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Egorov Theorem, semiclassical analysis, Heisenberg observable, long time propagation.
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\begin{document} 
\begin{center} 
{\bf \Large Uniform Semi-classical Estimates for the Propagation\\ of Quantum Observables} 
\vskip0.8cm 
A. Bouzouina\footnote{Permanent address~: D\'epartement de Math\'ematiques, 
Universit\'e de Reims, BP 1039,\\ F-51687 Reims Cedex 2.} 
\vskip0.2cm 
{\em School of Mathematics, University of Cardiff (Wales)\\ 
23, Senghennydd Road, 
CF2 4YH Cardiff, Great Britain.\\ 
E-mail~: Abdelkader.Bouzouina@univ-reims.fr} 
\vskip0.5cm 
D. Robert 
\vskip0.2cm 
CNRS, UMR 6629.\\ 
{\em D\'epartement de Math\'ematiques, 
Universit\'e de Nantes,\\ 
2, rue de la Houssini\`ere, 44322 Nantes cedex 03, France.\\ 
E-mail~: robert@math.univ-nantes.fr} 
\end{center} 
\vskip0.8cm 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%% ABSTRACT %%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{abstract} 
We prove here that the semi-classical asymptotic expansion 
for the propagation of quantum observables, 
for $C^\infty$-Hamiltonians growing at most 
quadratically at infinity, is uniformly dominated at any order, by an 
exponential term who's argument is linear in time. In 
particular, we recover the Ehrenfest time for the validity of the semi-classical 
approximation. This extends the result proved in \cite{bgp}. 
Furthermore, if the Hamiltonian and the initial observables are holomorphic in a complex 
neighborhood of the phase space, we prove that the quantum observable is 
an analytic semi-classical observable. Other results about the large time behavior of 
observables with emphasis on the classical dynamic are also given. 
In particular, precise Gevrey estimates are established for classically integrable systems. 
\end{abstract} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%% INTRODUCTION-RESULTS (section 1) %%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{Introduction and main results} 
According to the Bohr's correspondance principle, the quantum evolution of an observable 
is close to its classical evolution as the Planck constant $\hbar$ is small. In the mathematical 
literature this result is known as Egorov Theorem \cite{ho, ro}. But usually this result 
is proved for finite time and the asymptotics hold in the $C^\infty$ (Poincar\'e) sense. Here 
we put emphasis on large time behavior of asymptotic expansions in $\hbar$, 
in the $C^\infty$ case as well as in the analytic (or Gevrey) case. In the book \cite{iv} 
(section 2.3, Propagation of singularities along long bicharacteristics) one can find related 
results for the $C^\infty$ case. But our results are more explicite. 
Many years ago, physicists conjectured that semi-classical approximation is still valid 
in large time interval with length $T(\hbar) \approx \log(\hbar^{-1})$ which is called 
Erhenfest time \cite{ch, za}. That kind of result was rigorously proved for propagation 
of coherent states in \cite{cr} (and in \cite{hj} in the analytic case). In \cite{bgp} 
the authors have established long time estimates for the propagation of quantum 
observable. The main goal of this paper is to prove more accurate estimates 
in $\hbar$ and time. In particular, if the data are holomorphic in a complex neighborhood 
of the real phase space, then the semi-classical expansion of the time dependent quantum 
observable holds with an exponentially small remainder term, for time interval with length 
of order $\log(\hbar^{-1})$. 
Let us denote by $X = \R^n$ the configuration space of a classical mechanical system 
with $n$ degrees of freedom. The corresponding phase space $Z$ is identified with $\R^{2n}$ equipped 
with the symplectic form $\sigma$ 
defined by 
\b 
\sigma(z;z^\prime) = \langle Jz, z^\prime \rangle.\label{sympform} 
\e 
where $\langle,\; \rangle$ is the Euclidean scalar product and $J$ is the $2n\times 2n$ matrix 
\b 
J = \pmatrix{0 &\quad \Id \cr 
-\Id &\quad 0 \cr}. 
