\documentstyle[12pt]{article} \pagestyle{plain } \addtolength{\textheight}{3cm} \addtolength{\textwidth}{3cm} \addtolength{\oddsidemargin}{-1,5cm} \addtolength{\evensidemargin}{-1cm} \addtolength{\topmargin}{-1,5cm} \begin{document} \newcommand{\nc}{\newcommand} \newtheorem{lemma}{Lemma}[section] \newtheorem{theorem}[lemma]{Theorem} \newtheorem{proposition}[lemma]{Proposition} \newtheorem{corollary}[lemma]{Corollary} \newtheorem{remark}[lemma]{Remark} \newtheorem{example}[lemma]{Example} \newenvironment{Remark}{\begin{remark}\rm }{\end{remark}\medskip} \newtheorem{definition}[lemma]{Definition} \newenvironment{Definition}{\begin{definition}\rm }{\end{definition}\medskip} \nc{\QED}{\mbox{}\hfill \raisebox{-2pt}{\rule{5.6pt}{8pt}\rule{4pt}{0pt}} \medskip\par} \newenvironment{Proof}{\noindent \parindent=0pt\abovedisplayskip = 0.5\abovedisplayskip \belowdisplayskip=\abovedisplayskip{\bf Preuve. }}{\QED} \newenvironment{Example}{\begin{example}\rm }{\QED\end{example}\medskip} \font\cur=rsfs10 \nc{\sch} {\hbox{\cur S}} \nc{\N}{{\rm I\mkern-4.0mu N}} \nc{\R}{{\rm I\mkern-4.0mu R}} \nc{\Z}{{\sf Z\mkern-6.5mu Z}} \nc{\Id}{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}} \nc{\di}{\displaystyle} \nc{\beq}{\begin{equation}} \nc{\edq}{\end{equation}} \nc{\beqn}{\begin{eqnarray}} \nc{\edqn}{\end{eqnarray}} \nc{\C}{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}} \nc{\Sph}{{\mathchoice {\setbox0=\hbox{$\displaystyle \rm S$}\hbox{\raise0.5\ht0\hbox to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle \rm S$}\hbox{\raise0.5\ht0\hbox to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}} {\setbox0=\hbox{$\rm S$}\hbox{\raise0.5\ht0\hbox to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}} \title {Distribution of Matrix Elements and Level Spacings for Classically Chaotic Systems} \author{ Monique Combescure(\#) and Didier Robert(\P) \\ (\#) Laboratoire de Physique Th\'eorique et Hautes Energies, URA CNRS,\\ Universit\'e de Paris-Sud, 91405- Orsay. Cedex, France\\ (\P) Departement de Mathematiques, URA CNRS 758, Universit\'e de Nantes\\ 44072-Nantes-Cedex 03, France } \date{} \maketitle \begin{abstract} For quantum systems obtained by quantization of chaotic classical systems we prove some rigorous results concerning the semi-classical behaviour of matrix elements of observables on an orthonormal system of bound states of the Hamiltonian. \end{abstract} \section{Introduction} Our aim in this paper is to study the energy levels and the corresponding eigenstates for quantum Hamiltonians like Schr\"odinger: $P(\hbar)=-\hbar^2\Delta + V$ on the configuration space $X = \R^n$ (our proofs can be easily translated on some Riemannian compact manifold $X$ (a torus, see \cite{doli} for examples, or a compact manifold with constant negative curvature) such that the corresponding classical system is chaotic on some energy shell of the phase space (ergodic or mixing).\\ Let $I_{c\ell} \subset \R$ be a classical energy interval such that the spectrum of $P(\hbar)$ is purely discrete in $I_{c\ell}$. So we have $P(\hbar)\varphi_j=E_j(\hbar)\varphi_j$ where $\{\varphi_j\}_j$ is an orthonormal system of bound states of energies $E_j(\hbar) \in I_{c\ell}$. Let us denote by $p(x,\xi) \in C^{\infty}(T^*(X))$ the corresponding classical Hamiltonian and assume that on some energy shell $\Sigma_E:=\{(x,\xi)\in T^*(X);\;p(x,\xi)=E\},\;\; E\in I_{c\ell}$, the classical motion is ergodic (or mixing). Let us introduce a classical smooth observable $a(x,\xi) \in C^{\infty}(T^*(X))$, $A(\hbar)$ its quantum counterpart and the matrix elements $A_{jk}(\hbar):=$ (scalar product in $L^2(X)$)\\ The matrix elements are important at least for two reasons: firstly, in quantum mechanics they measure the transition probabilities between the states $j$ and $k$; secondly they appear naturally in the stationary perturbation theory (see any text book in quantum theory for details). Let us briefly recall here how do they appear. Let us consider in the abstract Hilbert space ${\cal H}$ a self adjoint operator $P$ with a discrete spectrum: $\{E_j\}_{j\in\N}$, without multiplicities for simplicity. We have an orthormal basis of eigenfunctions: $\{\varphi_j\}_{j\in\N}$, $P\varphi_j=E_j\varphi_j$. Let us consider a small perturbation $P_\gamma:=P+\gamma A$ of $P$ where $A$ is a bounded operator in ${\cal H}$ and $\gamma \in\R$, small. For a fixed $j\in\N$ we try to solve the eigenvalue problem: $P_\gamma\varphi_j^\gamma=E_j^\gamma\varphi_j^\gamma$ by the "ansatz": \beqn\label{e11} E_j^\gamma &=& E_j+\gamma\varepsilon_1+\gamma^2\varepsilon_2+\cdots\\ \varphi_j^\gamma &=& \varphi_j+\gamma\psi_1+\gamma^2\psi_2+\cdots \edqn Asking that $\psi_1$ is orthogonal to $\varphi_j$, we get easily: \fbox{$\varepsilon_1=\langle A\varphi_j,\varphi_j\rangle$} \fbox{$\varepsilon_2=\sum_{k\neq j}\frac{\vert\langle A\varphi_j, \varphi_k\rangle \vert^2}{E_k-E_j}$} So we see that the diagonal elements give the first order approximation and the non diagonal elements give the second order approximation. Now we come back to the quantum mechanics case in the configuration space $\R^n$. There is a huge physicist literature discussing the behaviour of the $A_{jk}(\hbar)$ as the Planck constant $\hbar \searrow 0 \; \rm{and} \;\; E_j(\hbar),\; E_k(\hbar) \rightarrow E \in I_{c\ell}$, in connexion with the chaotic properties of the classical dynamics on $\Sigma_E$. (see references \cite{mcka,mapr,robe1,prro,fepe}). In particular, if the classical dynamics is ergodic on $\Sigma_E$, then it is claimed that for the diagonal elements we have: \beq\label{eq12} \lim_{\hbar\searrow 0}A_{jj}(\hbar) = _E \; {\rm (the\; average\; of}\; a\;{\rm for\; the\; Liouville\; measure\; on}\;\Sigma_E) \edq and for the non diagonal elements: \beq\label{eq13} \lim_{[\hbar\searrow 0,\;j\neq k]}A_{jk}(\hbar) = 0 \edq Untill now theses claims are not completly proved. Following the works by Shnirelmann \cite{sh}, Zelditch \cite{ze1}, Colin de Verdi\ere \cite{co}, Helffer-Martinez-Robert \cite{hemaro} it can be proved that (\ref{eq12}) is true "almost everywhere". One of the main goals of this paper is to discuss the claim (\ref{eq13}) and in particular to extend and improve some results obtained by Zelditch \cite{ze2}. We will also discuss the variance of the statistical distribution of the series $\{A_{jk}(\hbar)\}_{jk}$ according to a definition proposed by Wilkinson \cite{wi}. We will give a rigorous proof of the semi-classical $\hbar$-expansion which appeared in \cite{wi}.\\ The unifying theme of our paper is the role of different "sum rules" (see \cite{mapr,fepe}). The idea is to consider sums like: \beq\label{e13} S_{\theta,j}(\hbar) = \sum_k \vert A_{jk}(\hbar)\vert^2 \theta \left(\frac{E_j(\hbar)-E_k(\hbar)}{\hbar} \right) \edq We first transform this sum using Parseval relation and try to control the classical limit by direct estimates or by the WKB method. We get in this way different results, according to the choice of the test functions $\theta$, which generally will depend on some extra parameters. In a forthcoming paper we will apply these techniques to check rigorously the classical limit of the geometrical Berry's phase for chaotic systems \cite{robe2} (for integrable systems see \cite{be}). \section{A Semi-Classical Analysis Background} In this section we introduce our technical assumptions and recall some more or less well known mathematical facts about semi-classical analysis in the phase space. For details see for exemple the book \cite{ro}.\\ On the configuration space $\R^n$ it is convenient to choose the so called Weyl quantization which is defined by the formula: \beq (op^w_\hbar b)\psi(x) = (2\pi \hbar)^{-n}\int\;\int{\rm e}^{\frac{i}{\hbar}} b(\frac{x+y}{2}, \xi)\psi(y)dyd\xi \edq We shall use as well the notations $op^w_{\hbar}b :=b^w_{\hbar}:=B(\hbar)$; $b$ is by definition the $\hbar$-Weyl symbol of the operator $B(\hbar)$. ($b$ is also the classical observable corresponding to the quantum observable $B(\hbar)$).\\ We start with a quantum Hamiltonian $P(\hbar)$ of $\hbar$-Weyl symbol $p(\hbar,x;\xi)$. We assume that $p(\hbar,x;\xi)$ has an asymptotic expansion: \beq p(\hbar,x;\xi) \asymp \sum_{0\leq j < +\infty}\hbar^jp_j(x,\xi) \edq with the following properties:\\ $(H_1)$ $p(\hbar,x,\xi)$ {\it is real valued }, $p_j \in C^{\infty}(\R^{2n})$ \\ $(H_2)$ {\it There exist $C>0;\; M\in\R$ such that}: $$(1+p_0(x,\xi)^2) \leq C(1+p_0(y,\eta)^2)(1+\vert x-y \vert + \vert \xi-\eta \vert)^M\;\;\; \forall x,y,\xi,\eta, \in \R^n$$ $(H_3)$ $\forall j \geq 0 \;\;\forall \alpha,\;\beta$ {\it multiindices $\exists c>0$ such that}: $\vert \partial^\alpha_x\partial^\beta_\xi p_j\vert \leq c(1+p_0^2)^{1/2}$\\ $(H_4)$ {\it $\forall N \geq N_0$, $\forall \alpha, \beta\;\exists c(N,\alpha,\beta)>0$ such that $\forall \hbar \in ]0,1],\;\forall (x, \xi)\in\R^{2n}$ we have}: $$\vert \partial^\alpha_x\partial^\beta_\xi\{p(\hbar;x,\xi) - \sum_{0\leq j \leq N}\hbar^jp_j(x,\xi) \} \vert \leq c(N,\alpha,\beta)\hbar^{N+1}$$ Under these assumptions it is well known that $P(\hbar)$ has a unique self-adjoint extension in $L^2(\R^n)$ (see for example\cite{ro}) and the propagator: $$U(t,\hbar):=\exp\left(-\frac{it}{\hbar}P(\hbar)\right)$$ is well defined as a unitary operator in $L^2(\R^n)$, for every real number $t$. In what follows the function and operator norms in $L^2(\R^n)$ will be denoted by $\Vert . \Vert$.