ Now (\ref{stimadghh}), (\ref{stimaDjg}) and Proposition \ref{stimeMo} (3) entail: \begin{eqnarray*} && |D^j_\hbar [g_\ell(\L_\om(\xi);\la\om,q\ra\hbar,\ep,\hbar)]^n|_{\rho_\ell}\leq n^j|g|_{\rho_\ell}^{n-j} \prod_{{k=1}\atop {j_1+\ldots+j_k=j}}^j 2(j_k+1)\left({2|q|}\right)^{j_k}\theta_{\ell,j_k}(\N,\ep)^2 \\ && \leq 2n^j (\theta_\ell(\N,\ep))^{n-j} [2(j+1)|q|]^j\theta_{\ell,j}(\N,\ep)^{2j}. \end{eqnarray*} This concludes the proof of the Lemma. \end{proof} \noindent To conclude the proof of the theorem, we must estimate the $\|\cdot\|_{\rho_{\ell+1},k}$ norm of the derivatives $\ds \partial^\gamma _\hbar\W_{\ell,n}(\L_\om(\xi),x;\ep,\hbar)$. Obviously: \be \label{serieW} \|\W_\ell(\xi,x;\ep,\hbar)\|_{\rho_\ell+1,k}\leq \sum_{n=0}^\infty\,\|\W_{\ell,n}(\xi,x;\ep,\hbar)\|_{\rho_{\ell+1,k}}. \ee \vskip 4pt\noindent For $n=0$: \begin{eqnarray*} && \|\W_{\ell,0}(\xi,x;\ep,\hbar)\|_{\rho_{\ell+1,k}}\leq \gamma\sum_{\gamma=0}^k\int_{\R\times\R^l}|\partial^\gamma_\hbar\widehat{\W}_{\ell,0}(p,s;\cdot)||s|^{\tau}\mu_{k-\gamma}(p\om,s)\,e^{\rho_{\ell+1} (|p|+|s|)}\,d\lambda(p,s) \\ && \leq \gamma\sum_{\gamma=0}^k\int_{\R\times\R^l}|\partial^\gamma_\hbar\widehat{\V}_{\ell,0}(p,s;\cdot)||s|^{\tau}\mu_{k-\gamma}(p\om,s)\,e^{\rho_{\ell+1} (|p|+|s|)}\,d\lambda(p,s)\leq \gamma\frac{\tau^\tau}{(ed_\ell)^\tau}\|\V_\ell\|{\rho_{\ell,k}} \end{eqnarray*} where the inequality follows again by the standard majorization \vskip 6pt\noindent $$ e^{\rho_{\ell+1} (|p|+|s|)}=e^{\rho_{\ell} (|p|+|s|)}e^{-d_\ell(|p|+|s|)}, \quad \sup_{s\in\R^l}[|s|^\tau e^{-d_\ell |s|}]\leq \gamma\frac{\tau^\tau}{(ed_\ell)^\tau} $$ \vskip 4pt\noindent on account of the small denominator estimate (\ref{DC}). For $n>0$ we can write, on account of (\ref{pm1},\ref{pm2}): \begin{eqnarray*} && \|\W_{\ell,n}(\xi,x;\cdot)\|_{\rho_{\ell+1},k}=\sum_{\gamma=0}^k\int_{\R\times\R^l}|\partial^\gamma_\hbar\widehat{\W}_{\ell,n}(p,s;\cdot)||s|^{\tau}\mu_{k-\gamma}(p\om,s)\,e^{\rho_{\ell+1} (|p|+|s|)}\,d\lambda(p,s)\leq \\ && \leq \gamma\frac{\tau^\tau}{(ed_\ell)^\tau}\sum_{\gamma=0}^k\sum_{j=0}^\gamma \,\binom{\gamma}{j}\,\int_{\R^l}{\mathcal Q}(s,\cdot)e^{\rho_\ell |s|}\,d\nu(s) \end{eqnarray*} where \begin{eqnarray*} {\mathcal Q}(s,\cdot):=\int_\R|[\partial^{\gamma-j}_\hbar \widehat{\V}_{\ell}(p;s;\cdot)]\ast [D^j_\hbar \widehat{g}^{\,\ast_n}_\ell(p;\la\om,s\ra\hbar,\cdot)] \mu_{k-\gamma}(p\om,s)\,e^{\rho_\ell |p|}\,dp \end{eqnarray*} Here $\ast$ denotes convolution with respect only to the $p$ variable, and $\widehat{g}^{\,\ast_i n}_\ell(p,\zeta,\cdot)$ denotes the $n-$th convolution of $\widehat{g}_\ell$ with itself, i.e. the $p$-Fourier transform of $g^n_\ell$. Now, by Assertion (3) of Proposition (\ref{stimeMo}) and the above Lemma: \begin{eqnarray*} && \int_{\R^l}{\mathcal Q}(s,\cdot)e^{\rho_\ell |s|}\,d\nu(s)= \\ && =\int_{\R\times\R^l}|[\partial^{\gamma-j}_\hbar \widehat{\V}_{\ell}(p;s;\cdot)]\ast_\xi [D^j_\hbar g^{\ast_\xi n}_\ell(p;\la\om,s\ra\hbar,\cdot)] \mu_{k-\gamma}(p\om,s)\,e^{\rho_\ell(|p|+|s|)}\,d\lambda(p,s) \\ && \leq \int_{\R^l}\left[\int_{\R}|[\partial^{\gamma-j}_\hbar \widehat{\V}_{\ell}(p;s;\hbar)]\ast [D^j_\hbar \widehat{g}^{\,\ast_ n}(p;\la\om,s\ra\hbar,\cdot)]|\mu_{k-\gamma}(p\om,s)\,e^{\rho_\ell |p|}\,dp\right]e^{\rho_\ell |s|} \,d\nu(s) \\ && \leq 2A(j)^j\theta_\ell(\N,\ep)^{n-j}\int_{\R^l}\int_{\R}|\partial^{\gamma-j}_\hbar \widehat{\V}_{\ell}(p;s;\cdot)|\mu_{k-\gamma}(p\om,s)\,e^{\rho_\ell |p|}|s|^{j}e^{\rho_\ell |s|} \,\,dp d\nu(s), \end{eqnarray*} with \[ A(j):= 2n (j+1) \theta_{\ell,j}(\N,\ep)^{2}. \] This yields, with $\delta_\ell$ defined by (\ref{ddll}): \begin{eqnarray*} && \|\W_{\ell,n}(\xi,x;\cdot)\|_{\rho_\ell+1,k}\leq \gamma\frac{\tau^\tau}{(ed_\ell)^\tau}\sum_{\gamma=0}^k\int_{\R\times\R^l}|\partial^\gamma_\hbar\widehat{\W}_{\ell,n}(p,s;\cdot)\mu_{k-\gamma}(p\om,s)\,e^{\rho_\ell(|p|+|s|)}\,d\lambda(p,s)\leq \\ && \leq \frac{\gamma \tau^\tau (k+1)(2A(k))^k}{(ed_\ell)^\tau}\theta_\ell(\N,\ep)^{n-j}\sum_{\gamma=0}^k\int_{\R\times \R^l} |\partial^{\gamma}_\hbar \widehat{\V}_{\ell}(p;s;\cdot)|\cdot \mu_{k-\gamma}(p\om,s)\,e^{\rho_\ell |p|}|s|^{j}e^{\rho_\ell |s|} \,\,d\lambda(p,s) \\ && \leq \frac{\gamma \tau^\tau (k+1)(2A(k))^k}{(ed_\ell)^\tau}\frac{k^{k}}{(e\delta_\ell)^{k}}\theta_\ell(\N,\ep)^{n-j}\sum_{\gamma=0}^k\int_{\R^l}\int_{\R}|\partial^{\gamma}_\hbar \widehat{\V}_{\ell}(p;s;\cdot)| \mu_{k-\gamma}(p\om,s)e^{\rho |p|}e^{\rho |s|}\,d\lambda(p,s) \\ && \leq \gamma\frac{\tau^\tau}{(ed_\ell)^\tau}\frac{(k+1)k^{k}}{(e\delta_\ell)^{k}} 2(2n)^k(\theta_\ell(\N,\ep))^{n-j}(k+1)^k\theta_{\ell,k}^{2k} \|\V_\ell\|_{\rho,k}. \end{eqnarray*} \vskip 4pt\noindent Therefore, by (\ref{serieW}): \begin{eqnarray*} && \|{\W}_\ell(\xi;x;\ep,\hbar)\|_{\rho_{\ell+1},k} \leq \sum_{n=0}^\infty\,{\W}_{\ell,n} (\xi;x;\ep,\hbar)\|_{\rho_{\ell+1},k} \leq \\ && \leq \gamma \frac{\tau^\tau}{(ed_\ell)^\tau}\|\V_\ell\|_{\rho_\ell,k}\left[1+\frac{2^{k+1}(k+1)^{k+1}k^k}{(e\delta_\ell)^{k}} \theta_{\ell,k}^{2k}\sum_{n=1}^\infty\, n^k (\theta_\ell(\N,\ep))^{n-j}\right] \\ && \leq \gamma \frac{\tau^\tau}{(ed_\ell)^\tau}\|\V_\ell\|_{\rho_\ell,k}\left[1+\frac{2^{k+1}(k+1)^{k+1}k^k}{(e\delta_\ell)^{k}} \theta_{\ell,k}^{2k-j}\sum_{n=1}^\infty\, n^k (\theta_\ell(\N,\ep))^{n}\right] \\ && \leq\gamma \frac{\tau^\tau}{(ed_\ell)^\tau}\|\V_\ell\|_{\rho_\ell,k}\left[1+\frac{2^{k+1}(k+1)^{2(k+1)}k^k}{(e\delta_\ell)^{k}[(1- \theta_\ell(\N,\ep)^{k+1}]} \theta_{\ell,k}^{k+1}\right]. \end{eqnarray*} \vskip 4pt\noindent because $j\leq k$, and \begin{eqnarray*} && \sum_{n=1}^\infty \,n^kx^n\leq \sum_{n=1}^\infty\,(n+1)\cdots (n+k) x^n=\frac{d^k}{dx^k}\,\sum_{n=1}^\infty x^{n+k} \\ && =\frac{d^k}{dx^k}\frac{x^{k+1}}{1-x}=(k+1)!