\e 
A generic point in $Z$ is denoted $z$ and its coordinates by $(q, p)$ where $q, p \in \R^n$.\\ 
A classical Hamiltonian is a smooth real function $H:Z\rightarrow \R$. Our basic 
example will be $H(q,p) = \frac{\Vert p\Vert^2}{2m} + V(q)$ ($m>0$) 
where $\Vert p\Vert^2 = <p,p>$. We could also consider a Riemannian manifold $(X,g)$. 
In what follows we put emphasis on the case $X = \R^n$.\\ 
The motion of the classical system is determined by the system of Hamilton's equations 
\b\label{ham} 
\frac{dq}{dt} = \frac{\partial H}{\partial p}(q,p),\;\; 
\frac{dp}{dt} = -\frac{\partial H}{\partial q}(q,p). 
\e 
The equations (\ref{ham}) generate a flow $\Phi^t$ on the phase space $Z$, defined by \\ 
$\Phi^t(q(0),p(0)) = (q(t),p(t))$; $\Phi^0 = \Id$. $\Phi^t$ exists locally 
by the Cauchy-Lipchitz Theorem for O.D.E. But we need more assumptions 
on $H$ to define $\Phi^t$ globally on $Z$. $\Phi^t$ defines a symplectic 
diffeomorphism (canonical transformation) group of transformations on $Z$. 
Let us consider a classical observable $A$, i.e $A$ a smooth complex valued function 
defined on phase space $Z$. 
The time evolution of $A$ can be easily computed 
\b\label{cl.prog} 
\frac{d}{dt}A(\Phi^t(z)) = \{H, A\}(\Phi^t(z)),\;\; z=(q,p) 
\e 
where $\{H, A\}$ is the Poisson bracket defined by 
\b 
\{H, A\} = \partial_qH\cdot\partial_pA - \partial_pH\cdot\partial_qA.\label{poisson} 
\e 
Here we have used the notation $\partial_q = \frac{\partial}{\partial q}$. 
Now let us assume that $H, A$ are quantizable. That means that we can associate to them 
the quantum observables $\hat{H}$ and $\hat{A}$ i.e self-adjoint operators in 
$L^2(X)$. By solving formally the Schr\"odinger equation~: 
$i\hbar\partial_t\psi_t = \hat{H}\psi_t$, we can define the one parameter group 
of unitary operators $U(t) = \exp\left(-\frac{it}{\hbar}\hat{H}\right)$. 
The quantum time evolution of $\hat{A}$ 
is then given by $\hat{A}(t) = U(-t)\hat{A}U(t)$ which satisfies 
the Heisenberg-von Neumann equation 
\b\label{he.vn} 
\frac{d\hat{A}(t)}{dt} = \frac{i}{\hbar}[\hat{H}, \hat{A}], 
\e 
where $[K, B] = KB - BK$ is the commutator of $K, B$. 
We shall use here the $\hbar$-Weyl quantization defined for $A\in{\cal S}(Z)$ (the space of Schwartz functions) 
by the following formula, 
with $\psi\in{\cal S}(X)$, 
\b\label{we} 
{\hat A}\psi(x) = (2\pi \hbar)^{-n}\int\!\!\int_Z A\left(\frac{x+y}{2}, p\right) 
{\rm e}^{i\hbar^{-1}\langle x-y, p\rangle} \psi(y)dydp. 
\e 
Let us now introduce a more general set of classical observables for which the 
$\hbar$-Weyl quantization is well defined and have nice properties \cite{ho, ro}. 
\begin{definition} 
(i) $A\in {\cal O}(m)$, $m\in\R$, if and only if $Z\stackrel{A}{\rightarrow}\C$ 
is $C^\infty$ in $Z$ and for every multi-index $\gamma\in\N^{2n}$ there exists $C>0$ such that 
$$ 
\vert \partial^\gamma_z A(z)\vert 
\leq C\langle z\rangle^m,\;\forall z\in Z. 
$$ 
(ii) We say that $A$ is a $C^\infty$-semi-classical observable of weight $m$ if 
there exists $\hbar_0 >0$ and a sequence $A_j\in {\cal O}(m)$, $j\in\N$, 
so that $A$ is a map from $]0, \hbar_0]$ into ${\cal O}(m)$ satisfying the 
following asymptotic condition~: for every $N\in\N$ and every $\gamma\in\N^{2n}$ 
there exists $C_N>0$ such that for all $\hbar \in ]0, \hbar_0[$ we have 
\b 
\sup_Z\langle z\rangle^{-m}\left\vert \partial^\gamma_z 
\left(A(\hbar,z) - \sum_{0\leq j\leq N}\hbar^jA_j(z)\right)\right\vert 
\leq C_N \hbar^{N+1}.\label{semiobs} 
\e 
$A_0$ is called the principal symbol, $A_1$ the sub-principal symbol of $\hat{A}$. 