\\ Because we are interested in bound states, let us consider a classical energy interval $I_{c\ell} = ]\lambda_- , \lambda_+[; \;\; \lambda_- < \lambda_+$ and assume:\\ $(H_5)$ $p_0^{-1}(I_{c\ell})$ {\it is a bounded set of the phase space $\R^{2n}$.\\ This imply that for every closed interval $J_{c\ell}:=[E_-, E_+]\subset I_{c\ell}$, and for $\hbar > 0$ small enough, the spectrum of $P(\hbar)$ in $J_{c\ell}$ is purely discrete (\cite{hero1}). In what follows we fix such an interval $J_{c\ell}$.\\ For a fixed energy level $E\in ]E_-, E_+[$, we assume}:\\ $(H_6)$ $E$ {\it is a regular value of $p_0$. That means: $p_0(x,\xi)= E \Rightarrow \nabla_{(x,\xi)}p_0(x,\xi) \neq 0)$\\ So, the Liouville measure $d\sigma_E$ is well defined on the energy shell } $$\Sigma_E:=\{(x,\xi) \in \R^{2n},\; p_0(x,\xi)=E \}$$ Let us recall that: $$d\sigma_E=\left(\int_{\Sigma_E}\frac{d\Sigma_E}{\vert \nabla p_0\vert}\right)^{-1} \frac{d\Sigma_E}{\vert \nabla p_0\vert}$$ where $d\Sigma_\lambda$ is the Euclidean measure on $\Sigma_\lambda$. \\ Let us introduce also the Hamiltonian vector field ${\cal H}_{p_0}:=(\nabla_\xi p_0, -\nabla_x p_0)$ and the Hamiltonian flow: $\Phi^t(x,\xi):=\exp(t{\cal H}_{p_0}(x,\xi))$. We are mainly concerned here by the dynamical system $\left(\Sigma_E, d\sigma_E, \Phi^t \right)$ and its connections with the spectrum of $P(\hbar)$ close to $E$. There is a huge litterature on this subject in physics and in mathematics; but there are rather few mathematical rigorous result about quantum consequences of classical chaos.\\ For later purpose, let us recall some well established results concerning semi-classical asymptotics of bound states:\\ $(R_1)$ \underline {the general Weyl formula} \cite{hero1,iv})\\ Under the assumptions $(H_1)$ to $(H_5)$, if $\lambda,\; \mu \in I_{c\ell},\; \lambda < \mu$, are regular values for $p_0$, then we have: \beq \#\{j,\; E_j(\hbar)\in [\lambda, \mu] \} = (2\pi\hbar)^{-n}Vol_{\R^{2n}}\{p_0^{-1}[\lambda, \mu]\} + O(\hbar^{1-n}) \edq $(R_2)$ \underline {the "generic" Weyl formula} \cite{pero,iv}) \\ Furthermore, under the same assumptions as above, if we add the following condition $(H_7)$, for $E=\lambda$ and $E=\mu$ :\\ $(H_7)$ {\it The Liouville measure of the closed trajectories on $\Sigma_E$ is zero.}\\ (that means: $\sigma_E\{(x,\xi)\in \Sigma_E,\; \exists t\neq0,\;\Phi^t(x,\xi)=(x,\xi)\}=0$).\\ Then we have a two terms asymptotic expansion: \beq \# \{j,\; E_j\in [\lambda, \mu]\} = (2\pi\hbar)^{-n}Vol_{\R^{2n}}\{p_0^{-1}[\lambda, \mu]\} + c_1 \hbar^{1-n} + o(\hbar^{1-n}) \edq where $c_1$ was computed in \cite{pero} ($c_1$=0 if $p_1=0$)\\ The result $(R_2)$ can be seen as a weak form of chaos! (see also section 6 for a weaker form)\\ A more suggestive result is the following: let us consider $\hbar$-dependent energy intervals: $I(\hbar)=[\alpha(\hbar), \beta(\hbar)]$, $\alpha(\hbar)0$. Let us denote: $\Lambda(\hbar)=\{j,\; E_j(\hbar) \in I(\hbar)\}$ and by ${\cal B}_\infty$ the space of smooth functions $a$: $\R^{2n} \rightarrow \C$ such that all derivatives $\partial^\alpha_x\partial^\beta_\xi a(x, \xi)$ are bounded in $\R^{2n}$. Under the same assumptions as in $(R_2)$, we have:\\ $(R_3)$ (see \cite{hemaro}) \beq \lim_{\hbar \searrow 0}\frac{\sum_{j\in\Lambda(\hbar)}A_{jj}(\hbar)}{\#\Lambda(\hbar)}= \int_{\Sigma_E}ad\sigma_E \edq Moreover we have the following asymptotic formula for the number of bound states of $P(\hbar)$ in $I(\hbar)$, under the assumption $(H_7)$ (see Appendix (A)), \beq \lim_{\hbar\searrow 0}\left(\frac{(2\pi\hbar)^n(\#\Lambda(\hbar))}{\beta(\hbar)-\alpha(\hbar)}\right) = \int_{\Sigma_E}\frac{d\Sigma_E}{\vert \nabla p_0\vert} :=\vert\Sigma_E\vert \edq The above asymptotic result is an easy corollary of \cite{hemaro} (Th\'eor\eme 1.1, p.315). For a particular case see also (\cite{brur})\\ Let us remark here that if we have $$\lim_{\hbar\searrow 0}\left(\frac{\hbar}{\beta(\hbar)-\alpha(\hbar)}\right)=0$$ then (11) is still valid without $(H_7)$, simply by using the general Weyl formula $(R_1)$.\\ Let us introduce a first chaotic assumption:\\ $(H_8)${\it The dynamical system $\left(\Sigma_E, d\sigma_E, \Phi^t\right)$ is {\em ergodic} which means: for every continuous function $a$ on $\Sigma_E$, we have, for almost all $(x, \xi)\in \Sigma_E$: $$\lim_{T\nearrow +\infty}\frac{1}{T}\int_0^T a(\Phi^t(x, \xi))dt = \int_{\Sigma_E}ad\sigma_E$$} We will use in this paper the following basic result about the semi-classical behaviour of the diagonal matrix elements: \begin{theorem}\label{hmr}(\cite{hemaro}) Under the assumptions $(H_1)$ to $(H_8)$, $n\geq 2$, for every $\hbar>0$, there exists $M(\hbar) \subseteq \Lambda(\hbar)$, depending only on the Hamiltonian $P(\hbar)$, such that: \beq \lim_{\hbar\searrow 0}\left(\frac{\#M(\hbar)}{\#\Lambda(\hbar)}\right)=1,\;\;{\rm and}\;\; \lim_{[\hbar\searrow 0,\;j\in M(\hbar)]}A_{jj}(\hbar) = \int_{\Sigma_E}ad\sigma_E,\;\; \forall a \in {\cal B}_\infty \edq \end{theorem} \begin{remark} The following question is still open: can we take $M(\hbar)=\Lambda(\hbar)$ in the conclusion of the above therem, if $n\geq 2$? \end{remark} \section{New Results for the Non Diagonal Matrix Elements} We begin with a crude estimate but it explains further restrictions on energy localization. \begin{proposition} Under the assumptions $(H_1)$ to $(H_5)$, for every $a \in {\cal B }_\infty$ there exists $c_0>0$ such that we have: \beq \vert A_{jk}(\hbar) \vert \leq c_0\frac{\hbar}{\vert E_k(\hbar)-E_j(\hbar)\vert} \;\;\; \forall E_j,\;E_k \in J_{c\ell}, \;E_j(\hbar)\neq E_k(\hbar) \edq \end{proposition} {\bf Proof}: Let $\chi$ be a smooth cutoff, $\chi=1$ on $J_{c\ell}$ and compactly supported in $\R$. We have clearly: $$\langle[A(\hbar), \chi(P(\hbar)).P(\hbar)]\varphi_j, \varphi_k\rangle = (E_j(\hbar)-E_k(\hbar))\langle A(\hbar)\varphi_j, \varphi_k \rangle$$ But from the $\hbar$-Weyl calculus (see for example \cite{ro}) we have the well known commutator estimate: $$\Vert [A(\hbar), \chi(P(\hbar)).P(\hbar)]\Vert = O(\hbar)\; {\rm as}\; \hbar \searrow 0$$ The proposition follows. \QED \begin{remark} (i) The proof of the proposition (3.1) can be iterated to get for every $N$ the estimate: $A_{jk}(\hbar)=O\left(\frac{\hbar}{\vert E_j-E_k\vert}\right)^N$.\\ (ii) The proposition shows that it is sufficient to study $A_{jk}(\hbar)$ for level spacings of order $\hbar$ (only this case is considered in the physics litterature). \end{remark} Let us formulate a second crude result coming easily from Theorem(\ref{hmr}): \begin{proposition} Let us assume $(H_1)$ to $(H_7)$ and $n \geq 2$. Then for every $\hbar>0$ there exists $\Box(\hbar)\subseteq \Lambda(\hbar)\times\Lambda(\hbar)$ such that \beq \lim_{\hbar\searrow 0}\frac{\#\Box(\hbar)}{\#\Lambda(\hbar)^2}=1,\;\; {\rm and} \lim_{[\hbar\searrow 0,\; (j,k)\in \Box(\hbar)]}A_{jk}(\hbar) = 0 \edq \end{proposition} {\bf Proof:} Using Parseval equality for orthonormal systems in Hilbert spaces we get: \beqn \sum_{(j,k)\in \Lambda(\hbar)^2}\vert A_{jk}(\hbar)\vert^2 &=& \sum_{j\in \Lambda(\hbar)} \langle \Pi_P(I(\hbar)).A\varphi_j, A\varphi_j\rangle \nonumber\\ & \leq & \sum_{j\in \Lambda(\hbar)}\langle A^2\varphi_j, \varphi_j\rangle \edqn But we know that ${\di \lim_{\hbar\searrow 0}\#\Lambda(\hbar) = +\infty}$ (see \cite{pero} and (11)) So, using $(R_3)$, we get: $$\lim _{\hbar\searrow 0}\frac{1}{\#\Lambda(\hbar)^2}\sum_{(j,k)\in \Lambda(\hbar)^2} \vert A_{jk}(\hbar)\vert^2 = 0$$ and we get the proposition using the following lemma whose proof is implicit in \cite{hemaro} (part.3-p.319) \QED \begin{lemma} Let us consider a mapping: $$]0, +\infty[ \ni\hbar \mapsto \Omega(\hbar) \in {\cal F }(\N)$$ where ${\cal F}(\N)$ is the set of finite part of integers, and finite families of complex numbers: $\{a_j(\hbar)\}_{j\in \Omega(\hbar)}$ such that: $$\lim_{\hbar\searrow 0}\frac{1}{\#\Omega(\hbar)}\sum_{j\in \Omega(\hbar)}\vert a_j(\hbar)\vert = 0$$ then there exists $\tilde{\Omega}(\hbar)\subseteq\Omega(\hbar)$ such that $$\lim_{\hbar\searrow 0}\frac{\#\tilde{\Omega}(\hbar)}{\#\Omega(\hbar)} = 1\;\; {\rm and}\;\; \lim_{[\hbar\searrow 0,\;j\in\tilde{\Omega}(\hbar)]}a_j(\hbar) = 0$$ \end{lemma} Now we can formulate our main results concerning the non diagonal matrix elements. \begin{theorem} We assume that the assumptions $(H_1)$ to $(H_8)$ are fufilled.\\ (i) For every $\varepsilon > 0$ there exists $T_\varepsilon > 0$ and $\hbar_\varepsilon > 0$ such that: \beq \forall j\in M(\hbar),\;\forall k \in \Lambda(\hbar),\; 0 < \hbar\leq \hbar_\varepsilon;\; \vert E_j(\hbar)-E_k(\hbar) \vert \leq \frac{\pi\hbar}{2T_\varepsilon} \Rightarrow \vert A_{jk}(\hbar)\vert \leq \varepsilon \edq (ii) For every family of matrix elements $\{A_{jk}(\hbar)\}_{(j,k)\in \Omega(\hbar)}$ such that:\\ ($\alpha$) $\Omega(\hbar)\subseteq \Lambda(\hbar)^2$ and $(j,k) \in \Omega(\hbar) \Rightarrow j\neq k$\\ ($\beta$) $\di{\lim_{[\hbar\searrow 0,\; (j,k)\in \Omega(\hbar)]} \left(\frac{E_j(\hbar)-E_k(\hbar)}{\hbar}\right) = 0}$\\ ($\gamma$) ${\di \liminf_{\hbar\searrow 0}\left(\frac{\#\Omega(\hbar)}{\#\Lambda(\hbar)}\right)}>0$\\ there exists $\tilde{\Omega}(\hbar) \subseteq \Omega(\hbar)$ such that: \beq \lim_{\hbar\searrow 0}\frac{\#\tilde{\Omega}(\hbar)}{\#\Omega(\hbar)}=1; \; \lim_{\hbar\searrow 0}A_{jk}(\hbar) = 0,\;\; {\rm uniformly\; for}\;\; (j,k)\in \tilde{\Omega}(\hbar) \edq The above statement means: for all $\epsilon>0$, there exits $\hbar_\epsilon>0$, such that for every $0<\hbar<\hbar_\epsilon$ and for every $(j,k)\in \tilde{\Omega}(\hbar)$ we have $\vert A_{jk}(\hbar)\vert \leq \epsilon$.