\sum_{j=0}^{k+1}\left(k+1-j\atop j\right) \frac{x^{k+1-j}}{(1-x)^j}\leq \frac{2^{k+1}(k+1)!}{(1-x)^{k+1}}. \end{eqnarray*} By the Stirling formula this concludes the proof of the Theorem. \end{proof} \vskip 2pt\noindent \subsection{Towards KAM iteration}\label{towkam} Let us now prove the estimate which represents the starting point of the KAM iteration: \begin{theorem} \label{resto} Let $\F_\ell$ and $V_\ell$ be as in Theorem \ref{homeq}, and let $W_\ell$ be the solution of the homological equation (\ref{heq}) as constructed and estimated in Theorem \ref{homo}. Let (\ref{epbar}) hold and let furthermore \be \label{condepell} |\ep|<\overline{\ep}_\ell, \quad \overline{\ep}_\ell:=\left(\frac{d_\ell}{\|\W_\ell\|_{\rho_{\ell+1},k}}\right)^{2^{-\ell}}. \ee Then we have: \be \label{resto1} e^{i\ep_\ell W_\ell/\hbar}(\F_\ell(L_\om)+\ep_\ell V_\ell)e^{-i\ep_\ell W_\ell/\hbar}=(\F_\ell+\ep_\ell N_\ell)(L_\om)+\ep_\ell^2V_{\ell+1,\ep} \ee where, $\forall\,0<2d_\ell<\rho_\ell$ and $k=0,1,\ldots$: \bea && \label{resto2} \|V_{\ell+1,\ep}\|_{\rho_\ell-2d_\ell,k}\leq C(\ell,k,\ep) \frac{\|\V_\ell\|^2_{\rho_\ell,k}} {1-{|\ep_\ell |}A(\ell,k,\ep) \|\V\|_{\rho_\ell,k}/{d_\ell}} \\ && \nonumber {} \\ \label{Cdrk} && C(\ell,k,\ep):=\frac{(k+1)^2 4^{2k}}{(ed_\ell)^3}{A(\ell,k.\ep)}\left[2+|\ep_{\ell} |\frac{(k+1) 4^{k}}{(ed_\ell)^2 }{A(\ell,k.\ep)\|\V_\ell\|_{\rho_\ell,k}}{} \right] \eea \vskip 6pt\noindent Here $A(\ell,k,\ep)$ is defined by (\ref{Adrk}). \end{theorem} \begin{remark} We will verify in the next section (Remark \ref{verifica} below) that (\ref{condepell}) is actually fulfilled for $|\ep|<1/|\V|_\rho$. \end{remark} \begin{proof} To prove the theorem we need an auxiliary result, namely: \begin{lemma} \label{RResto4} For $\ell=0,1,\ldots$ let $\rho_\ell>0, \rho_0:=\rho$, $A\in J_k(\rho)$, $W_\ell\in J_k(\rho_\ell)$, $k=0,1,\ldots$. Let $W_\ell^\ast=W_\ell$, and define: \be \label{resto5} A_{\ep}(\hbar):=e^{i\ep_\ell W_\ell/\hbar}Ae^{-i\ep_\ell W/\hbar}. \ee Then, for $\ds |\ep|< (d^\prime_\ell/\|\W\|_{\rho_{\ell+1},k})^{2^{-\ell}}$, and $\forall\,00$ such that the commutator expansion for $A_{\ep}(\hbar)$: \vskip 4pt\noindent $$ A_{\ep}(\hbar)=\sum_{m=0}^\infty \frac{(i\ep_\ell)^m}{ \hbar^m m!}[W_\ell,[W_\ell,\ldots,[W_\ell,A]\ldots] $$ \vskip 4pt\noindent is norm convergent for $|\ep|<\ep_0$ if $\hbar\in]0,1[$ is fixed. The corresponding expansion for the symbols is \vskip 4pt\noindent $$ \A_{\ep}(\hbar)=\sum_{m=0}^\infty \frac{(\ep_\ell)^m}{m!}\{\W_\ell,\{\W,\ldots,\{\W_\ell,{\mathcal A}\}_M\ldots\}_M $$ \vskip 4pt\noindent Now we can apply once again Corollary \ref{multipleM}. We get, with the same abuse of notation of Theorem 4.1: \be \frac{1}{m!}\|\{\W_\ell,\{\W_\ell,\ldots,\{\W_\ell,{\mathcal A}\}_M\ldots\}_M\|_{\rho-d^\prime_\ell,k} \leq \frac{(k+1)4^{k}}{ed_1}\left(\frac{\|\W_\ell\|_{\rho_\ell,k}}{d^\prime_\ell}\right)^m \|{\mathcal A}\|_{\rho_\ell,k} \ee Therefore \vskip 4pt\noindent $$ \|A_{\ep}(\hbar)\|_{\rho_\ell-d^\prime_\ell,k}\leq \frac{(k+1)4^{k}}{ed^\prime_\ell}\|{\mathcal A}\|_{\rho_\ell,k}\sum_{m=0}^\infty |\ep|^m [\|\W\|_{\rho_{\ell+1},k}/d^\prime_\ell]^m=\frac{(k+1)4^{k}}{ed^\prime_\ell}\frac{\|{\mathcal A}\|_{\rho_\ell,k}}{1-|\ep_\ell| \|\W\|_{\rho_{\ell+1},k}/d^\prime_\ell} $$ \vskip 4pt\noindent and this concludes the proof. \end{proof} $W_\ell$ solves the homological equation (\ref{heq}). Then by Theorem \ref{homo} $W_\ell=W_\ell^\ast\in J_k(\rho_\ell-d_\ell)$, $k=0,1,\ldots$; in turn, by Assertion (3) of Corollary \ref{corA} the unitary operator $\ds e^{i\ep_\ell W_\ell/\hbar}$ leaves $H^1(\T^l)$ invariant. Therefore the unitary image of $H_\ep$ under $\ds e^{i\ep_\ell W/\hbar}$ is the real-holomorphic operator family in $L^2(\T^l)$ \be \label{S} \ep\mapsto S_{\ep}:=e^{i\ep_\ell W_\ell/\hbar}(\F_\ell(L_\om)+\ep_\ell V_\ell)e^{-i\ep\_ell W/\hbar}, \quad D(S(\ep))=H^1(\T^l) \ee Computing its Taylor expansion at $\ep_\ell=0$ with second order remainder we obtain: \begin{eqnarray}\label{lemmm} && S_{\ep}u=\F_\ell(L_\om)u+\ep_\ell N_\ell(L_\om)u+ \ep_\ell^2 V_{\ell+1,\ep}u, \quad u\in H^1(\T^l) \\ \nonumber && {} \\ && V_{\ell+1,\ep}=\frac12\int_0^{\ep_\ell} (\ep_\ell -t)e^{i t W_\ell/\hbar}\left(\frac{[N_\ell,W_\ell]}{i\hbar}+\frac{[W_\ell,V_\ell]}{i\hbar}+t \frac{[W_\ell,[W_\ell,V_\ell]]}{(i\hbar)^2}\right)e^{-itW_\ell/\hbar}\,dt \end{eqnarray} To see this, first remark that $S_0=\F(L_\om)$. Next, we compute, as equalities between continuous operators in $L^2(\T^l)$: \begin{eqnarray*} && S^\prime_{\ep}=e^{i\ep_\ell W/\hbar}([\F_\ell(L_\om),W_\ell]/i\hbar +V_\ell+\ep_\ell [V,W]/i\hbar)e^{-i\ep_\ell W/\hbar}= \\ && e^{i\ep_\ell W/\hbar}(N_\ell+\ep_\ell [V_\ell,W_\ell]/i\hbar)e^{i\ep_\ell W_\ell/\hbar}; \qquad S^\prime_0= N_\ell \\ && S^{\prime\prime}_{\ep}=e^{i\ep_\ell W_\ell/\hbar}([N_\ell,W_\ell]/i\hbar + [V_\ell,W_\ell]/i\hbar +\ep_\ell [W_\ell,[W_\ell,V_\ell]]/(i\hbar)^2)e^{-i\ep_\ell W_\ell/\hbar}, \end{eqnarray*} and this proves (\ref{lemmm}) by the second order Taylor's formula with remainder: $$ S_{\ep}=S(0)+\ep S^\prime_0+\frac12\int_0^{\ep_\ell} (\ep-t)S^{\prime\prime}(t),dt $$ The above formulae obviously yield \be \label{stimar2} \| {V}_{l+1,\ep}\|\leq |\ep_\ell |^2 \max_{0\leq |t|\leq |\ep_\ell |}\|S^{\prime\prime}(t)\| \ee Set now: \be \label{R1} R_{\ell+1,\ep}:=[N_\ell,W_\ell]/i\hbar + [V_\ell,W_\ell]/i\hbar +\ep_\ell [W_\ell,[W_\ell,V_\ell]]/(i\hbar)^2 \ee %By the above Lemma, $R_{\ell+1,\ep}$ is a continuous operator in $L^2$, corresponding to the symbol \be \label{simbR1} {\mathcal R}_{\ell+1,\ep}(\L_\om(\xi),x;\hbar)=\{\N_\ell,\W_\ell\}_M+\{\V_\ell,\W_\ell\}_M+\ep_\ell\{\W_\ell,\{\W_\ell,\V_\ell\}_M\}_M \ee Let us estimate the three terms individually. By Theorems \ref{homo} and \ref{stimeMo} we can write, with $A(\ell,k,\ep)$ given by (\ref{Adrk}): \begin{eqnarray*} && \|[N_\ell,W_\ell]/i\hbar\|_{\rho_\ell-d_\ell,k}\leq \|\{\N_\ell,\W_\ell\}_M\|_{\rho_\ell-d_\ell,k}\leq \frac{(k+1)4^k}{(ed_\ell)^2}\|\W_\ell\|_{\rho_{\ell+1},k}\|\N_\ell\|_{\rho_\ell,k} \\ && \leq \frac{(k+1)4^k}{(ed)^2} A(\ell,k,\ep)\|\V_\ell\|^2_{\rho_\ell,k} \\ && \|[V_\ell,W_\ell]/i\hbar\|_{\rho_\ell-d_\ell,k}\leq\|\{\V_\ell,\W_\ell\}_M\|_{\rho_\ell-d_\ell,k}\leq \frac{(k+1)4^k}{(ed_\ell)^2}\|\V_\ell\|_{\rho_\ell,k}\|\W_\ell\|_{\rho_{\ell+1},k}\leq \\ && \leq \frac{(k+1)4^k}{(ed_\ell)^2}A(\ell,k.\ep)\|\V_\ell\|^2_{\rho_\ell,k} \\ && \|[W_\ell,[W_\ell,V_\ell]]/(i\hbar)^2\|_{\rho_\ell-d_\ell,k}\leq \|\{\W_\ell,\{\W_\ell,\V_\ell\}_M\}_M\|_{\rho_\ell-d_\ell,k}\leq \frac{(k+1)^2 4^{2k}}{(ed_\ell)^4} \|\W_\ell\|_{\rho_{\ell+1},k}^2 \|\V_\ell\|_{\rho_\ell,k} \\ && \leq \frac{(k+1)^2 4^{2k}}{(ed_\ell)^4}A(\ell,k,\ep)^2\|\V_\ell\|_{\rho_\ell,k}^3 \end{eqnarray*} \vskip 6pt\noindent We can now apply Lemma \ref{RResto4}, which yields: \vskip 2pt\noindent \begin{eqnarray*} && \|e^{i\ep_\ell W_\ell/\hbar}[N_\ell,W_\ell] e^{-i\ep_\ell W_\ell/\hbar}/i\hbar\|_{\rho_\ell-d_\ell-d^\prime_\ell,k}\leq \frac{(k+1)^2 4^{2k}}{(ed_\ell)^2 ed^\prime_\ell}\Xi(\ell,k) \\ && \|e^{i\ep_\ell W_\ell/\hbar}[V_\ell,W_\ell] e^{-i\ep_\ell W_\ell/\hbar}/i\hbar\|_{\rho_\ell-d_\ell-d^\prime_\ell,k}\leq \frac{(k+1)^2 4^{2k}}{(ed_\ell)^2 ed^\prime_\ell}\Xi(\ell,k) \\ && \|e^{i\ep_\ell W_\ell/\hbar}[W_\ell,[W_\ell,V_\ell]] e^{-i\ep_\ell W_\ell/\hbar}/(i\hbar)^2\|_{\rho_\ell-d_\ell-d^\prime_\ell,k}\leq \frac{(k+1)^3 4^{3k}}{(ed_\ell)^4 ed^\prime_\ell}\Xi_1(\ell,k) \end{eqnarray*} where \bea && \label{Xi} \Xi(\ell,k):= A(\ell,k)\cdot\frac{\|\V_\ell\|^2_{\rho_\ell,k}} {1-|\ep_\ell |\|\W\|_{\rho_{\ell+1},k}/d^\prime_\ell} \\ && \label{Xi1} \Xi_1(\ell,k)=A(\ell,k,\ep)^2\cdot \frac{\|\V\|^3_{\rho_\ell,k}} {1-|\ep_\ell |\|\W\|_{\rho_{\ell+1},k}/d^\prime_\ell} \eea \vskip 6pt\noindent Therefore, summing the three inequalities we get \vskip 3pt\noindent \begin{eqnarray*} && \|V_{\ell+1,\ep}\|_{\rho_\ell-d_\ell-d^\prime_\ell,k}\leq \frac{(k+1)^2 4^{2k}}{(ed_\ell)^2 ed^\prime_\ell}A(\ell,k,\ep)\times \\ && \times \frac{\|\V_\ell\|^2_{\rho_\ell,k}} {1-|\ep_\ell |\|\W_\ell\|_{\rho_{\ell+1},k}/d^\prime_\ell}\left[2+|\ep_\ell|\frac{(k+1) 4^{k}}{(ed_\ell)^2 }A(\ell,k,\ep){\|\V_\ell\|_{\rho_\ell,k}} \right] \end{eqnarray*} \vskip 8pt\noindent If we choose $d^\prime_\ell=d_\ell$ this is (\ref{resto2}) on account of Theorem \ref{homo}. This concludes the proof of Theorem \ref{resto}. \end{proof} \vskip 1cm\noindent %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% \section{Recursive estimates}\label{recesti} \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}} \renewcommand{\theremark}{\thesection.\arabic{remark}} \def\P{{\mathcal P}} \setcounter{equation}{0}% \setcounter{theorem}{0}% Consider the $\ell$-th step of the KAM iteration. Summing up the results of the preceding Section we can write: \begin{eqnarray*} && \bullet\ S_{\ell,\ep}:=e^{i\ep_\ell W_\ell/\hbar}\cdots e^{i\ep_2 W_1/\hbar}e^{i\ep W_0/\hbar}(\F(L_\om)+\ep V)e^{-i\ep W_0/\hbar}e^{-i\ep_2 W_1/\hbar}\cdots e^{-i\ep_\ell W_\ell/\hbar} \\ && = e^{i\ep_\ell W_\ell/\hbar}(\F_{\ell,\ep}(L_\om)+\ep^{2^\ell} V_{\ell,\ep})e^{-i\ep_\ell W_\ell/\hbar} =\F_{\ell+1,\ep}(L_\om)+\ep_{\ell +1} V_{\ell +1,\ep}, \\ && \bullet\ \F_{\ell,\ep}(L_\om)=\F(L_\om)+\sum_{k=1}^{\ell-1} \ep_kN_k(L_\om), \quad [\F_{\ell}(L_\om),W_\ell]/i\hbar +V_{\ell,\ep} =N_\ell(L_\om,\ep) \\ && \bullet V_{\ell+1,\ep}=\frac12\int_0^{\ep_\ell} (\ep_\ell -t)e^{i t W_\ell/\hbar}R_{\ell+1,t}e^{-itW_\ell/\hbar}\,dt \\ && \bullet\ R_{\ell+1,\ep}:=[N_{\ell},W_{\ell}]/\hbar+[W_{\ell},V_{\ell,\ep}]/\hbar+\ep_{\ell} [W_{\ell},[W_{\ell},V_{\ell,\ep}]]/\hbar^2 \end{eqnarray*} We now proceed to obtain recursive estimates