\\ Furthermore, the set $M(\hbar)$ is the same as in Theorem (2-1) and the set $\tilde{\Omega}(\hbar)$ of (ii) can also be chosen independently of the observable $A(\hbar)$. \end{theorem} \begin{remark} (1)There exists a lot of non diagonal families satisfying the assumptions of Theorem (3.5) $(ii)$ (see Appendix (A))\\ (2) Let us consider the Harmonic oscillator in one degree of freedom. For $E>0$ it is not difficult to construct $A(\hbar)$ such that $\langle A(\hbar)\varphi_j, \varphi_{j+1}\rangle \rightarrow 1$ and $(2j+1)\hbar\rightarrow E$ as $\hbar\searrow 0$ (take $a(x,\xi)=x\;{\rm for}\; \vert x \vert \leq \sqrt{E+1}$ ). We can compare this fact with (16) \end{remark} To give further results we introduce a stronger chaotic assumption:\\ $(H_9)$ {\it The dynamical system $\left(\Sigma_E, d\sigma_E, \Phi^t\right)$ is mixing, that means:} $$\lim_{t\nearrow +\infty}\left(\int_{\Sigma_E} a(\Phi^t(z)).a(z)d\sigma_E(z)\right) = \left(\int_{\Sigma_E}a(z)d\sigma_E(z)\right)^2$$ \begin{theorem} Let us assume $(H_1)$ to $(H_9)$. \\ (i) There exists $M(\hbar) \subseteq \Lambda(\hbar)$ ($M(\hbar)$ is the same as in Theorems (2.1) and (3.5) ) such that: $$\lim_{\hbar\searrow 0}\frac{\#M(\hbar)}{\#\Lambda(\hbar)}=1,\;\;\; \lim_{[\hbar\searrow, 0\;j\in M(\hbar),\;k\in \Lambda(\hbar),\;j\neq k]}A_{jk}(\hbar) =0, \;\; \forall aJ\in {\cal B}_\infty$$ (ii) For every family of matrix elements $\{A_{jk}(\hbar)\}_{(j,k)\in \Omega(\hbar)}$ such that:\\ ($\alpha$) $\Omega(\hbar)\subseteq \Lambda(\hbar)^2$ and $(j,k) \in \Omega(\hbar) \Rightarrow j\neq k$\\ ($\beta$) $\exists \tau \in\R$ such that $\di{\lim_{[\hbar\searrow 0,\; (j,k)\in \Omega(\hbar)]} \left(\frac{E_j(\hbar)-E_k(\hbar)}{\hbar}\right) = \tau}$\\ ($\gamma$) ${\di \liminf_{\hbar\searrow 0}\left(\frac{\#\Omega(\hbar)}{\#\Lambda(\hbar)}\right)}>0$\\ there exists $\tilde{\Omega}(\hbar) \subset \Omega(\hbar)$ such that: \beq \lim_{\hbar\searrow 0}\frac{ \#\tilde{\Omega}(\hbar)}{\#\Omega(\hbar)}=1 \; {\rm and}\; \lim_{\hbar\searrow 0}A_{jk}(\hbar) = 0,\;\; {\rm uniformly\; for}\;\; (j,k)\in \tilde{\Omega}(\hbar) \edq the set $\tilde{\Omega}(\hbar)$ of (ii) can also be chosen independently of the observable $A(\hbar)$. \end{theorem} \begin{remark} The results given in Theorems (3.5) and (3.7) can be extended in two directions as we can see in Appendix. \\ Firstly the assumption "$a\in {\cal B}_\infty$" is not essential for the validity of the above results (and also for the following ones). Because we consider the action of $op^w_\hbar(a)$ on states essentially localized in a compact set of the phase space we can take as well unbounded observables satisfying, for some $m\in\N$, $\partial_x^\alpha\partial_\xi^\beta a \in {\cal B}_\infty$ for $\vert\alpha+\beta\vert \geq m$. This can be applied for example to the position or momentum observables (conductivity). We will give in the appendix B the technical details.\\ Secondly, some results hold also for non smooth symbols ( in the Borel class may be sufficient) by replacing Weyl quantization by anti-Wick quantization (which coincides with an error $O(\hbar)$ in the smooth case see Appendix (C)). \end{remark} {\bf COMPARAISON WITH PREVIOUS RESULTS (\cite{ze2})}\\ {\em In \cite{ze2}, S.Zelditch proved analogous results in the high energy "r\'egime", for the Laplace-Beltrami operator $\Delta$ on Riemannian compact manifold $M$. Our results are more acurate for the following reasons. In the case considered in \cite{ze2} the semi-classical parameter is $\hbar=\lambda_j^{-1/2}$, the $\lambda_j$ being the eigenvalues of $\Delta$. Our methods can be applied also to this case, using known results in spectral analysis on manifolds (\cite{dugu,iv}). In \cite{ze2}, theorems A and B, the order of magnitude of eigenvalues families considered is at least $O(\hbar^{-n})$ but ours is at least $O(\hbar^{1-n})$, which is the order of the mean level spacing in quantum mechanics (in agreement with the remainder term in the Weyl formula). Furthermore our proofs show that the number of non controlled non diagonal matrix elements is independent of the observable $A$ }.\\ Let us remark now that the Proposition (1.1) of \cite{ze2} concerning the so called "coherent non vanishing families" $A_{jk}(\hbar)$ can also be extended in our setting:\\ ${\cal B }_\infty\ni a \rightarrow A_{jk}(\hbar)$ is a Schwartz distribution on the phase space $T^*(\R^n)$. By modifying the quantification, we can replace $A(\hbar)$ by $\tilde{A}(\hbar)$ (anti-Wick procedure, see\cite{vo,hemaro} and Appendix (C)), such that $\Vert A(\hbar)-\tilde{A}(\hbar)\Vert=O(\hbar)$ and ${\cal B}_\infty\ni a \stackrel{\mu_{jk}}{\rightarrow} \tilde{A}_{jk}(\hbar)$ define complex valued Radon measures $d\mu_{jk}$ on $\R^{2n}$. It is proved in \cite{hemaro} that $d\mu_{jj}$ are positive bounded Radon measures and the non diagonal case follows easily by the parallelogram identity (Appendix (C)). Following \cite{ze2} we have: \begin{proposition} Let assume $(H_1)$ to $(H_7)$ If $\Omega(\hbar) \subseteq\;\Lambda(\hbar)^2$ is such that there exists a non vanishing Radon measure $\mu$ on $T^*(\R^n)$ satisfying: $$\forall a\in C_0^\infty(\R^{2n})\;\;\; \lim_{[\hbar\searrow 0\;(j,k)\in \Omega(\hbar)]}\int_{\R^{2n}} a(z)d\mu_{jk}(z)= \int_{\R^{2n}} a(z)d\mu(z)$$ then we have:\\ (i) $$\lim_{[\hbar\searrow 0,\;(j,k)\in\Omega(\hbar)]}\left(\frac{E_k(\hbar)-E_j(\hbar)}{\hbar}\right) =\tau\;\; {\rm with}\; \tau:=i^{-1}\frac{\int\{a,p_0\}(z)d\mu(z)}{\int a(z)d\mu(z)}$$ Moreover the limiting value $\tau$ is also an eigenvalue of the Hamiltonian flow i.e: $$\forall t\in\R,\;\forall a\in C_0^\infty(\R^{2n}),\; \int_{\R^{2n}} a(\Phi^t(z))d\mu(z)={\rm e}^{it\tau}\int_{ \R^{2n}}a(z)d\mu(z)$$ and:\\ (ii) $\#\Omega(\hbar)=O(\#\Lambda(\hbar))$ \end{proposition} {\bf Proof:} Using the rules on calculus for $\hbar$-admissible operators, in particular the connections between commutators and Poisson bracket, and between the quantum flow and the classical flow (semi-classical Egorov Theorem, see\cite{ro}), we get: \beqn \langle \frac{1}{\hbar}[A(\hbar), P(\hbar)]\varphi_j,\varphi_k\rangle &=& \left(\frac{E_j(\hbar)-E_k(\hbar)}{\hbar}\right) A_{jk}(\hbar) \nonumber\\ &=&i\langle op_\hbar^w(\{a, p_0\}).\varphi_j,\varphi_k\rangle +O(\hbar)\\ \langle A(\hbar,t)\varphi_j,\varphi_k\rangle &=& \exp\left(\frac{it}{\hbar}(E_k(\hbar)-E_j(\hbar))\right)A_{jk}(\hbar)\nonumber \\ &=& \langle op_\hbar^w(a(\Phi^t))\varphi_j,\varphi_k\rangle+ O(\hbar) \edqn So, if we choose $a\in C_0^\infty(\R^{2n})$ such that $\int_{\R^{2n}} a(z)d\mu(z) \neq 0$ we get: $$\lim_{[\hbar\searrow 0,\;(j,k)\in \Omega(\hbar)]} \left(\frac{E_k(\hbar)-E_j(\hbar)}{\hbar}\right) = i^{-1}\frac{\int \{a, p_0\}d\mu}{\int ad\mu}$$ The part $(i)$ of the proposition follows. \\ \underline {Proof of $(ii)$:} Let $a$ be such as above. There exists $\kappa > 0$ and a $\hbar_\kappa >0$ such that for $0<\hbar\leq\hbar_\kappa$ we have: \beq \lim_{\hbar\searrow 0}\frac{1}{\#\Omega(\hbar)}\sum_{(j,k)\in \Omega(\hbar)} \vert A_{jk}(\hbar)\vert^2 \geq \kappa \edq But we have: \beqn\label{*} \frac{1}{\#\Lambda(\hbar)}\sum_{(j,k)\in \Lambda(\hbar)^2}\vert A_{jk}(\hbar)\vert^2 &=& \frac{1}{\#\Lambda(\hbar)}\sum_{j\in \Lambda(\hbar)} \langle \Pi_{P(\hbar)}(I(\hbar)).A\varphi_j, A\varphi_j\rangle \nonumber\\ & \leq & \frac{1}{\#\Lambda(\hbar)} \sum_{j\in \Lambda(\hbar)}\langle A^2(\hbar)\varphi_j, \varphi_j\rangle \edqn then, using a variant of Th\'eor\eme (1.3) in\cite{hemaro} we get: \beq\label{**} \lim_{\hbar\searrow 0} \frac{1}{\#\Lambda(\hbar)}\sum_{(j,k)\in \Lambda(\hbar)^2}\vert A_{jk}(\hbar)\vert^2 \leq \int_{\Sigma_E}\vert a\vert^2d\sigma_E \edq So, we get that for $\hbar$ small enough, there exists $K>0$ such that: \beq K(\#\Lambda(\hbar))\geq\sum_{(j,k)\in \Lambda(\hbar)^2}\vert A_{jk}(\hbar)\vert^2 \geq\sum_{(j,k)\in \Omega(\hbar)}\vert A_{jk}(\hbar)\vert^2 \geq\kappa(\#\Omega(\hbar)) \edq and $(ii)$ of the proposition follows. \QED Our third main result in this paper is a mathematically rigorous version of a Wilkinson's result concerning the variance of the statistical distribution of the matrix elements $A_{jk}(\hbar)$ (\cite{wi}).\\ In physics literature (see for example \cite{wi,pe}) it is conjectured that for classically chaotic systems, the matrix elements $A_{jk}(\hbar)$ are independent, Gaussian, with mean zero when $j\neq k$. The last statement is corroborated by our above theorems. Wilkinson\cite{wi} proposed the following definition for the variance: \beqn S_{(f,g)}(\hbar,E,\Delta E)&=& \nonumber\\ \sum_{[E_j(\hbar),E_k(\hbar)\in J_{c\ell}]}\vert A_{jk}(\hbar)\vert^2 f_{\hbar}\left(E-\frac{1}{2}(E_j(\hbar)+E_k(\hbar)\right)&&. g_{\hbar}\left(\Delta E-(E_j(\hbar)-E_k(\hbar))\right) \edqn where $E$ is inside the interval $J_{c\ell} \subset I_{c\ell}$ and $f_\hbar,\;g_\hbar$ are Gaussian regularisations of the Dirac $\delta$ distribution. For technical convenience, we choose for $f_\hbar,\;g_\hbar$ smooth functions, compactly supported in Fourier variable. Let $f,\;g$ be smooth functions on $\R$ with Fourier transform of compact support: $\hat{f}(v)=\int_{\R}{\rm e}^{-iuv}f(u)du$. Then we define $f_\hbar(u):=\frac{1}{\hbar}f(\frac{u}{\hbar})$. \begin{theorem} Let us assume that Supp($\hat{f}) \subseteq\; ]-T_0, T_0[$ with $T_0>0$ small enough and Supp($g$) compact. Then under assumptions $(H_1)$ to $(H_6)$ we have the following asymptotic expansion, mod($O(\hbar^\infty)$), as $\hbar \searrow 0$: \beq S_{(f,g)}(\hbar,E,\Delta E)\asymp \hbar^{-n-1}. \left(\sum_{j\geq 0}\Gamma_j\left(E,\frac{\Delta E}{\hbar}\right)\hbar^j\right) \edq where $\Gamma_j(E,\tau)$ is smooth in $E$ and $\tau$. In particular we have: \beq \Gamma_0(E,\tau)=\hat{f}(0)\int\hat{g}(t){\rm e}^{it\tau}C_a(E,t)dt \edq where $C_a(E,t)$ is the classical auto-correlation function: $$C_a(E,t) := \int_{\Sigma_E}a(z)a(\Phi^t(z))d\sigma_E(z)$$ \end{theorem} \begin{remark} No chaotic assumption is needed for the validity of the above theorem because we choose Supp($\hat{f}$) small around 0. We could get an analogous of Gutzwiller trace formula \cite{gu} by taking Supp($\hat{f}$) compact but arbitrary large. In clear, if the flow $\Phi^t$ restricted to $\Sigma_E$ is clean, that is to say:\\ (i) ${\cal P }_E:=\{(t,z)\in \R\times\Sigma_E, \Phi^t(z)=z \}$ is a smooth manifold\\ (ii) $\forall (t_0,z_0)\in {\cal P }_E$ the tangent space $T_{(t_0,z_0)}({\cal P}_E)$ at $(t_0,z_0)$ satifies $$T_{(t_0,z_0)}({\cal P }_E)= \{(\tau,\zeta)\in \R\times T_{z_0}(\Sigma_E),\; \tau{\cal H }_{p_0}(z_0) + D\Phi^{t_0}(z_0)(\zeta) = \zeta \}$$ then we can get: $$S_{(f,g)}(\hbar,E,\Delta E) \asymp \sum_{j\geq 0}\hbar^{j-n-1} \gamma_j(\hbar;\hat{f},\hat{g},E,\Delta E)$$ where $\hat{f} \rightarrow \gamma_j(\hbar;\hat{f},\hat{g},E,\Delta E)$ are distributions on $\R$, supported by the set of periods of the flow $\Phi^t$ on $\Sigma_E$ and can be made more explicit under a non degenerescence condition on the closed path of $\Phi^t$ on $\Sigma_E$. ( for details see \cite{guur,me}) But it seems difficult to get rigorous results for Supp($\hat{f}$) non compact even very fast decreasing! \end{remark} \section{ Proofs for Theorems (3.5) and (3.7)} Let $a\in{\cal B}_\infty$ be such that $_E=0$ (for the proofs of Theorems (3.5) and (3.7) it is clearly sufficient to consider this case). With the notations of sections 1 and 2, we have: \beq\label{eq21} \sum_{E_k\in J_{c\ell}}\vert A_{jk}(\hbar)\vert^2 = \Vert \Pi_{P(\hbar)}(J_{c\ell})A(\hbar)\varphi_j\Vert^2 \leq \Vert A(\hbar)\varphi_j\Vert^2 \edq We apply this estimate replacing $A(\hbar)$ with $A_{\theta_T}(\hbar)$ which is defined by: $$A_{\theta_T}(\hbar)=\int_\R \theta_T(t)A(\hbar,t)dt,\; {\rm with}\; A(\hbar,t):=U(-t,\hbar)A(\hbar)U(t,\hbar)$$ Let us choose $\theta_T=\frac{1}{2T}1_{[-T,T]}$ so we have: $\hat{\theta}_T(\lambda)=\frac{\sin(T\lambda)}{T\lambda}$. It is known that $A(\hbar,t)$ is an $\hbar$-admissible operator (see for example \cite{ro}) with a $\hbar$-Weyl principal symbol $a(t;x,\xi)=a\left(\Phi^t(x,\xi)\right)$. In particular we have: \beq \Vert A(\hbar,t)-op^w_\hbar(a(t))\Vert = O(\hbar) \;\;{\rm as}\;\hbar\searrow 0\; {\rm uniformly \;for }\; t\; {\rm in\; every\; bounded\;interval} \edq Let us denote: $\omega_{jk}(\hbar)=\frac{E_k(\hbar)-E_j(\hbar)}{\hbar}$ (energies transition). We have: \beq \langle A_{\theta_T}(\hbar)\varphi_j, \varphi_k\rangle = 2\pi\hat{\theta}_T(\omega_{jk}(\hbar))A_{jk}(\hbar) \edq We shall use the elementary inequality: $\frac{\sin(u)}{u} \geq \frac{2}{\pi}$ for $0\leq u \leq \frac{\pi}{2}$.(*)\\ Hence we get from (28) \beq \sum_{E_k \in J_{c\ell}}4\pi^2\left(\frac{\sin(T\omega_{jk})}{T\omega_{jk}}\right)^2 \vert A_{jk}(\hbar)\vert^2 \leq \langle A^*_{\theta_T}A_{\theta_T}\varphi_j,\varphi_j\rangle \edq Fix an $\varepsilon >0$ and $T>0$. Using Theorem(\ref{hmr}), there exists $\hbar_{\varepsilon,T} > 0$ such that for $0<\hbar<\hbar_{\epsilon,T}$ and $j\in M(\hbar)$ we have : \beq\label{eq22} \vert\langle A^*_{\theta_T}A_{\theta_T}\varphi_j,\varphi_j\rangle - \int_{\Sigma_E}\left\vert\frac{1}{2T}\int_{-T}^{T}a(\Phi^t(z))dt\right\vert^2d\sigma_E(z)\vert \leq 8\varepsilon^2 \edq Now, using that $_E=0$ and $(H_8)$ we choose $T=T_\varepsilon$ large enough such that: $$\int_{\Sigma_E}\left\vert\frac{1}{2T_\varepsilon} \int_{-T_\varepsilon}^{T_\varepsilon}a(\Phi^t(z))dt\right\vert^2d\sigma_E(z) \leq 8\varepsilon^2$$ and $\hbar_\varepsilon$ small enough such that: $\vert\omega_{jk}(\hbar)\vert\leq \frac{\pi}{2T_\varepsilon}\;\; \forall (j,k) \in \Omega(\hbar),\; 0<\hbar\leq\hbar_\varepsilon$. Then the conclusions of the first part of theorem (3.5) follow.\\ For proving the second part we follow \cite{ze2} by estimating the variances ( we always assume $=0$) : $${\cal V}_{\Omega(\hbar)}(A):=\frac{1}{\#\Omega(\hbar)}\sum_{(j,k)\in \Omega(\hbar)} \vert A_{jk}(\hbar) \vert^2$$ Using $(*)$, for every $T>0$ there exists $\hbar_T>0$ such that: \beq \label{eq23} {\cal V}_{\Omega(\hbar)}(A) \leq \frac{\pi^2}{4(\#\Omega(\hbar))} \sum_{\omega_{jk}(\hbar)T<\pi/2, E_j\in I(\hbar), E_k\in J_{c\ell}}\vert A_{jk}(\hbar) \vert^2 \left\vert\frac{\sin(\omega_{jk}T)}{\omega_{jk}T}\right\vert^2 \edq for $\hbar\in ]0, \hbar_T[$.\\ Using (\ref{eq22}), (\ref{eq21}) we can see easily that it exists $c>0$ such that for $\hbar$ small enough we have: \beq\label{eq27} {\cal V}_{\Omega(\hbar)}(A) \leq \frac{c}{\#\Lambda(\hbar)}.\sum_{j\in \Lambda(\hbar)} \langle A_{\theta_T}^*.A_{\theta_T},\varphi_j,\varphi_j\rangle +O(\hbar) \edq with $O(\hbar)$ uniform in $T$.\\ The proof of the second part of Theorem (3.5) follows from (\ref{eq27}) by the same argument used in the first part (see \ref{eq22}) and using the lemma (3.4). To prove that we can choose $\tilde{\Omega}(\hbar)$ independently of $A(\hbar)$ we use the construction of \cite{hemaro}, page 321. \QED Now we begin the proof of the Theorem (3.7). We will use the following lemma concerning sum rules which appeared in the physics literature for example in Feingold-Peres \cite{fepe} and in Prosen-Robnik \cite{prro}. \begin{lemma} Let $J_{m,c\ell}:=[\alpha_m, \beta_m],\;m=1,2$ two closed classical energy intervals such that: $\lambda_- <\alpha_1 <\alpha_2 0$ we denote $\theta_T(t)=\frac{1}{T}\theta(\frac{t}{T})$. Here we choose $\hat{\theta}(\lambda)=\left(\frac{\sin\lambda}{\lambda}\right)^2$ \\ So from (\ref{eq24}) we have: \beq\label{eq28} \sum_{E_k\in J_{c\ell}}\vert A_{jk}(\hbar)\vert^2\hat{\theta}_T(\omega_{jk}(\hbar)-\tau) =\int_{\R}{\rm e}^{it\tau}\theta_T(t)\langle\varphi_j,A(t,\hbar))\chi(P(\hbar))A(\hbar) \varphi_j\rangle dt + O(\hbar) \edq with $O(\hbar)$ uniform in $T>0$, $\tau\in \R$ and $j\in\Lambda(\hbar)$.\\ The calculus on $\hbar$-admissible operators (\cite{ro}) shows that $A(t,\hbar))\chi(P(\hbar))A(\hbar)$ has for principal symbol: $a(\Phi^t(x,\xi))\chi(p_0(x,\xi))a(x,\xi)$ . Let us recall that $C_a(E,t) = \int_{\Sigma_E}a(z)a(\Phi^t(z))d\sigma_E(z)$ (autocorrelation function). So fixing $\varepsilon>0,\;T>0$ and using Theorem (\ref{hmr}) again we get that for every $T>0$ there exists $\hbar_{\varepsilon,T}>0$ such that: \beqn\label{eq29} \vert \langle\varphi_j,A(t,\hbar))\chi(P(\hbar))A(\hbar)\varphi_j\rangle - C_a(E,t)\vert < \frac{\varepsilon}{2} \\ \forall \hbar\in ]0, \hbar_{\varepsilon,T}[, \;\forall t\in [-T^2, T^2]\; {\rm and}\; j\in M(\hbar) \edqn Then we use (\ref{eq28}), splitting the integral according $\vert t \vert < T^2$ and $\vert t \vert > T^2$, and using (\ref{eq29}) we get for some $\gamma > 0$, independent of $\hbar, \varepsilon, T$: \beqn\label{eq210} \vert A_{jk}(\hbar)\vert^2\hat{\theta}_T(\omega_{jk}(\hbar)-\tau) &\leq &\\ \int_{\R}\vert \theta(u).C_a(E,Tu)\vert du + \frac{\varepsilon}{2} &+& \gamma\int_{\{\vert u \vert \geq T \}} \vert \theta (u) \vert du; \;\; \ \forall \hbar\in ]0,\hbar_{\varepsilon,T}[,\;\forall \tau\in \R. \edqn Now we will use the mixing assumption $(H_9)$ and $_E=0$. That gives by dominated Lebesgue convergence theorem : $$\lim_{T\nearrow +\infty}\int\vert \theta (u)C_a(E,uT) \vert du = 0,\; \lim_{T\nearrow +\infty}\int_{\{\vert u \vert \geq T \}} \vert \theta (u) \vert du = 0$$ hence there exits $T_\varepsilon>0$ such that: \beq\label{eq211} \vert A_{jk}(\hbar)\vert^2\hat{\theta}_{T_{\varepsilon}} (\omega_{jk}(\hbar)-\tau) \leq \varepsilon, \;\; \ \forall \hbar\in ]0,\hbar_\varepsilon[\;\;(\hbar_\varepsilon:=\hbar_{\varepsilon,T_\varepsilon});\; \forall (j,k)\in M(\hbar)\times \Lambda(\hbar),\; \forall \tau\in\R \edq The estimates (\ref{eq211}) being uniform in $\tau$ we can take $\tau=\omega_{jk}(\hbar)$ so we get \beq \vert A_{jk}(\hbar)\vert^2 \leq \epsilon, \;\; \ \forall \hbar\in ]0,\hbar_\epsilon[,\;\; \forall (j,k)\in M(\hbar)\times \Lambda(\hbar) \edq so that we have proved the first part of the theorem.