for the above quantities in the $\|\cdot\|_{\rho_\ell,k}$ norm. Consider (\ref{resto2}) and denote: \vskip 6pt\noindent \bea && \label{stimaPsi} \Psi(\ell,k)=\frac{(k+1)^24^k}{(ed_\ell)^3}\Pi(\ell,k); \quad \Pi(\ell,k):= \frac{[2(k+1)^2]^{k+1}k^k}{e^{k}\delta_\ell^{k}} \\ \label{Pll} && P(\ell,k,\ep):=\frac{\theta_{\ell,k}(\N,\ep)^{k+1}}{[1-\theta_\ell(\N,\ep)]^{k+1}} \eea \vskip 6pt\noindent where $\theta_{\ell,k}(\N,\ep)$ is defined by (\ref{Theta}). (\ref{stimaPsi}) and (\ref{Pll}) yield \be \label{alk} A(\ell,k,\ep)= \gamma \frac{\tau^\tau}{(ed_\ell)^\tau}[1+\Pi(\ell,k)P(\ell,k,\ep)]. \ee Set furthermore: \bea && \label{resto22} E({\ell}, k,\ep) := \frac{\Psi(\ell,k)B(\ell,k,\ep)[2+ |\ep_{\ell}| e\Psi(\ell,k)A(\ell,k,\ep)\|\V_{\ell,\ep}\|_{\rho_\ell,k}]}{1-|\ep_\ell |A(\ell,k,\ep)\|\V_{\ell,\ep}\|_{\rho_\ell,k}/d_\ell} \eea Then we have: \begin{lemma} \label{stimaVl+1} Let: \be \label{stimadenE} |\ep_\ell |A(\ell,k,\ep)\|\V_{\ell,\ep}\|_{\rho_\ell,k}/d_\ell<1. \ee Then: \be \label{restoll} \|V_{\ell+1,\ep}\|_{\rho_{\ell+1},k}\leq E({\ell}, k,\ep)\|V_{\ell,\ep}\|^{2}_{\rho_{\ell},k} \ee \vskip 3pt\noindent \end{lemma} \begin{remark} The validity of the assumption (\ref{stimadenE}) is to be verified in Proposition \ref{estl} below. \end{remark} \begin{proof} Since $d_\ell <1$, by (\ref{Cdrk}), (\ref{stimaPsi}) and (\ref{alk}) we can write: \be C(\ell,k,\ep)\leq \Psi(\ell,k)A(\ell,k,\ep))\left[2+ |\ep_{\ell}| e\Psi(\ell,k)A(\ell,k,\ep)\|\V_{\ell,\ep}\|_{\rho_\ell,k}\right] \ee and therefore, by (\ref{resto2}): \begin{eqnarray*} && \|V_{\ell+1,\ep}\|_{\rho_\ell-2d_\ell,k}\leq C(\ell,k,\ep) \frac{\|\V_\ell\|^2_{\rho_\ell,k}} {1-|\ep_\ell|A(\ell,k,\ep)\|\V\|_{\rho_\ell,k}/d_\ell} \\ && {} \\ && \leq \frac{\Psi(\ell,k)A(\ell,k,\ep)\left[2+ |\ep_{\ell}| e\Psi(\ell,k)A(\ell,k,\ep)\|\V_{\ell,\ep}\|_{\rho_\ell,k}\right]}{1-|\ep_\ell |A(\ell,k,\ep)\|\V_{\ell,\ep}\|_{\rho_\ell,k}/d_\ell}\|\V_\ell\|^2_{\rho_\ell,k}= E(\ell,k,\ep)\|\V_\ell\|^2_{\rho_\ell,k}. \end{eqnarray*} \vskip 6pt\noindent This yields (\ref{restoll}) and proves the Lemma. \end{proof} Now recall that the sequence $\{\rho_j\}$ is decreasing. Therefore: \be \|\N_{j,\ep}\|_{\rho_\ell,k}\leq \|\N_{j,\ep}\|_{\rho_j,k}= \|\overline{\V}_{j,\ep}\|_{\rho_j,k} \leq \|{\V}_{j,\ep}\|_{\rho_j,k}, \quad \;j=0,\ldots,\ell-1. \ee \vskip 4pt\noindent At this point we can specify the sequence $d_\ell, \ell=1,2,\ldots$, setting: \vskip 4pt\noindent \be \label{ddelta} d_\ell:=\frac{\rho}{(\ell+1)^2}, \qquad \ell=0,1,2,\ldots \ee \vskip 4pt\noindent Remark that (\ref{ddelta}) yields $$ d- \sum_{\ell=0}^\infty d_\ell=\rho-\frac{\pi^2}{6}>\frac{\rho}{2}. $$ as well as the following estimate \be \label{stimapigreco} \Pi(\ell,k)\leq \frac{[2(k+1)^2]^{k+1}}{e^{k}\rho^{k}} \ee \vskip 2pt\noindent We are now in position to discuss the convergence of the recurrence (\ref{restoll}). \begin{proposition} \label{estl} Let: \be \label{condep} |\ep|< \ep^\ast(\gamma,\tau,k):= \frac{1}{e^{24(3+2\tau)}(k+2)^{2\tau}\|\V\|_{\rho,k}} \ee \vskip 4pt\noindent \be \label{condrho} \rho>\lambda(k):=1+8\gamma\tau^\tau [2(k+1)^2]. \ee Then the following estimate holds: \vskip 8pt\noindent \begin{equation} \label{rec2} \|\V_{\ell,\ep}\|_{\rho_\ell,k} \leq \left(e^{8(3+2\tau)} \|V_0\|_{\rho,k}\right)^{2^{\ell}}=\left(e^{8(3+2\tau)} \|\V_0\|_{\rho,k}\right)^{2^{\ell}}, \quad \ell=0,1,2,\ldots \quad V_0:=V. \end{equation} \end{proposition} \vskip 4pt\noindent \begin{proof} We proceed by induction. The assertion is true for $\ell=0$. Now assume inductively: \vskip 6pt\noindent \be \label{Hell} |\ep_j|\|\V_{j,\ep}\|_{\rho_j,k}\leq (k+2)^{-2\tau( j+1)}, \qquad\quad 0\leq j\leq \ell. \ee \vskip 6pt\noindent Out of (\ref{Hell}) we prove the validity of (\ref{rec2}) and of (\ref{stimadenE}); to complete the induction it will be enough to show that (\ref{rec2}) implies the validity of (\ref{Hell}) for $j=\ell+1$. \vskip 6pt Let us first estimate $\theta_\ell(\N,\ep)$ as defined by \eqref{theta} assuming the validity of \eqref{Hell} . We obtain: \begin{eqnarray*} && \theta_\ell(\N,\ep)\leq \theta_{\ell,k}(\N,\ep) \leq \sum_{s=0}^{\ell-1}|\ep_s|\|\V\|_{\rho_s,k}/d_s = \frac{1}{\rho}\sum_{s=0}^{\ell-1}\,(s+1)^2(k+2)^{-2\tau (s+1)}= \\ && \frac{1}{4\rho}\frac{d^2}{d\tau^2}\sum_{s=0}^{\ell-1}\,(k+2)^{-2\tau (s+1)} =\frac{1}{4\rho}\frac{d^2}{d\tau^2}[(k+2)^{-2\tau}\frac{1-(k+2)^{-2\tau \ell}}{1-(k+2)^{-2\tau}} \leq \frac{1}{\rho}(k+2)^{-2}\leq \frac{1}{\rho} \end{eqnarray*} because $\tau>l-1\geq 1$. Now $\rho>1$ entails that \be \label{dentheta} \frac1{1-\theta_\ell}<\frac{\rho}{\rho-1}. \ee \vskip 4pt\noindent Hence we get, by (\ref{Pll}) and (\ref{Theta}), the further $(\ell,\ep)-$in\-de\-pen\-dent estimate: \be \label{Hells} P(\ell,k,\ep)\leq \frac{\rho^{k+1}}{(\rho-1)^{k+1}}\left((k+2)^2{\rho}\right)^{-k-1}=\left(\frac{1}{(\rho-1)(k+2)^2}\right)^{k+1}. \ee whence, by (\ref{alk}): \begin{eqnarray} \nonumber && A(\ell,k,\ep)\leq \gamma\frac{\tau^\tau (\ell+1)^{2\tau}}{(e\rho)^\tau}[1+[2(k+1)^2]^{k+1}\left[(\rho-1)(k+2)^2\right]^{-(k+1)}(e\rho^3)^{-k}] \\ \label{stimaAell} && \leq \gamma\frac{\tau^\tau (\ell+1)^{2\tau}}{(e\rho)^\tau}[1+\frac{2}{(\rho-1)^{k+1}}(e\rho^3)^{-k}]. \end{eqnarray} \vskip 5pt\noindent Upon application of the inductive assumption we get: \vskip 5pt\noindent \begin{eqnarray*} && |\ep_\ell | \Psi_{\ell,k}A(\ell,k,\ep)\|\V\|_{\rho_\ell,k}/d_\ell\leq \frac{ 4^k [2(k+1)^2]^{k+3}}{e^{k+3}\rho^{k+4}}(\ell+1)^{2\tau+8}|\ep_\ell | A(\ell,k,\ep)\|\V\|_{\rho_\ell,k} \\ && \leq \gamma\frac{\tau^{\tau} (\ell+1)^{2(\tau+4)}}{(e\rho)^{\tau}}[1+\frac{2}{(\rho-1)^{k+1}}(e\rho^3)^{-k}]\frac{ 4^k [2(k+1)^2]^{k+3}}{e^{k+3}\rho^{k+4}} (k+2)^{-2(\ell+1)\tau} \\ && \leq \left(\frac{2(\tau+4)}{2\tau\ln{(k+2)}}\right)^{2(\tau+4)}(k+2)^{-\frac{4(\tau+4)}{2\tau\ln{(k+2)}}}\frac{ 4^k [2(k+1)^2]^{k+3}}{e^{k+3}\rho^{k+4}}\frac{\gamma\tau^{\tau}} {(e\rho)^{\tau}}[1+\frac{2}{(\rho-1)^{k+1}}(e\rho^3)^{-k}] \end{eqnarray*} \vskip 5pt\noindent because $$ \sup_{\ell\geq 0} (\ell+1)^{2(\tau+4)}(k+2)^{-2(\ell+1)\tau} =\left(\frac{2(\tau+4)}{2\tau\ln{(k+2)}}\right)^{2(\tau+4)}(k+2)^{-\frac{4(\tau+4)}{2\tau\ln{(k+2)}}}. $$ \vskip 4pt\noindent Hence: \be \label{stimaBpsi} |\ep_\ell | \Psi_{\ell,k}A(\ell,k,\ep)\|\V\|_{\rho_\ell,k}/d_\ell\leq \frac1{2e} \ee provided \be \label{2condep} \rho\geq \lambda(k); \qquad \lambda(k)=1+8\gamma\tau^\tau [2(k+1)^2]. \ee \vskip 4pt\noindent Since $\Psi_{\ell,k}\geq 1$, if (\ref{2condep}) holds, (\ref{stimaBpsi}) a fortiori yields \vskip 4pt\noindent $$ |\ep_\ell | A(\ell,k,\ep)\|\V\|_{\rho_\ell,k}/d_\ell\leq \frac1{2}. $$ \vskip 4pt\noindent Therefore, by (\ref{resto22}): $$ E(\ell,k,\ep) \leq 3 \Psi_{\ell,k}A(\ell,k,\ep) \leq 6 \gamma\frac{\tau^\tau (\ell+1)^{2\tau}}{(e\rho)^\tau} \Psi_{\ell,k} $$ and (\ref{restoll}) in turn entails: $$ \|\V_{\ell+1}\|_{\rho_\ell+1,k}\leq \Phi_{\ell,k} \|\V_\ell\|_{\rho_\ell,k}^2, \quad \Phi_{\ell,k}:=6 \gamma\frac{\tau^\tau (\ell+1)^{2\tau}}{(e\rho)^\tau} \Psi_{\ell,k}. $$ This last inequality immediately yields \be \label{rec3} \|\V_{\ell+1}\|_{\rho_\ell,k} \leq [\|\V\|_{\rho,k}]^{2^{\ell+1}}\prod_{m=0}^{\ell}\Phi_{\ell -m,k}^{2m}. \ee \vskip 3pt\noindent Now: \begin{eqnarray*} \Phi_{\ell,k}= 6 \gamma\frac{\tau^\tau (\ell+1)^{2\tau}}{(e\rho)^\tau} \frac{(k+1)^24^{2k}}{ed_{\ell}^3}\frac{[2(k+1)^2]^{k+1}}{e^{k+\tau}d_\ell^{\tau}\delta_\ell^{k}}\leq \gamma\nu(k,\tau,\rho)(\ell+1)^{6+4\tau} \end{eqnarray*} \vskip 5pt\noindent \begin{eqnarray*} \label{nu} && \nu(k,\tau,\rho):=6\frac{\tau^{\tau} 4^{2k}[2(k+1)^2]^{k+2}}{e^{k+\tau+1}\rho^{k+\tau+3}}\leq 6\frac{\tau^{\tau} 4^{2k}[2(k+1)^2]^{k+2}}{e^{k+\tau+1}\lambda(k)^{k+\tau+3}} \leq \\ && \leq 6\frac{\tau^{\tau} 4^{2k}[2(k+1)^2]^{k+2}}{e^{k+\tau+1}[8\gamma\tau^\tau 2(k+1)^2]^{k+\tau+3}} \leq 6\left(\frac{2}{e}\right)^k\frac{1}{e^{\tau+1}\gamma^{k+\tau+3}[2(k+1)^2]^{\tau+1}} \leq \\ && \leq \frac{6}{\gamma^{\tau+3}\tau^{\tau^2+2}(2e)^{\tau+1}} \end{eqnarray*} \vskip 5pt\noindent Therefore \vskip 4pt\noindent \be \label{gammanu} \gamma\nu(k,\tau,\rho)\leq \frac{6}{\gamma^{\tau+2}\tau^{\tau^2+2}(2e)^{\tau+1}} <1 \ee \vskip 6pt\noindent because $\tau>1$ and $\gamma>1$. As a consequence, since $\Phi_{j,k}\leq \Phi_{\ell,k}, j=1,\ldots$, we get: \vskip 5pt\noindent \begin{eqnarray*} && \prod_{m=1}^{\ell}\Phi^{2m}_{\ell+1-m,k} \leq [\Phi_{\ell,k}]^{\ell(\ell+1)}\leq [\gamma\nu(k,\tau,\rho)]^{\ell(\ell+1)} (\ell+1)^{(6+4\tau)\ell(\ell+1)}\leq (\ell+1)^{(6+4\tau)\ell(\ell+1)} \end{eqnarray*} \vskip 3pt\noindent Now $\ell(\ell+1)<2^{\ell+1}$, $\forall\,\ell\in\N$. Hence we can write: $$ (\ell+1)^{(6+4\tau)\ell(\ell+1)} < [e^{(24+16\tau)}]^{2^{\ell+1}}. $$ The following estimate is thus established \bea \label{stimapsi} && \prod_{m=0}^{\ell}\Psi^{2m}_{\ell -m,k} \leq [e^{8(3+2\tau)}] ^{2^{\ell+1}}. \eea If we now define: \bea \label{mu} && \mu :=e^{8(3+2\tau)}, \qquad \mu_\ell:=\mu^{2^\ell} \eea then (\ref{rec3}) and (\ref{stimapsi}) yield: \vskip 6pt\noindent \bea && \label{GVS} \|\V_{\ell+1,\ep}\|_{\ell+1,k} \leq \left[\mu_\ell\|\V_\ell\|_{\rho_\ell,k}\right]^{2}\leq \left[\|\V\|_{\rho,k}\,\mu\right]^{2^{\ell+1}} \\ \label{GVSS} && \ep_{\ell+1}\|\V_{\ell+1,\ep}\|_{\ell+1,k} \leq \left[ \|\V\|_{\rho_\ell,k}\,\mu_\ell\ep_\ell\right]^{2} \leq \left[ \|\V\|_{\rho,k}\,\mu\ep\right]^{2^{\ell+1}} \eea \vskip 5pt\noindent Let us now prove out of (\ref{GVS},\ref{GVSS}) that the condition (\ref{Hell}) preserves its validity also for $j=\ell+1$. We have indeed, by the inductive assumption (\ref{Hell}) and (\ref{GVS}): \begin{eqnarray*} \label{verifica} && |\ep_{\ell+1}|\V_{\ell+1,\ep}\|_{\ell+1,k} \leq \left[ \|\V\|_{\rho_\ell,k}\,\mu_\ell\ep_\ell\right]^{2}\leq (k+2)^{-2\tau(\ell+1)}\ep_\ell(\mu_\ell)^2\|\V\|_{\rho_\ell,k} \\ && \leq (k+2)^{-2\tau(\ell+1)}\left[\ep\mu^3\|\V\|_{\rho,k}\right]^{2^\ell}\leq (k+2)^{-2\tau(\ell+2)} \end{eqnarray*} provided \vskip 4pt\noindent \be \label{epsast} |\ep|< \frac{1}{\mu^3\|\V\|_{\rho,k}(k+2)^{2\tau}}= \frac{1}{e^{24(3+2\tau)}\|\V\|_{\rho,k}(k+2)^{2\tau}}:=\ep^\ast(\gamma,\tau,k) \ee \vskip 8pt\noindent where the last expression follows from (\ref{mu}). This proves (\ref{condep}), and concludes the proof of the Proposition. \end{proof} \noindent %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{final}[Final estimates of $W_\ell$, $N_\ell$, $V_\ell$] %%%%%%%%%%%%%%%%%%% %Final estimates %%%%%%%%%%%%%%%% \newline Let $\V$ fulfill Assumption (H2-H4). Then the following estimates hold, $\forall \ell\in\Na$: \vskip 4pt\noindent \begin{eqnarray} \label{stimafw} \ep_\ell \|W_{\ell,\ep}\|_{\rho_{\ell+1},k}\leq \gamma\left(\frac{\tau}{e}\right)^\tau (\ell+1)^{2\tau}(1+8\gamma\tau^\tau [2(k+1)^2])^{-\tau}\cdot (\mu \ep \|\V\|_{\rho})^{2^{\ell}}. \end{eqnarray} \begin{eqnarray} \label{stimafn} \ep_\ell \|N_{\ell,\ep}\|_{\rho_\ell,k}\leq \ep_\ell \|\V_{\ell,\ep}\|_{\rho_\ell,k}\leq \left[ \|\V\|_{\rho}\,\ep \mu\right]^{2^{\ell}}. \end{eqnarray} \begin{eqnarray} \label{stimafv} \ep_{\ell+1} \|V_{\ell+1,\ep}\|_{\rho_{\ell+1},k}\leq \left[ \|V\|_{\rho}\,\ep \mu\right]^{2^{\ell+1}}. \end{eqnarray} \end{theorem} \begin{proof} Since $\V$ does not depend on $\hbar$, obviously $ |\V\|_{\rho,k}\equiv \|\V\|_{\rho}$. Then formula (\ref{Thm5.1}) yields, on account of (\ref{stimaAell}), (\ref{dentheta}), (\ref{2condep}), (\ref{GVS}), (\ref{GVSS}) and of the obvious inequalities $e\rho^{-3}<1$, $\rho/(\rho -1) >1$ when $\rho >\lambda(k)$: \vskip 3pt\noindent \begin{eqnarray*} \nonumber && \label{stimaWfl} \ep_\ell \|W_{\ell,\ep}\|_{\rho_{\ell},k} \leq \gamma\frac{\tau^\tau (\ell+1)^{2\tau}}{(e\rho)^\tau}[1+\frac{2}{(\rho-1)^{k+1}}(e\rho^3)^{-k}](\mu \ep \|\V\|_{\rho})^{2^{\ell}} \\ && \leq 2 \gamma\frac{\tau^\tau (\ell+1)^{2\tau}}{(e\rho)^\tau}(\mu \ep \|\V\|_{\rho})^{2^{\ell}}\leq \gamma\left(\frac{\tau}{e}\right)^\tau (\ell+1)^{2\tau}(1+8\gamma\tau^\tau [2(k+1)^2])^{-\tau}\cdot (\mu \ep \|\V\|_{\rho})^{2^{\ell}}. \end{eqnarray*} \vskip 5pt\noindent because of the straightforward inequality $$ [1+\frac{2}{(\rho-1)^{k+1}}(e\rho^3)^{-k}] <1 $$ which in turn follows from $\gamma>1$. This proves (\ref{stimafw}). Moreover, since $\N_{\ell,\ep}=\overline{\V}_{\ell,\ep}$, again by (\ref{GVS}), (\ref{GVSS}): \be\nonumber \label{stiman} \ep_\ell \|\N_{\ell,\ep}\|_{\rho_\ell,k}= \ep_\ell \|\overline{\V}_{\ell,\ep}\|_{\rho_\ell,k}\leq \left[ \|\V\|_{\rho}\,\ep \mu\right]^{2^{\ell}}. \ee The remaining assertion follows once more from (\ref{GVSS}). This concludes the proof of the Theorem. \end{proof} \begin{remark} \label{verifica1} (\ref{stimafw}) yields, with $\ds K:= \gamma\left(\frac{\tau}{e}\right)^\tau (1+8\gamma\tau^\tau [2(k+1)^2])^{-\tau}$: \vskip 4pt\noindent $$ \ep_\ell \frac{\|W_{\ell,\ep}\|_{\rho_{\ell+1},k}}{d_\ell}\leq K\ep^{2^\ell}(\ell+1)^{2(\tau+1)}\|\V\|_\rho^{2^\ell} $$ This yields: $$ |\ep|\left(\frac{\|W_{\ell,\ep}\|_{\rho_{\ell+1},k}}{d_\ell}\right)^{2^{-\ell}}\leq [K(\ell+1)^{2(\tau+1)}]^{2^{-\ell}}\|\V\|_{\rho}\to \|\V\|_\rho, \quad \ell\to\infty $$ so that (\ref{condepell}) is actually fulfilled for $\ds |\ep|< \frac1{\|\V\|_\rho}.$ \end{remark} \begin{corollary} \label{maincc} In the above assumptions set: \be \label{Un} U_{n,\ep}(\hbar):= \prod_{s=0}^ne^{i\ep_{n-s}W_{n-s,\ep}}, \quad n=0,1,\ldots. \ee Then: \begin{enumerate} \item $U_{n,\ep}(\hbar)$ is a unitary operator in $L^2(\T^l)$, with $$ U_{n,\ep}(\hbar)^\ast=U_{n,\ep}(\hbar)^{-1}=\prod_{s=0}^ne^{-i\ep_{s}W_{s,\ep}} $$ \item Let: \be S_{n,\ep}(\hbar):=U_{n,\ep}(\hbar)(\L_\om+\ep V)U_{n,\ep}(\hbar)^{-1} \ee Then: \bea S_n&=&D_{n,\ep}(\hbar)+\ep_{n+1}V_{n+1,\ep} \\ D_{n,\ep}(\hbar)&=&L_\om+\sum_{s=1}^n\ep_sN_{s,\ep} \eea The corresponding symbols are: \bea && {\mathcal S}_n(\xi,x;\hbar)=\D_{n,\ep}(\L_\om(\xi),\hbar)+\ep_{n+1}V_{n+1,\ep}(\L_\om(\xi),x;\hbar) \\ \label{sumD} && \D_{n,\ep}(\L_\om(\xi),\hbar)=\L_\om(\xi)+\sum_{s=1}^n \ep_s\N_{s,\ep}(\L_\om (\xi),\hbar). \eea Here the operators $W_{s,\ep}$, $N_{s,\ep}$, $V_{\ell+1,\ep}$ and their symbols $\W_{s,\ep}$, $\N_{s,\ep}$, $\V_{\ell+1,\ep}$ fulfill the above estimates. \item Let $\ep^\ast$ be defined as in (\ref{condep}). Remark that $ \ep^\ast(\cdot,k)> \ep^\ast(\cdot,k+1), \,k=0,1,\ldots$. Then, if $|\ep|<\ep(k,\cdot)$: \be \lim_{n\to\infty}\D_{n,\ep}(\L_\om(\xi),\hbar)=\D_{\infty,\ep}(\L_\om(\xi),\hbar) \ee where in the convergence takes place in the $C^k([0,1];C^\om (\rho/2))$ topology, namely \be \label{limD} \lim_{n\to\infty}\|\D_{n,\ep}(\L_\om(\xi),\hbar)-\D_{\infty,\ep}(\L_\om(\xi),\hbar)\|_{\rho/2,k}=0. \ee \end{enumerate} \end{corollary} \begin{proof} Since Assertions (1) and (2) are straightforward, we limit ourselves to the simple verification of Assertion (3). If $|\ep|<\ep^\ast(\cdot,k)$ then $\ds \|V\|_{\rho,k}\mu \ep < \Lambda<1$. Recalling that $\|\cdot\|_{\rho,,k} \leq \|\cdot\|_{\rho^\prime,k}$ whenever $\rho\leq \rho^\prime$, and that $\rho_\ell <\rho/2$, $\forall\,\ell \in {\Bbb N}$, (\ref{stimafv}) yields: \begin{eqnarray*} && \ep_{n+1}\|\V_{n+1,\ep}\|_{\rho/{2},k}\leq \ep_{n+1}\|\V_{n+1,\ep}\|_{\rho_{n+1},k}\leq \\ && \left[\|V\|_{\rho,k}\mu \ep\right]^{2^{n+1}}\to 0, \quad n\to\infty, \;k\;{\rm fixed}. \end{eqnarray*} In the same way, by (\ref{stimafn}): \begin{eqnarray*} && \|\N_{n,\ep}\|_{\rho/{2},k}\leq \|\N_{n,\ep}\|_{\rho_{n},k}= \|\overline{\V}_{n,\ep}\|_{\rho_{n},k}\leq \|\V_{n,\ep}\|_{\rho_{n},,k}\leq \\ && \left[\|V\|_{\rho,k}\mu \ep\right]^{2^{n}}\to 0, \quad n\to\infty, \;k\;{\rm fixed}. \to 0, \quad n\to\infty, \;k\;{\rm fixed}. \end{eqnarray*} This concludes the proof of the Corollary. \end{proof} \vskip 1cm\noindent \section{Convergence of the iteration and of the normal form.