\\ For the second part we use the same method as in Theorem (3.5). With the same notations, we have: \beq \label{eq212} {\cal V}_{\Omega(\hbar)}(A) \leq \frac{\pi^2}{4}\frac{1}{\#\Omega(\hbar)} \sum \vert A_{jk}(\hbar) \vert^2 \left\vert\frac{\sin((\omega_{jk}(\hbar)-\tau)T)}{(\omega_{jk}(\hbar)-\tau)T}\right\vert^2 \edq for $\hbar\in ]0, \hbar_T[$, where the sum is computed for ${(\omega_{jk}(\hbar)-\tau)T<\pi/2, E_j\in I(\hbar), E_k\in J_{c\ell}}$\\ Then we get, using again the lemma (4.1): \beq\label{eq213} {\cal V}_{\Omega(\hbar)}(A) \leq c.\frac{1}{\#\Lambda(\hbar)}\sum_{j\in \Lambda(\hbar)} \int_{\R}{\rm e}^{it\tau}\theta_T(t)\langle\varphi_j,A(t,\hbar))\chi(P(\hbar))A(\hbar) \varphi_j\rangle dt + O(\hbar) \edq with $O(\hbar)$ uniform in $T$.\\ >From (\ref{eq213}) the second part of the theorem (3.7) is obtained using the arguments of the first part. \QED By the same kind of estimates as those used in the proof of the Theorem (3.7), we can get something which seems connected with a well known phenomenon called the levels repulsion \cite{ch}. Let us introduce the following assumption on the autocorrelation function:\\ $(H_9^*)$ $\Gamma_E:=\int_{\R}\vert C_a(E,t) \vert dt <+\infty$ Indeed, using the function $\theta(t)={\rm e}^{-\omega\vert t \vert},\; \omega>0$, we get, as above, that: \beqn\label{eq8} \forall \varepsilon>0,\;\forall \omega>0,\; \exists \hbar_{\varepsilon,\omega}>0\; {\rm such\;that}\nonumber\\ \vert A_{jk}(\hbar)\vert^2\frac{\omega}{\omega^2+\omega_{jk}^2} \leq \pi\int_{\R}\vert C_a(E,t)\vert dt + \varepsilon, \;\;\forall \hbar\in ]0, \hbar_{\varepsilon,\omega}[,\;\forall j\in M(\hbar),\; \forall k\in \Lambda(\hbar) \edqn From (\ref{eq8}) we get easily the proposition: \begin{proposition} Under the conditions of (\ref{eq8}), for every $K>\pi\Gamma_E-\epsilon$, we have: \beq \{\vert A_{jk}(\hbar)\vert^2 \geq K\omega \} \Rightarrow \{\vert E_j(\hbar)-E_k(\hbar)\vert^2\geq \left(\frac{K-\pi\Gamma_E-\epsilon}{(\pi\Gamma_E+\epsilon)}\right)(\hbar\omega)^2 \} \edq \end{proposition} \begin{remark} The assumption $(H_9^*)$ is not completely absurd. In \cite{po,ru} the authors consider dynamical systems such that the Fourier transforms of autocorrelation function have analytic extension in horizontal trips in the complex plane. The geodesic flow on constant negative curvature compact manifolds (like the Poincar\'e half plane) satisfies this property; so in this case the autocorrelations are exponentially fast decreasing ($O({\rm e}^{-\delta\vert t \vert})$ for some $\delta>0$). \end{remark} %\documentstyle{article} %\pagestyle{empty} %\begin{document} %\input{new.def} \section{Asymptotic Sum Rule for the Variance of Matrix Elements} Let us assume that Supp($\hat{f})\; \subseteq\; ]-T_0, T_0[$ with $T_0>0$ small enough, and Supp($\hat{g}$) compact. We assume the $P(\hbar)$ satisfies $(H_1)$ to $(H_5)$. The variance considered by Wilkinson \cite{wi} is defined by: $$S_{(f,g)}(\hbar,E,\Delta E):=\sum_{E_j,E_k\in J_{c\ell}}\vert A_{j,k}(\hbar)\vert^2 f_{\hbar}(E-\frac{1}{2}(E_j+E_k)) g_{\hbar}\left(\Delta E-(E_j-E_k)\right)$$ In fact, Wilkinson considers the case where $\hat{f},\;\hat{g}$ are Gaussians which seems difficult to treat mathematically (If Supp($\hat{f}$) is compact but not small see Remark (3.11) ).\\ Let us begin by the following lemma which localizes the sum $S_{(f,g)}(\hbar,E,\Delta E)$ to energies which are close to $E$ and replace the "abrupt" energy cut off in $J_{c\ell}$ by a smooth one. \begin{lemma} Let $\varepsilon_0>0$ be such that $[E-\varepsilon_0, E+\varepsilon_0]\subset J_{c\ell}$ and $\chi \in C^\infty_0(J_{c\ell})$, $\chi=1$ on $[E-\varepsilon_0, E+\varepsilon_0]$, $0\leq \chi \leq 1$. Let us denote $A_\chi(\hbar):=\chi(P(\hbar))A(\hbar)\chi(P(\hbar))$ and by $A_{\chi,jk}(\hbar)$ the corresponding matrix elements. Then we have: \beq S_{(f,g)}(\hbar,E,\Delta E)= \sum_{j,k}\vert A_{\chi,jk}(\hbar)\vert^2 f_{\hbar}(E-\frac{1}{2}(E_j+E_k) g_{\hbar}\left(\Delta E-(E_j-E_k)\right) + O(\hbar^{\infty}) \edq \end{lemma} {\bf Proof:} We have: \beqn\label{e31} S_{(f,g)}(\hbar,E,\Delta E)- \sum_{j,k}\vert A_{\chi,jk}(\hbar)\vert^2 f_{\hbar}(E-\frac{1}{2}(E_j+E_k)) g_{\hbar}\left(\Delta E-(E_j-E_k)\right) = \nonumber\\ \sum_{j,k}(1-\chi(E_j)^2\chi(E_k)^2) \vert A_{jk}(\hbar)\vert^2 f_{\hbar}(E-\frac{1}{2}(E_j+E_k) g_{\hbar}\left(\Delta E-(E_j-E_k)\right) \edqn We split the r.h.s of the last equality into two terms according $\vert E_k-E_j\vert > \varepsilon_0$ or $\vert E_k-E_j\vert \leq \varepsilon_0$. First, remark that, using the proposition 3.1 and the remark 3.2 , the contribution of $\vert E_k-E_j\vert > \varepsilon_0$ in the sum is $O(\hbar^{\infty})$. We have $\chi(E_j)^2\chi(E_k)^2 <1$. Assume for example that $\chi(E_j) < 1$. Then we have $\vert E-E_j\vert > \varepsilon_0$ hence if $\vert E_k-E_j\vert \leq \varepsilon_0$ we have: $$\vert E-\frac{1}{2}(E_j+E_k) \vert \geq \vert E-E_j \vert - \frac{1}{2}\vert E_j-E_k \vert \geq \varepsilon_0/2$$ But $f \in {\cal S }(\R)$ (the Schwartz space) so we see that the contribution of $\vert E_k-E_j\vert \leq \varepsilon_0$ is $O(\hbar^\infty)$ and (\ref{e31}) gives the lemma.\QED We continue to denote by $S_{(f,g)}(\hbar,E,\Delta E)$ the approximation of the variance mod($O(\hbar^{\infty})$) given by the above lemma.\\ By inverse Fourier transform, we get: \beqn\label{e32} f_\hbar(E-\frac{1}{2}(E_j+E_k))&=& \frac{1}{2\pi\hbar}\int_\R \hat{f}(\Delta t) \exp\left(i\frac{\Delta t}{\hbar}(E-\frac{1}{2}(E_j+E_k))\right)d\Delta t \\ g_\hbar(\Delta E-(E_j-E_k))&=&\frac{1}{2\pi\hbar}\int_\R \hat{g}(t) \exp\left(i\frac{t}{\hbar}(\Delta E-(E_j-E_k))\right)dt \edqn Now by plugging (\ref{e32}) and (50) in (\ref{e31}), and using Parseval relation for the orthonormal system $\{\varphi_j\}$, we get after computations: \beqn\label{e33} S_{(f,g)}(\hbar,E,\Delta E) = \frac{1}{(2\pi\hbar)^2}{\rm tr} \left\{\int\int\hat{g}(t)\exp(\frac{i}{\hbar}t\Delta E)A_\chi(\hbar)\hat{f}(\Delta t)\right. . \nonumber\\ \left. .\exp(\frac{-i\Delta t}{\hbar}(P(\hbar)-E))A_\chi(\hbar,\frac{\Delta t}{2}-t)(\hbar) dtd\Delta t\right\} \edqn To achieve the proof of the theorem (3.10) we use the W.K.B method along the same lines as in \cite{hero1,ro,pero} where similar quantities were studied (i.e a regularization of the Fourier transform of spectral density). So, we will give here only a sketchy proof. Let us first recall that the operators: $\chi(P(\hbar))A(\hbar)$ and $A_\chi(\hbar,\frac{\Delta t}{2}-t)$ are $\hbar$-admissible operators with weight 1 (see \cite{ro}). Secondly if $\tilde{\chi} \in C_0^\infty(J_{c\ell})$ is an other cut-off such that $\tilde{\chi} = 1$ on Supp($\chi$) then we have : $$\chi(P(\hbar))\exp(\frac{i\tau}{\hbar}P(\hbar)) = \chi(P(\hbar))\exp(\frac{i\tau}{\hbar}\tilde{\chi}(P(\hbar)).P(\hbar))$$ So nothing is changed if we replace in the exponent $P(\hbar)$ by $\tilde{\chi}(P(\hbar)).P(\hbar)$ which satisfies the assumptions $(H_1)$ to $(H_5)$ with $\tilde{p}_0 \in {\cal B}_\infty$; this last property is important to have uniform estimates in W.K.B method when solving the eikonal equation: \beqn \partial_\tau \phi(\tau,x,\eta) + p_0(x,\partial_x \phi(\tau,x,\eta)) & = & 0 \nonumber \\ \phi(0,x,\eta) & = & x.\eta \edqn and the transport equations (see \cite{ro}) \\ In this way we get, for $\vert \tau \vert$ small enough, accurate approximations $U_{app}(\tau,\hbar)$ for $U(\tau,\hbar)$ such that: $$\Vert\exp\left(\frac{-i\tau}{\hbar}\tilde{\chi}(P(\hbar)).P(\hbar)\right) - U_{app}(\tau,\hbar)\Vert = O(\hbar^\infty)$$ where $U_{app}(\tau,\hbar)$ is constructed as oscillating integral kernel: \beq\label{e34} U_{app}(\tau,\hbar)(x,y) = (2\pi\hbar)^{-n} \int \exp\left(\frac{i}{\hbar}(\phi(\tau,x,\eta) - x.\eta\right) \left(\sum_{j\geq 0}\hbar^ju_j(\tau,x,\eta)\right)d\eta \edq Hence using the computation rules on $\hbar$-admissible operators \cite{ro}, finally, the proof of the theorem (3.10) is achieved by applying the stationary phase theorem to integrals like: $$\int\;\int\;\int_{\R_t\times\R_x^n\times\R_\xi^n} b_t(\tau,x;\xi) \exp\left(\frac{i}{\hbar}(\phi(\xi,x,\xi) - x.\xi\right) \hat{f}(\tau)d\tau dxd\xi$$ In particular, for the dominant term $S_0(\hbar,E,\Delta E)$, we have: \beqn\label{e35} S_0(\hbar,E,\Delta E) = \nonumber\\ (2\pi\hbar)^{-n }\int\;\int\;\int\hat{g}(t)\exp(\frac{i}{\hbar}t\Delta E) \chi^4(p_0(x,\partial_x\phi(\tau;x,\eta)))\nonumber \\ \times a(x,\partial_x\phi(\tau;x,\eta))a\left(\Phi^{\tau/2-t}(x,\partial_x\phi(\tau;x,\eta)\right) \nonumber\\ \times \hat{f}(\tau)\exp\left(\frac{i}{\hbar}(\phi(\tau,x,\eta) - x.\eta + \tau E)\right)d\tau dxd\eta dt \edqn $Supp(\hat{f})$ being small enough, the stationary points in the integral (\ref{e35}) in the variables $(\tau,x,\eta)$ are defined by the equations: $$\{\tau = 0,\;\;\;\;p_0(x,\eta) = 0\}$$ So, the computation of the first term in the stationary phase theorem \cite{ro} gives: \beqn S_0(\hbar,E,\Delta E) = (2\pi\hbar)^{-n-1}\hat{f}(0)\int_\R \left(\int_{\Sigma_E}a(x,\eta)a(\Phi^t(x,\eta))d\sigma_E\right).\nonumber \\ .\hat{g}(t)\exp\left(\frac{it}{\hbar}\Delta E\right)dt\;\; +\;\; \sum_{j\geq 1}\hbar^{-n-1+j}c_j\left(E,\frac{\Delta E}{\hbar}\right) \edqn \QED %\documentstyle{article} %\pagestyle{empty} %\begin{document} %\input{new.def} \section{Other related results} {\bf A-A very weak form of chaos} Here, we shall prove a quantum mechanical analog of a simple and beautiful result due to Helton \cite{he} (see also \cite{gu}) for elliptic operators on compact manifolds. \begin{theorem} Under the assumptions $(H_1)$ to $(H_6)$ for $P(\hbar)$, assume furthermore that there exits on $\Sigma_E$ a non periodical trajectory for the flow $\Phi^t$. Then for every $c>0,\;\delta>0$ and every $\hbar_0>0$ the set: $${\cal T}_{E,\delta}:= \{\omega_{jk}(\hbar),\; E_j(\hbar), E_k(\hbar) \in [E-c\hbar^{1-\delta}, E+c\hbar^{1-\delta}],\; 0<\hbar\leq\hbar_0\}$$ is dense in $\R$; where $\omega_{jk}(\hbar)=\frac{E_j(\hbar)-E_k(\hbar)}{ \hbar}$. \end{theorem} {\bf Proof:} Let $f \in C^\infty_0(\R)$ be such that $f=0$ on ${\cal T}_{E,\delta}$. We have to show that $f\equiv 0$ on $\R$. Let us introduce $\chi\in C_0^\infty(]-c, c[)$ with $\chi(0)=1$. Following Helton\cite{he} we consider the operator: \beq\label{eq41} A_{E,f}(\hbar)=\int\hat{f}(t)U(-t,\hbar)A_E(\hbar)U(t,\hbar)dt \edq with $A_E(\hbar)=\chi(\frac{P(\hbar)-E}{h^{1-\delta}})op^w_\hbar(a) \chi(\frac{P(\hbar)-E}{h^{1-\delta}})$, $a\in C_ 0^\infty(\R^{2n})$. By inverse Fourier transform, we have also: \beq\label{eq42} A_{E,f}(\hbar)=2\pi\sum_{j,k} f(\omega_{jk}(\hbar))\chi\left(\frac{E_j(\hbar)-E}{\hbar^{1-\delta}}\right)\chi \left(\frac{E_k(\hbar)-E}{\hbar^{1-\delta}}\right) \Pi_kA(\hbar)\Pi_j \edq where $\Pi_k$ is the projection on the state $\varphi_k$.\\ >From (\ref{eq41},\ref{eq42}) we have: \beq\label{eq43} \int\hat{f}(t)U(-t,\hbar)A_E(\hbar)U(t,\hbar)dt = 0,\;\forall a \in {\cal B}_\infty \edq >From (\ref{eq43}) we would like to prove that $f\equiv 0$ by going to the classical limit $\hbar \searrow 0$. To do that we first use the semi-classical Egorov theorem\cite{ro} and functional calculus with parameter for pseudodifferential operators \cite{daro}. We test (\ref{eq43}) with the trace functional of operators with an arbitary $op^w_\hbar(b),\;b\in {\cal B}_\infty$. we get easily: \beq\label{eq44} \lim_{\hbar\searrow 0}(2\pi\hbar)^n.Tr\left( A_{E,f}(\hbar).op^w_\hbar(b)\right) = \int\int\hat{f}(t)a\left(\Phi^t(z))\right).b(z)dzdt = 0, \; \forall b\in {\cal B}_\infty \edq So we get \beq\label{eq45} \int\hat{f}(t)a\left(\Phi^t(z))\right)dt = 0, \;\forall z\in \Sigma_E \edq Now, choose $z_0 \in \Sigma_E$ such that $t\rightarrow \Phi^t(z_0)$ is not periodic, we should like to deduce from (\ref{eq45}) that $\hat{f}\equiv 0$. Using the same arguments as in \cite{hero2} (p.866-867) we can get easily the following: \begin{lemma} For $T>0$ we can find $\rho_T>0$ such that the mapping: $\Phi: (t,z)\stackrel{F}{\mapsto} (t,\Phi^t(z))$ is a diffeomorphism from $]-T, T[\times D_{\rho_T}(z_0)$ onto an open neighborhood ${\cal N_T}$ of the curve: $\{\Phi^t(z_0),\;-T0$ such that: \beq\label{doz} {\rm spectrum}[P(\hbar)]\cap [E-c\hbar^{1-\delta}, E+c\hbar^{1-\delta}] \subseteq \bigcup_{k\in \Z}[\gamma_0+\gamma_1k\hbar-C\hbar^{1+\varepsilon}, \gamma_0+\gamma_1k\hbar-C\hbar^{1+\varepsilon}] \edq Clearly (\ref{doz}) entails that ${\cal T}_{E,\delta}$ is not dense in $\R$. \end{remark} {\bf B-Sum rules and classical limits}\\ We want here to revisit the physics literature on this subject and make the connection with the technique used in part {\bf 4} of this paper.\\ Under the assumptions $(H_1)$ to $(H_5)$, a useful distribution to consider is: $${{\cal R}}_{A,B;j,\hbar}(\Delta E) = \sum_{E_k(\hbar)\in J_{c\ell}}A_{jk}(\hbar)B_{kj}(\hbar)\delta(\omega_{jk}(\hbar)-\Delta E)$$ where $A(\hbar), B(\hbar)$ are two quantum observables, with $\hbar$-Weyl symbols in ${\cal B}_\infty$. For $A=B$, ${\cal R}_{A,A;j,\hbar} ( \Delta)$ is the response function of some atomic kernel, in the state $\varphi_j$, to the action of $A$. \\ Let us denote by $\hat{{\cal R}}_{A,B;j,\hbar}(t)$ the Fourier transform in $\Delta E$ of ${\cal R}_{A,B;j,\hbar}(\Delta E)$. >From Parseval identity we have also: $$\hat{{\cal R}}_{A,B;j,\hbar}(t)= \langle \varphi_j, A(t,\hbar)\Pi_{P(\hbar)}(J_{c\ell})B(\hbar)\varphi_j\rangle$$ As above (see part.4) it is convenient to smooth the spectral projector $\Pi_{P(\hbar)}(J_{c\ell})$. \begin{lemma} Let us consider an interval $\tilde{J}_{c\ell} \subset J_{c\ell}$ and a smooth cutoff $\chi \in C_0^\infty(J_{c\ell}),\;\chi\equiv 1 \;{\rm on\;a \; neighborhood\;of}\; \tilde{J}_{c\ell}$. Then we have: \beq \hat{{\cal R}}_{A,B;j,\hbar}(t)=\langle \varphi_j, (A(t,\hbar)\chi(P(\hbar))B(\hbar)\varphi_j\rangle + O(\hbar^\infty), \;\; {\rm for}\; E_j(\hbar) \in \tilde{J}_{c\ell}, \edq the $O(\hbar^\infty)$ being uniform in $j$ such that $E_j(\hbar) \in \tilde{J}_{c\ell}$ and in $t$ such that $\vert t \vert \leq T$ for some $T>0$. \end{lemma} {\bf Proof:} We denote $J_{c\ell}=[\alpha, \beta]$ and $\tilde{J}_{c\ell}=[\lambda,\mu]$ with $\alpha < \lambda < \mu < \beta$. We fix some $\varepsilon > 0$, small enough. Then we construct a family of smooth cutoff functions as follows: $\chi=\chi_0,\; \chi\equiv1$ on $[\lambda-\varepsilon, \mu+\varepsilon]$ and for every $N\geq 1$ we construct $\chi_N$ such that $\chi_N\in C_0^\infty(]\lambda-\frac{\varepsilon}{2^{N-1}}, \mu+\frac{\varepsilon}{2^{N-1}}[)$, $\chi_N\equiv 1$ on $[\lambda-\frac{\varepsilon}{2^N}, \mu+\frac{\varepsilon}{2^N}]$\\ In what follows we skip the $\hbar$-dependence for simplicity, although it is everywhere present.\\ In the first step we have: \beq\label{e61} \langle \varphi_j, A(t)\Pi_P(J_{c\ell})B\varphi_j\rangle = \langle \varphi_j, A(t)\chi_0(P)B\varphi_j\rangle +\langle \varphi_j, A(t)\Pi_P(J_{c\ell})[B,\chi_0(P)]\varphi_j\rangle \edq the last term in (\ref{e61}) is $O(\hbar)$. To improve this estimate we apply (\ref{e61}) to it remarking that $B$ is replaced by $[B,\chi_0(P)]$. So we get: \beqn\label{e62} \langle \varphi_j, A(t)\Pi_P(J_{c\ell})[B,\chi_0(P)]\varphi_j\rangle= \langle \varphi_j, A(t)\chi_1(P)[B,\chi_0(P)]\varphi_j\rangle + \nonumber\\ \langle \varphi_j, A(t)\Pi_P(J_{c\ell}[[B,\chi_0(P)],\chi_1]\varphi_j\rangle \edqn Using standard rules for the $\hbar$ admissible calculus \cite{ro} and property of supports for $\chi_0$ and $\chi_1$ we have $\chi_1(P)[B,\chi_0(P)]=O(\hbar^\infty)$ in the operator norm. Furthermore, thanks to the double commutator, the last term in (\ref{e62}) is $O(\hbar^2)$. Clearly, the procedure can be iterated and at the step N we get: \beq \langle \varphi_j, A(t)\Pi_P(J_{c\ell})B\varphi_j\rangle = \langle \varphi_j, A(t)\chi_0(P)B\varphi_j\rangle + O(\hbar^N) \edq \QED Let us introduce the classical correlation: $$C_{a,b}(E,t) = \int_{\Sigma_E}\bar{b}(z)a\left(\Phi^t(z)\right)d\sigma_E(z)$$ By applying \cite{hemaro} we can get: \begin{theorem}\label{smr} Under the assumptions $(H_1)$ to $(H_7)$, for every real $T>0$ and every integer $\ell$ we have: \beq\label{e63} \frac{d^\ell}{dt^\ell}\hat{{\cal R}}_{A,B;j,\hbar}(t) = \frac{d^\ell}{dt^\ell}C_{a,b}(E,t) + o(\hbar),\;{\rm as}\;\hbar\searrow 0\; {\rm for}\; j\in M(\hbar)\; {\rm and \; uniformly \;in} t\in[-T, T] \edq \end{theorem} {\bf Proof:}\\ We have only to check the uniformity in $t\in [-T, T]$ . It is sufficient to consider the case $\ell=0$. The conclusion comes easily from the following elementary lemma whose proof is an exercise about application of compactness by an $\varepsilon/3$-argument! \begin{lemma} Let us consider a family of probability measures $(\mu_{\hbar,j})_{j\in M(\hbar)}$ on $\R^m$, weakly convergent to some probability measure $\mu$ as $\hbar \searrow 0$. Let us consider a continuous mapping: $t \mapsto g_t$ from $[-T, T]$ into the Banach space ${\cal C}_b(\R^m)$ (bounded and continuous functions on $\R^m$ with the supremum norm). Then we have: $$\lim_{[\hbar \searrow 0,\;j\in M(\hbar)]}\int_{R^m} g_t(z)d\mu_{\hbar,j}(z) = \int_{\R^m}g_t(z)d\mu(z),\;\; {\rm uniformly\;in}\; t \in [-T, T]$$ \end{lemma} \begin{remark} The semi-classical sum rules \cite{fepe,prro} concerning $$\sum_{E_k\in J_{c\ell}}\left(\frac{(E_k(\hbar)-E_j(\hbar)}{\hbar}\right)^\ell \vert A_{jk}(\hbar)\vert^2$$ are obvious consequences of (\ref{e63}) by computing $\frac{d^\ell}{dt^\ell}C_{a,b}(E,t)$. For example we have: \beqn \frac{d}{dt}C_{a,b}(E,t)=\int_{\Sigma_E} \bar{b}(z)\{a,p_0\}(\Phi^t(z))d\sigma_E(z)\\ \frac{d^2}{dt^2}C_{a,b}(E,t)=-\int_\Sigma\overline{\{b,p_0\}}(z).\{a,p_0\}(\Phi^t(z))d\sigma_E(z) \edqn where $\{a,p_0\}$ denotes the classical Poisson bracket. \end{remark} \begin{remark} In \cite{robe1} the authors consider the quantum correlation function: $$Q_j(t,\hbar) := \Re\left\{ \frac{1}{2i}[\hat{{\cal R}}_{A,B;j,\hbar}(t)-\hat{{\cal R}}_{A,B;j,\hbar}(-t)]\right\}$$ We have also: $$Q_j(t,\hbar) = \sum A_{jk}B_{jk}\sin(\omega_{jk}(\hbar)t)$$ They remark that all the moments: $\int t^r Q_j(t,\hbar)dt=0,\;\forall r\in \N$. These integrals indeed exist as generalized integrals. It is well known that we have: $$\int_0^{+\infty}t^r{\rm e}^{it\omega}dt=i^{-r}\frac{d^r}{d\omega^r} \left(\pi\delta(\omega)+iPV(\frac{1}{\omega})\right)$$ where $PV$ is the Cauchy principal value.\\ By considering the cases $r$ even and $r$ odd we get easily that, for $\omega\neq 0$, $\int_{-\infty}^{+\infty} t^r.