} \label{iteration} \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}} \renewcommand{\theremark}{\thesection.\arabic{remark}} \setcounter{equation}{0}% \setcounter{theorem}{0}% Let us first prove the uniform convergence of the unitary transformation sequence as $n\to\infty$. Recall that $\ep^\ast(\cdot,k)> \ep^\ast(\cdot,k+1), \; k=0,1,\ldots$, and recall the abbreviation $\|\cdot\|_{\rho,0}:=\|\cdot\|_{\rho}$. Define moreover: \be \label{epszero} \ep^\ast:=\ep^\ast_0=\ep^\ast(\gamma,\tau,0). \ee where $\ep^\ast(\gamma,\tau,0)$ is defined by (\ref{epsast}). Then: \begin{lemma} \label{Wsequence} Let $\hbar$ be fixed, and $\ds |\ep|<\ep^\ast_0$. Consider the sequence $\ds \{U_{n,\ep}(\hbar)\}$ of unitary operators in $L^2(\T^l)$ defined by (\ref{Un}). Then there is a unitary operator $U_{\infty,\ep}(\hbar)$ in $L^2(\T^l)$ such that $$ \lim_{n\to\infty}\|U_{n,\ep}(\hbar)-U_{\infty,\ep}(\hbar)\|_{L^2\to L^2}=0 $$ \end{lemma} \begin{proof} Without loss we can take $\hbar=1$. We have, for $p=1,2,\ldots$: \begin{eqnarray*} && U_{n+p,\ep}-U_{n,\ep}=\Delta_{n+p,\ep}e^{i\ep_n W_n}\cdots e^{i\ep W_1}, \quad \Delta_{n+p,\ep}:=(e^{i\ep_{n+p}W_{n+p}}\cdots e^{i\ep_{n+1}W_{n+1}}-I) \\ && \|U_{n+p,\ep}-U_{n,\ep}\|_{L^2\to L^2}\leq 2\|\Delta_{n+p,\ep}\|_{L^2\to L^2} \end{eqnarray*} Now we apply the mean value theorem and obtain $$ e^{i\ep_\ell W_{\ell,\ep}}=1+\beta_{\ell,\ep} \quad \beta_{\ell,\ep}:=i\ep_\ell W_{\ell,\ep} \int_0^{\ep_\ell}e^{i\ep^\prime_\ell W_{\ell,\ep}}\,d\ep^\prime_\ell , $$ whence, by (\ref{stimafw}) in which we make $k=0$: \be \label{stimaesp} \|\beta_{\ell,\ep}\|\leq \ep_\ell \|W_{\ell,\ep}\|_{\rho_{\ell}}\leq \ep_\ell \|W_{\ell,\ep}\|_{\rho_{\ell},k} \leq \gamma\tau^\tau (\ell+1)^{2\tau}\frac{(1+8\gamma\tau^\tau [2(k+1)^2])^{2-\tau}}{64\gamma^2\tau^{2\tau}[2(k+1)^2]^4}\cdot (\mu \ep \|\V\|_{\rho})^{2^{\ell}} \leq A^\ell \ee for some $A<1$. Now: \begin{eqnarray*} && \Delta_{n+p,\ep}=[(1+\beta_{n+p,\ep}\ep_{n+p})(1+\beta_{n+p-1,\ep}\ep_{n+p-1})\cdots (1+\beta_{n+1,\ep}\ep_{n+1})]=\sum_{j=1}^p\beta_{n+j,\ep}\ep_{n+j} \\ && +\sum_{j_10 \end{eqnarray*} \vskip 5pt\noindent Hence $\{U_{n,\ep}(\hbar)\}_{n\in{\Bbb N}}$ is a Cauchy sequence in the operator norm, uniformly with respect to $|\ep|<\ep^\ast_0$, and the Lemma is proved. \end{proof} We are now in position to prove existence and analyticity of the limit of the KAM iteration, whence the uniform convergence of the QNF. %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \vskip 0.3cm\noindent {\bf Proof of Theorems \ref{mainth} and \ref{regolarita}} \newline The operator family $H_\ep$ is self-adjoint in $L^2(T^l)$ with pure point spectrum $\forall\,\ep\in\R$ because $V$ is a continuous operator. By Corollary \ref{maincc}, the operator sequence $\{D_{n,\ep}(\hbar)\}_{n\in {\Bbb N}}$ admits for $|\ep|<\ep^\ast_0$ the uniform norm limit $$ D_{\infty,\hbar}(L_\om,\hbar)=L_\om+\sum_{m=0}^\infty\ep^{2^m}N_{m,\ep}(L_\om,\hbar) $$ of symbol $\D_{\infty,\hbar}(\L_\om(\xi))$. The series is norm-convergent by (\ref{stimafn}). By Lemma (\ref{Wsequence}), $D_{\infty,\hbar}(L_\om,\hbar)$ is unitarily equivalent to $H_\ep$. The operator family $\ep\mapsto D_{\infty,\ep}(\hbar)$ is holomorphic for $|\ep|<\ep^\ast_0$, uniformly with respect to $\hbar\in[0,1]$. As a consequence, $D_{\infty,\ep}(\hbar)$ admits the norm-convergent expansion: $$ D_{\infty,\ep}(L_\om,\hbar)=L_\om+\sum_{s=1}^\infty B_s(L_\om,\hbar)\ep^s, \quad |\ep|<\ep^\ast_0 $$ which is the convergent quantum normal form. On the other hand, (\ref{limD}) entails that the symbol $\D_{\infty,\ep}(\L_\om(\xi),\hbar)$ is a $\J(\rho/2)$-valued holomorphic function of $\ep$, $|\ep|<\ep^\ast_0$, continuous with respect to $\hbar\in [0,1]$. Therefore it admits the expansion \be \label{fnormale} \D_{\infty,\ep}(\L_\om(\xi),\hbar)=\L_\om(\xi)+\sum_{s=1}^\infty{\mathcal B}_s(\L_\om(\xi),\hbar)\ep^s, \quad |\ep|<\ep^\ast \ee convergent in the $\|\cdot\|_{\rho/2}$-norm, with radius of convergence $\ep^\ast_0$. Hence, in the notation of Theorem \ref{mainth}, $\D_{\infty,\ep}(\L_\om(\xi),\hbar)\equiv \B_{\infty,\ep}(\L_\om(\xi),\hbar)$. By construction, ${\mathcal B}_s(\L_\om(\xi),\hbar)$ is the symbol of $B_s(L_\om,\hbar)$. $\B_{\infty,\ep}(\L_\om(\xi),\hbar)$ is the symbol yielding the quantum normal form via Weyl's quan\-ti\-za\-tion. Likewise, the symbol $\W_{\infty,\ep}(\xi,x,\hbar)$ is a $J(\rho/2)$-valued holomorphic function of $\ep$, $|\ep|<\ep^\ast$, continuous with respect to $\hbar\in [0,1]$, and admits the expansion: \be \label{fgen} \W_{\infty,\ep}(\xi,x,\hbar)=\la\xi,x\ra+\sum_{s=1}^\infty{\mathcal W}_s(\xi,x,\hbar)\ep^s, \quad |\ep|<\ep^\ast_0 \ee convergent in the $\|\cdot\|_{\rho/2}$-norm, once more with radius of convergence $\ep^\ast_0$. Since Since $\|\B_s\|_1 \leq \|\B_s\|_{\rho/2}$, $\|{\mathcal W}_s \|_1\leq \|{\mathcal W}_s\|_{\rho/2} $ $\forall\,\rho>0$. By construction, $\B_{\infty,\ep}(\xi,x,\hbar)=\B_{\infty,\ep}(t,x,\hbar)|_{t=\L_\om(\xi)}$. Theorem \ref{mainth} is proved . Remark that the principal symbol of $\B_{\infty,\ep}(\L_\om(\xi),\hbar)$ is just the convergent Birkhoff normal form: $$ \B_{\infty,\ep}=\L_\om(\xi)+\sum_{s=1}^\infty{\mathcal B}_s(\L_\om(\xi))\ep^s, \quad |\ep|<\ep^\ast_0 $$ Theorem (\ref{regolarita}) is a direct consequence of (\ref{limD}) on account of the fact that $$ \sum_{\gamma=0}^r\max_{\hbar\in [0,1]} \|\partial^\gamma_\hbar \B_\infty(t;\ep,\hbar) \|_{\rho/2}\leq \|\B_\infty\|_{\rho/2,k} $$ Remark indeed that by (\ref{limD}) the series (\ref{fnormale}) converges in the $\|\cdot\|_{\rho/2,r}$ norm if $|\ep|<\ep^\ast(\cdot,r)$. Therefore $\B_s(t,\hbar)\in C^r([0,1];C^\om(\{t\in\C\,|\,|\Im t|<\rho/2\})$ and the formula (\ref{EQF}) follows from (\ref{fnormale}) upon Weyl quantization. This concludes the proof of the Theorem. \vskip 1.0cm\noindent %\newpage %%%%%%%%%%%%%%%%%%%%%% \begin{appendix} \section{The quantum normal form} \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}} \renewcommand{\theremark}{\thesection.\arabic{remark}} \setcounter{equation}{0}% \setcounter{theorem}{0}% %\end{center} \noindent The quantum normal form in the framework of semiclassical analysis has been introduced by Sj\"ostrand \cite{Sj}. We follow here the presentation of \cite{BGP}. \vskip 6pt\noindent {\bf 1. The formal construction} Given the operator family $\ep\mapsto H_\ep=L_\om+\ep V$, look for a unitary transformation $\ds U(\om,\ep,\hbar)=e^{i W(\ep)/\hbar}: L^2(\T^l)\leftrightarrow L^2(\T^l)$, $W(\ep)=W^\ast(\ep)$, such that: \be \label{A1} S(\ep):=UH_\ep U^{-1}=L(\om)+\ep B_1+\ep^2 B_2+\ldots+ \ep^k R_k(\ep) \ee where $[B_p,L_0]=0$, $p=1,\ldots,k-1$. Recall the formal commutator expansion: \be %\nonumber S(\ep)=e^{it W(\ep)/\hbar}He^{-it W(\ep)/\hbar}=\sum_{l=0}^\infty t^lH_l,\quad H_0:=H,\quad H_l:=\frac{[W,H_{l-1}]}{i\hbar l}, \;l\geq 1 \label{A2} \ee and look for $W(\ep)$ under the form of a power series: $W(\ep)=\ep W_1+\ep^2W_2+\ldots$. Then (\ref{A2}) becomes: \be \label{A3} S(\ep)=\sum_{s=0}^{k-1}\ep^s P_s +\ep^{k}{R}^{(k)} \ee where \be \label{A4} P_0=L_\om;\quad {P}_s:=\frac{[W_s,H_0]}{i\hbar}+V_s,\quad s\geq 1, \;V_1\equiv V \ee \begin{eqnarray*} V_s =\sum_{r=2}^s\frac{1}{r!}\sum_{{j_1+\ldots+j_r=s}\atop {j_l\geq 1}}\frac{[W_{j_1},[W_{j_2},\ldots,[W_{j_r},H_0]\ldots]}{(i\hbar)^r} +\sum_{r=2}^{s-1}\frac{1}{r!}\sum_{{j_1+\ldots+j_r=s-1}\atop {j_l\geq 1}}\frac{[W_{j_1},[W_{j_2},\ldots,[W_{j_r},V]\ldots]}{(i\hbar)^r} \end{eqnarray*} \begin{eqnarray*} {R}^{(k)}=\sum_{r=k}^\infty\frac{1}{r!}\sum_{{j_1+\ldots+j_r=k}\atop {j_l\geq 1}}\frac{[W_{j_1},[W_{j_2},\ldots,[W_{j_r},L_\om]\ldots]}{(i\hbar)^r} +\sum_{r=k-1}^{\infty}\frac{1}{r!}\sum_{{j_1+\ldots+j_r=k-1}\atop {j_l\geq 1}}\frac{[W_{j_1},[W_{j_2},\ldots,[W_{j_r},V]\ldots]}{(i\hbar)^r} \end{eqnarray*} Since $V_s$ depends on $W_1,\ldots,W_{s-1}$, (A1) and (A3) yield the recursive homological equations: \be \label{A5} \frac{[W_s,P_0]}{i\hbar} +V_s=B_s, \qquad [L_0,B_s]=0 \ee To solve for $S$, $W_s$, $B_s$, we can equivalently look for their symbols. The equations (\ref{A2}), (\ref{A3}), (\ref{A4}) become, once written for the symbols: \be \label{A6} \Sigma(\ep)=\sum_{l=0}^\infty {\H}_l,\quad {\H}_0:=\L_\om+\ep \V,\quad {\H}_l:=\frac{\{w,{\H}_{l-1}\}_M}{ l}, \;l\geq 1\ee \be \label{A7} \Sigma(\ep)=\sum_{s=0}^{k}\ep^s {\mathcal P}_s +\ep^{k+1}{\R}^{(k+1)} \ee where \be \label{A8} {\mathcal P}_0=\L_\om;\qquad {\mathcal P}_s :=\{\W_s,{\mathcal P}_0 \}_M+\V_s,\quad s=1, \ldots,\qquad \V_1\equiv \V_0=\V \ee \begin{eqnarray*} && \V_s :=\sum_{r=2}^s\frac{1}{r!}\sum_{{j_1+\ldots+j_r=s}\atop {j_l\geq 1}}\{\W_{j_1},\{\W_{j_2},\ldots,\{\W_{j_r},\L_\om\}_M\ldots\}_M + \\ && +\sum_{r=1}^{s-1}\frac{1}{r!}\sum_{{j_1+\ldots+j_r=s-1}\atop {j_l\geq 1}}\{\W_{j_1},\{\W_{j_2},\ldots,\{\W_{j_r},\V\}_M\ldots\}_M, \quad s>1 \end{eqnarray*} \begin{eqnarray*} && {\R}^{(k)}=\sum_{r=k}^\infty\frac{1}{r!}\sum_{{j_1+\ldots+j_r=k}\atop {j_l\geq 1}}\{\W_{j_1},\{\W_{j_2},\ldots,\{\W_{j_r},\L_\om\}_M\ldots\}_M+ \\ && \sum_{r=k-1}^{\infty}\frac{1}{r!}\sum_{{j_1+\ldots+j_r=k-1}\atop {j_l\geq 1}}\{\W_{j_1},\{\W_{j_2},\ldots,\{\W_{j_r},\V\}_M\ldots\}_M\end{eqnarray*} In turn, the recursive homological equations become: \be \label{A9} \{\W_s,\L_{\om}\}_M +\V_s=\B_s, \qquad \{\L_{\om},\B_s\}_M =0 \ee \vskip 6pt\noindent {\bf 2. Solution of the homological equation and estimates of the solution} \vskip 3pt\noindent The key remark is that $\{\A,\L_\om\}_M=\{\A,\L_\om\}$ for any smooth symbol $\A(\xi;x;\hbar)$ because $\L_\om$ is linear in $\xi$. The homological equation (A.9) becomes therefore \be \label{A10} \{\W_s,\L_\om\} +\V_s=\B_s, \qquad \{\L_\om,\B_s\} =0 \ee We then have: %\par\noindent %{\bf Proposition A.1} \begin{proposition} Let $\V_s(\xi,x;\hbar)\in\J(\rho_s)$. Then the equation \be \label{A11} %\label{cchomsq} \{\W_s,\L_\om\} +\V_s=\B_s, \qquad \{\L_\om,\B_s\} =0 \ee admits $\forall\,0