\sin(\omega t)dt =0$. If the classical system is mixing in the classical limit, applying the theorem (\ref{smr}), we have: \beq \lim_{[\hbar\searrow 0,\;j\in M(\hbar)]}Q_j(t,\hbar)=\int_{\Sigma_E}(a\left(\Phi^t(z)\right) b(z) -b\left(\Phi^t(z)\right) a(z))d\sigma_E(z):=\Gamma_{a,b}(t,E)\\ \edq $\Gamma_{a,b}(.,E)$ is an odd function. If furthermore we have exponential decay for the correlations as in \cite{po,ru} then the Fourier transform of $\Gamma_{a,b}(.,E)$ is analytic in a complex neighborhood of the real axis. So if for every $r\in\N$ we have $\int t^r \Gamma_{a,b}(t,E)dt = 0$ then $\Gamma_{a,b}(.,E)\equiv 0$. So, as claimed in \cite{robe1}, if $\Gamma_{a,b}(t,E)\neq 0$ for some time $t$, this shows that for very large times quantum and classical evolutions are very different. \end{remark} %\documentstyle[12pt]{article} %\pagestyle{plain } %\addtolength{\textheight}{3cm} %\addtolength{\textwidth}{3cm} %\addtolength{\oddsidemargin}{-1,5cm} %\addtolength{\evensidemargin}{-1cm} %\addtolength{\topmargin}{-1,5cm} %\input{new.def} %\begin{document} \appendix \section{Families of energy transitions with a classical limit} The aim of this section is to constuct examples of non diagonal matrix elements satisfying assumptions $(\alpha), (\beta), (\gamma)$ of Theorems (3.5) and (3.7). We assume that the quantum Hamiltonian $P(\hbar)$ satisfies the hypotheses $(H_1)$ to $(H_7)$ . Let us recall the following notations and assumptions: \begin{eqnarray*} I(\hbar) &=& [\alpha(\hbar), \beta(\hbar)],\;{ \rm with}\; \alpha(\hbar)0 \\ \Lambda(\hbar) & = & \{j,\; E_j(\hbar) \in I(\hbar)\}; \; \omega_{jk}(\hbar)= \left(\frac{E_k(\hbar)-E_j(\hbar)}{\hbar}\right) \\ \vert \Sigma_E \vert & = & \int_{\Sigma_E}\frac{d\Sigma_E}{\vert \nabla p_0\vert} \end{eqnarray*} We first give estimates as $\hbar\searrow 0$ for the size of the sets: $$\Omega(\tau,\delta,\hbar):=\{(j,k); E_j(\hbar), E_k(\hbar) \in I(\hbar), \vert \omega_{jk}(\hbar)-\tau \vert \leq \frac{\delta}{2} \}$$ Our main tool will be the following result stated in \cite{hemaro}: \begin{theorem}(see also \cite{pero}). There exist $\gamma >0$, depending only on the fixed energy $E$ and on the classical Hamiltonian $p$ such that: \beqn \forall \varepsilon>0,& & \exists \eta_\varepsilon>0, \exists C_\varepsilon >0,\; {\rm such \;that\;for\;all\; interval}\; I\subseteq ]E-\eta, E+\eta[, \forall \hbar\in ]0, 1], {\rm we \; have}: \nonumber\\ & &\vert \#\{j;E_j(\hbar)\in I \} - (2\pi\hbar)^{-n}Vol_{\R^{2n}}p_0^{-1}(I) \vert \leq \gamma\varepsilon\hbar^{1-n} + C_\varepsilon\hbar^{2-n} \edqn \end{theorem} We first prove the following lemma: \begin{lemma} There exists $c_0 >0$ small enough such that for every $0<\delta< c_0{\varepsilon_2}$ there exists $\hbar_\delta>0$ such that: \beq \frac{1}{2}(2\pi\hbar)^{-2n}\delta\hbar\lambda(\hbar)\vert \Sigma_E\vert^2 \leq \#\Omega(\tau,\delta,\hbar) \leq \frac{3}{2}(2\pi\hbar)^{-2n}\delta\hbar\lambda(\hbar)\vert \Sigma_E\vert^2 \edq under the conditions: $$0\leq \hbar\leq \hbar_\delta,\; (\vert \tau \vert +\frac{\delta}{2})\hbar\leq \frac{4}{5}\lambda(\hbar)$$ \end{lemma} {\bf Proof:} We will establish the lower bound only and for $\tau> 0$. The other cases can be checked in the same way. We have: \beqn \#\Omega(\tau,\delta,\hbar) \geq& &\nonumber \\ \#\{(j,k); E_j(\hbar)\in I(\hbar)\cap [\alpha(\hbar)-(\tau-\frac{\delta}{2})\hbar,& & \beta(\hbar)-(\tau +\frac{\delta}{2})\hbar], \vert \omega_{jk}(\hbar)-\tau\vert \leq \frac{\delta}{2} \} \edqn In what follows $\gamma$ is a "generic constant" independent of $\hbar$ and $\varepsilon$.\\ Using Theorem (A1) we have the following estimate: \beq\label{eqA1} \#\{k; \vert \omega_{jk}(\hbar)-\tau\vert \leq \frac{\delta}{2}\} \geq (2\pi\hbar)^{-n}(\delta\hbar\vert \Sigma_E\vert - \gamma\delta\hbar\lambda(\hbar) -\gamma\varepsilon\hbar) \edq under the conditions: $$\hbar\in ]0, \hbar_\varepsilon],\; E_j(\hbar) \in I(\hbar)\cap [\alpha(\hbar)-(\tau-\frac{\delta}{2})\hbar, \beta(\hbar)-(\tau +\frac{\delta}{2})\hbar]$$ where we have used that under the above conditions we have, by the fondamental theorem on calculus: $$Vol_{\R^{2n}}(p_0^{-1}[E_j(\hbar)+(\tau-\frac{\delta}{2})\hbar, E_j(\hbar)+(\tau+\frac{\delta}{2})\hbar] \geq\delta\hbar\vert\Sigma_E\vert - \gamma\delta\hbar\lambda(\hbar)$$ In the same way we have also: \beq\label{eqA2} \#\{j; E_j(\hbar)\in [\alpha(\hbar), \beta(\hbar)-(\tau+\frac{\delta}{2})\hbar]\} \geq (2\pi\hbar)^{-n}(\lambda(\hbar)\vert\Sigma_E\vert - (\tau+\frac{\delta}{2})\hbar \vert\Sigma_E\vert - \gamma\lambda(\hbar)^2) \edq Now putting together (\ref{eqA1}) and (\ref{eqA2}) and choosing $\varepsilon=\frac{\delta}{\Gamma}$ with $\Gamma$ large enough, we get the lower bound: \beq \#\Omega(\tau,\delta,\hbar) \geq \frac{1}{2}(2\pi\hbar)^{-2n}\delta\hbar\lambda(\hbar)\vert \Sigma_E\vert^2 \edq under the conditions of the lemma. \QED By using the same technique as above (being a little bit more accurate in the estimates) it is not difficult to prove the following asymptotic result: \begin{proposition} Let us assume that $(H_1)$ to $(H_7)$ are fulfilled and furthermore that we have: $\di{\lim_{\hbar\searrow 0}\frac{\hbar}{\lambda(\hbar)}=0}$. Then, for every $\tau\in \R$ and every $\delta >0$ we have for $\hbar\searrow 0$: \beq \#\Omega(\tau,\delta,\hbar) = (2\pi\hbar)^{-2n}\delta\hbar\lambda(\hbar)\vert \Sigma_E\vert^2 + o(\hbar^{1-2n}\lambda(\hbar)) \edq \end{proposition} Now we come to the main goal of this section. Let us choose a large enough integer $N_0$ and two real numbers $C_1,\; C_2$ such that $\frac{C_1}{C_2} >4$. Let us introduce, for $N \geq N_0$: $$\Omega_N(\hbar):=\{(j,k); E_j(\hbar), E_k(\hbar)\in I(\hbar),\; \frac{C_2}{N}\leq \vert \omega_{jk}(\hbar)-\tau\vert \leq \frac{C_1}{N} \}$$ Then, using the Lemma (A2) with $\delta=\frac{C_i}{N}$, we get that there exists $C_3>0$ such that for every $N\geq N_0$ there exists $\hbar_N$ such that for all $\hbar\in ]0, \hbar_N]$ we have: \beq\label{eqA3} \#\Omega_N(\hbar) \geq \frac{C_3}{N}\hbar^{1-2n}\lambda(\hbar) \edq Now, choose a decreasing sequence $\tilde{\hbar}_N >0$, such that $\di{\lim_{N\rightarrow \infty}\tilde{\hbar}_N = 0}$ and $$\tilde{\hbar}_N \leq \min\left\{ \hbar_N, N^{\frac{1}{1-n}}\right\}$$ >From (\ref{eqA3}), for all $\hbar\in ]0,\tilde{\hbar}_N]$ we have \beq\label{eqA4} \#\Omega_N(\hbar) \geq C_3\hbar^{-n}\lambda(\hbar) \edq Let us define: $$\Omega(\hbar):= \Omega_N(\hbar) \;\;{\rm if}\; \tilde{\hbar}_ {N+1}<\hbar\leq \tilde{\hbar}_N$$ Clearly $\Omega(\hbar)$ satisfies the assumptions $(\alpha)$ and $(\beta)$. To check the assumption $(\gamma)$ we remark that from Theorem (A1) we get easily the asymptotic formula: \beq\label{eqA5} \lim_{\hbar\searrow 0}\frac{\#\Lambda(\hbar)}{\hbar^{-n}\lambda(\hbar)}= (2\pi)^{-n}\vert\Sigma_E \vert \edq \QED \section{Smooth unbounded observables} We want here to give a rigorous meaning to the matrix elements $A_{jk}(\hbar)$ for $A(\hbar):=op^w_\hbar(a)$ not necessarry bounded. The answer follows easily from the following lemma: \begin{lemma} Let us assume the hypotheses $(H_1)$ to $(H_5)$ for the Hamiltonian $P(\hbar)$. Let $\chi_0\in C_0^\infty(I_{c\ell})$ and $a\in C^\infty(\R^{2n})$ satisfying the following condition: \beq [S(m)]\; \exists C>0,\;\exists m\in \N \;{\rm such\; that \;for} \vert\alpha\vert+\vert\beta\vert \geq m \;{\rm we\;have}\; \vert\partial^\alpha_x\partial^\beta_\xi a(x,\xi)\vert \leq C \edq Then there exists $\hbar_0>0$ small enough such that for every $\hbar\in ]0,\hbar_0]$, the operator $A(\hbar)\chi_0(P(\hbar))$ is bounded on $L^2(\R^n)$. \end{lemma} {\bf Proof:} We first prove the result for $m=1$. Let us consider $\chi\in C_0^\infty(I_{c\ell})$ such that $\chi\equiv 1$ on the support of $\chi_0$. We recall the following result concerning the functional calculus (see \cite{hero1,daro}). \beq\label{eqA5} \chi(P(\hbar))=op^w_\hbar(p_\chi) + \hbar op^w_\hbar(r_\chi(\hbar)) \edq where $p_\chi(x,\xi):=\chi(p_0(x,\xi))$ is a smooth compactly supported symbol, and : $$\vert\partial^\alpha_x\partial^\beta_\xi r_\chi(\hbar,x,\xi)\vert \leq C$$ where $C$ is independent of $\hbar$ and $(x,\xi)$.\\ For every $\varphi \in {\cal S}(\R^n)$ we have: \beqn\label{eqA6} A(\hbar)\chi_0(P(\hbar))\varphi=A(\hbar)\chi(P(\hbar))\chi_0(P(\hbar))\varphi=\\ \nonumber A(\hbar)op^w_\hbar(p_\chi)\chi_0(P(\hbar))\varphi + \hbar A(\hbar)op^w_\hbar(r_\chi(\hbar))\chi_0(P(\hbar))\varphi \edqn Let introduce a commutator: \beq\label{eqA7} A(\hbar)op^w_\hbar(r_\chi(\hbar))=op^w_\hbar(r_\chi(\hbar))A(\hbar) + [A(\hbar), op^w_\hbar(r_\chi(\hbar))] \edq Now, using the rule on the symbolic calculus for Weyl quantization (see \cite{ro,daro}), we get: \beq\label{eqA8} \hbar^{-1}[A(\hbar), op^w_\hbar(r_\chi(\hbar))]=op^w_\hbar (b(\hbar)) \edq (let us remark that the "first term" in the $\hbar$-expansion of $b(\hbar)$ is the Poisson bracket $\{a, r_\chi(\hbar)\}$) We have: $$\vert\partial^\alpha_x\partial^\beta_\xi b(\hbar, x,\xi)\vert \leq C$$ hence, using the Calderon-Vaillancourt theorem, we get for some constants $C_1,\;C_2$ independent of $\hbar$ and $\varphi$: \beq\label{eqA9} \Vert A(\hbar)\chi_0(P(\hbar))\varphi\Vert \leq C_1\Vert \varphi\Vert + C_2 \hbar\Vert A(\hbar)\chi_0(P(\hbar))\varphi\Vert \edq Then we get the result under the condition $C_2 \hbar \leq \frac{1}{2}$ on $\hbar$.\\ We can extend now the result for $m\geq 2$ by an induction argument. Indeed, if $a$ satisfies $[S(m)]$ then the symbol $b(\hbar)$ defined in (\ref{eqA8}) satisfies $[S(m-1)]$. So the induction is clear. \section{Non smooth observables} It is convenient here to use the anti-Wick quantization which can be defined in the following way: let us intoduce the fondamental normalized bound state of the harmonic oscillator: $$\Psi_\hbar(x):=(\pi\hbar)^{-n/4}{\rm e}^{-x^2/2\hbar}$$ The coherent states centered at the point $(y,\eta)\in \R^{2n}$ is defined by: $$\Psi_{\hbar,y,\eta}(x):= \left(\exp\frac{i}{\hbar}(\eta.x - y.P).\Psi_\hbar\right)(x)$$ where we denote : $P:=\frac{\hbar}{i}\nabla_x$ Then the anti-Wick quantization of a classical observable $a$ is given by: $$op^{AW}_\hbar(a)\varphi=(2\pi\hbar)^{-n}\int\int_{\R^{2n}}a(y,\eta) \langle \varphi, \Psi_{\hbar,y,\eta}\rangle \Psi_{\hbar,y,\eta}dyd\eta$$ we have the three following useful properties (see \cite{hemaro}):\\ (AW1) $a\geq 0 \Rightarrow op^{AW}_\hbar(a) \geq 0$\\ (AW2) $op^{AW}_\hbar(a)$ admits an $\hbar$-Weyl symbol $a_w(\hbar)$ given by: $$a_w(\hbar,x,\xi)=(\pi\hbar)^{-n}\int\int_{\R^{2n}}a(y,\eta)\exp(-\frac{1}{\hbar} [(x-y)^2+(\xi-\eta)^2])dyd\eta$$ (AW3) For every $a\in {\cal B}_\infty, \Vert op^{AW}_\hbar(a)-op^w_\hbar(a) \Vert = O(\hbar)$ as $\hbar\searrow 0$.\\ To state results we need a mild smoothness assumption: we say that a Borel function $a$ on $\R^{2n}$ satisfies the condition ($S$) if the following property holds: $$\forall \varepsilon>0, \exists a_1;\; a_2, \;{\rm continuous\; on} \R^{2n}\; {\rm such\;that:} a_1 \leq a \leq a_2, \; {\rm and}\; \int (a_2-a_1)d\sigma_E \leq \varepsilon$$ Let us remark that if $a$ is the caracteristic function of an open set $U$ then the condition ($S$) means that the boundary of $U$ is $d\sigma_E$-negligible. \begin{proposition} The theorems (3.5) and (3.7) (i) can be extended to any quantum observable $A(\hbar)=op^{AW}_\hbar(a)$ with any bounded Borel function $a$ satisfying ($S$) on the phase space $\R^{2n}$ and for $A(\hbar)= op^w_\hbar(a)$ with $a$ a bounded Borel function satisfying ($S$) and depending only on position variables or only on momentum variables.\\ More precisely we have:\\ (I) Under the conditions $(H_1)$ to $(H_8)$ for every $\varepsilon > 0$ there exists $T_\varepsilon > 0$ and $\hbar_\varepsilon > 0$ such that: \beq \forall (j,k)\in M(\hbar)\times M(\hbar),\; 0 < \hbar\leq \hbar_\varepsilon;\; \vert E_j(\hbar)-E_k(\hbar) \vert \leq \frac{\pi\hbar}{2T_\varepsilon} \Rightarrow \vert A_{jk}(\hbar)\vert \leq \varepsilon \edq (II) Let us assume $(H_1)$ to $(H_9)$. Then we have: \\ \beq \lim_{[\hbar\searrow, 0\;(j,k)\in M(\hbar)\times M(\hbar),\;j\neq k]}A_{jk}(\hbar) =0 \edq (for the moment we are not able to extend part $(ii)$ of the theorems for discontinuous classical observables!) \end{proposition} {\bf Proof:} Let us consider, for example, Theorem (3.7) $(i)$. Using the property (AW3), the conclusions of Theorem (3.7) $(i),\;(ii)$ hold for $A(\hbar)=op^{AW}_\hbar(a)$ with $a$ continuous and bounded on $\R^{2n}$. \\ It is sufficient to consider only real valued observables. We have the elementary identity: \beqn \langle A\varphi_j,\varphi_k\rangle &=& \\ \nonumber \frac{1}{4} (\langle A(\varphi_j+\varphi_k), \varphi_j+\varphi_k\rangle &-& \langle A(\varphi_j-\varphi_k), \varphi_j-\varphi_k\rangle) \edqn which gives an explicit decomposition of the measures $d\mu_{jk}$ into its positive and negative part: $$\int ad\nu_{jk}^{\pm}:=\langle op^{AW}_\hbar(a)(\varphi_j\pm\varphi_k), \varphi_j\pm\varphi_k\rangle$$ Hence we have clearly, under the condition $(j,k) \in M(\hbar)\times M(\hbar)$ and $\hbar \searrow 0$: \beq\label{eqA9} \{\int ad\mu_{jk}\rightarrow 0\} \Leftrightarrow \{ \int ad\nu_{jk}^{\pm}\rightarrow 2_E\} \edq $a$ satisfies ($S$), so for any $\varepsilon >0$ we can find $a_1, a_2 \in {\cal B}_\infty$ such that: \beq\label{eqA10} a_1 \leq a \leq a_2, \; {\rm and}\; \int (a_2-a_1)d\sigma_E \leq \varepsilon \edq So we have: \beq\label{eqA11} \int a_1d\nu_{jk}^{\pm} \leq \int ad\nu_{jk}^{\pm} \leq \int a_2d\nu_{jk}^{\pm} \edq But we know that for $i=1, 2$: \beq\label{eqA12} \int a_id\nu_{jk}^{\pm} \rightarrow 2\int a_id\sigma_E \;\;{\rm as}\; \hbar\searrow 0,\;(j,k)\in M(\hbar)\times M(\hbar) \edq Hence from (\ref{eqA10}, \ref{eqA11}, \ref{eqA12}) we get: $$\int ad\nu_{jk}^{\pm} \rightarrow 2\int ad\sigma_E \;\;{\rm as}\; \hbar\searrow 0,\;(j,k)\in M(\hbar)\times M(\hbar)$$ For $a(x,\xi)= f(x)$ or $a(x,\xi)=g(\xi)$, with the Weyl quantization, we use the same positivity arguments by approximating below and above $f$ and $g$ by smooth functions. \QED \begin{thebibliography}{60} \bibitem{sh} Shnirelman. A.I: Ergodic properties of eigenfunctions. Upehi Math.nauk,29, n.6, 181-182, 1974 \bibitem{co} Colin de Verdi\ere. Y: Ergodicit\'e et fonctions propres du Laplacien. CMP 102, 497-502, 1985 \bibitem{ze1} Zelditch. S: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math.J. 55, 919-941, 1987 \bibitem{ze2} Zelditch. S: Quantum transition amplititude for ergodic and for completely integrable systems. J.of Funct.Anal. 94, 2, 415-436, 1990 \bibitem{hemaro} Helffer. B, Martinez. A, Robert. D: Ergodicit\'e et limite semi-classique. CMP. 109, 313-326, 1987 \bibitem{ro} Robert. D: Autour de l'approximation semi-classique. Bikha\"user. PM 68. 1987 \bibitem{pero} Petkov. V, Robert. D: Asymptotique semi-classique du spectre d'hamiltoniens quantiques et trajectoires classiques p\'eriodiques. Com. in PDE 10(4), 365-390, 1985 \bibitem{iv}Ivrii. V: Semi-classical analysis and precise spectral asymptotics. Preprints 1,2,3. Ecole Polytechnique 1990/91. \bibitem{wi}Wilkinson. M: A semi-classical sum rule for matrix elements of classically chaotic systems. J.Phys.A: Math. Gen. 20 2415-2423, 1987 \bibitem{he} Helffer. B: Remarks on recent results in semi-classical analysis. publ. of Technische Universitat. Berlin. 1991 \bibitem{vo}Voros. A: D\'eveloppements semi-classiques. Th\`ese de doctorat de Paris-Orsay. 1977 \bibitem{me}Meinrenken. E: Semi-classical principal symbols and Gutzwiller's trace formula Publ. Univ; of Freiburg, Nov. 1991 \bibitem{gu}Gutzwiller. M: Periodic orbits and classical quantization conditions J. of Math. Phys.12, n.3, 343-358, 1971 \bibitem{guur}Guillemin. V, Uribe. A: Circular symmetry and the trace formula. Invent. Math. 96, 385-423, 1989 \bibitem{brur}Brummelhuis. R, Uribe. A: A trace formula for Schr\"odinger operators. Comm.Math. Phys. 136, 567-584, 1991. \bibitem{paur}Paul. T, Uribe. A: Sur la formule semi-classique des traces. Note CRAS 313 I, 217-222, 1991. \bibitem{fepe} Feingold. A, Peres; A: Distribution of matrix elements of chaotic systems Physical Review A, Vol.34, n1, 591-595, 1986 \bibitem{pe} Pechukas. P: Distribution of energy eigenvalues in the irregula spectrum Physical Review Letters, Vol. 51, n11, 943-950, 1983 \bibitem{robe1} Robbins. JM, Berry. MV: Discordance between quantum and classical correlation moments for chaotic systems. J. Phys. A:Math. Gene. 25; L961-L965, 1992. \bibitem{robe2} Robbins. JM, Berry. MV: The geometric phase for chaotic systems. Proc. R. Soc. Lond. A 436, 631-661, 1992 \bibitem{be} Berry. MV: Classical adiabatic angles and quantal adiabatic phase. J. Phys. A.18, 15-27, 1985. \bibitem{prro} Prosen. T, Robnik. M: Distribution and fluctuations of transition probabilities in a system between integrability and chaos. J. Phys. A:Math. Gene. 26 L319-L326, 1993 \bibitem{mcka}McDonald. S, Kaufman. A: Spectrum and eigenfunctions for a Hamiltonian with stochastic trajectories. Physical Review Letters, Vol.42, n18, 1189-1991, 1979 \bibitem{mapr} Marshalek. E, da Provid\^encia. J: Sum rules, Random-Phase-Approximations, and Constraint Self-Consistent Fields. Physical Review C Vol.7,n6, 2281-2293, 1973 \bibitem{doli} Donnay. V, Liverani. C: Potentials on the two-torus for which the Hamiltonian flow is ergodic. CMP, 135, 267-302, 1991 \bibitem{po} Pollicott. M: On the rate of mixing of axiom A flows. Invent. Math. 81, 413-426, 1985 \bibitem{ru}Ruelle. D: Resonances of chaotic dynamical systems. Physical Review Letters Vol.56, n5, 405-407, 1986 \bibitem{hero1}Helffer. B, Robert. D: Calcul fonctionnel par la transform\'ee de Mellin. J. of Funct.Anal. Vol.53 n3, 246-268, 1983 \bibitem{hero2}Helffer. B, Robert. D: Propri\'et\'es asymptotiques du spectre d'op\'erateurs pseudodiff\'erentiels sur $\R^n$. Comm. in PDE, 7(7), 795-882, 1982 \bibitem{ch}Chirikov. B.V: Time-dependant quantum systems. in "Chaos and quantum physics, Les Houches, 1989" p.518-519. North-Holland, 1991. \bibitem{dugu}Duistermaat. J.J, Guillemin. V: The spectrum of positive elliptic operators and periodic bicharacteristics. Inv. Math. 29. 39-79, 1975 \bibitem{daro}Dauge. M, Robert. D. Weyl's formula for a class of pseudodifferential operators with negative order on $L^2(\R^n)$. Lecture Notes in Math, n.1256, Springer-Verlag, 1986. \bibitem{hel} Helton. J.W: An operator algebra approach to partial differential equations. Indiana. Univ. Math. J. Vol . 26, n6, 997-1018, 1977 \bibitem{gui} Guillemin. V: Lectures on spectral theory of elliptic operators. Duke Math. J. 485-517, 1977. \bibitem{do} Dozias. S: Thesis to appear, DMI.ENS. Paris \end{thebibliography} \end{document}