The set of semi-classical observables of weight $m$ is denoted by ${\cal O}_{sc}(m)$. 
By the $\hbar$-Weyl quantization, its range 
in ${\cal L}({\cal S}(X))$ is denoted $\widehat{{\cal O}}_{sc}(m)$. 
\end{definition} 
\noindent{\bf Notation :} For any $A$ and $A_j$'s satisfying (\ref{semiobs}), 
we will write : $A(\hbar) \asymp \sum_{j\geq 0}\hbar^jA_j$ in ${\cal O}_{sc}(m)$. 
\vskip5pt 
Let us now recall the statement of the propagation Theorem 
which will be improved in this paper. The microlocal version 
of the result is due to Egorov \cite{eg}. R. Beals \cite{be} found a nice simple proof 
which is reproduced in \cite{ro}. 
\begin{theorem}\label{Propag1} 
Let us consider an Hamiltonian $H$ and an observable $A$ satisfying~:\\ 
\ba\label{asham} 
\vert\partial_z^\gamma H_j(z)\vert \leq C_{\gamma�j},\;{\rm for}\; 
\vert\gamma\vert+j\geq 2 \\ 
\hbar^{-2}(H-H_0-\hbar H_1) \in {\cal O}_{sc}(0), \\ 
\vert\partial_z^\gamma A(z)\vert\leq C_\gamma, \;{\rm for}\; \vert \gamma\vert \geq 1. 
\ea 
Then we have the following properties~:\\ 
(a) For $\hbar$ small enough, $\hat{H}, \hat{A}$ are essentially self-adjoint operators 
in $L^2(X)$, 
with core ${\cal S}(X)$, hence the quantum evolution \hbox {$U(t) = 
\exp(-\frac{it}{\hbar}\hat{H})$} 
is well defined for all $t\in\R$.\\ 
(b) For each $t\in\R$, $\hat{A}(t) = U(-t)\hat{A}U(t)\in \widehat{{\cal O}}_{sc}(1)$. 
Its symbol has an asymptotic expansion, such 
that $A(t) -A\circ\Phi^t \asymp \sum_{j\geq 1}\hbar^jA_j(t)$, holds in 
${\cal O}_{sc}(0)$, uniformly in $t$, for $t$ in a bounded interval. 
Moreover $A_j(t)$ can be 
computed by the following formulas 
\ba 
A_0(t,z) &=& A(\Phi^t(z)),\\ 
A_1(t,z) &=& \int_0^t\{A(\Phi^\tau), H_1\}\Phi^{t-\tau}(z)d\tau \label{eq:term1} 
\ea 
and for $j\geq 2$, by induction, 
\ba\label{termj} 
A_j(t,z) = \sum_{{\vert(\alpha,\beta)\vert+k+\ell=j+1}\atop {0\leq\ell \leq j-1} } 
\Gamma(\alpha,\beta) 
\int_0^t[(\partial_p^\alpha\partial_q^\beta H_k)(\partial_q^\alpha\partial_p^\beta A_{\ell}) 
(\tau)](\Phi^{t-\tau}(z))d\tau, 
\ea 
with 
$$ 
\Gamma(\alpha,\beta) = \frac{(-1)^{\vert\beta\vert} - (-1)^{\vert\alpha\vert}} 
{\alpha!\beta!2^{\vert\alpha\vert+\vert\beta\vert}}i^{-1-\vert(\alpha,\beta)\vert}. 
$$ 
where $\Phi^t$ is the classical flow defined by the principal term $H_0$. 
\end{theorem} 
Since our aim is to improve Theorem \ref{Propag1}, 
let us recall here briefly the method to prove it. 
We admit here that $\hat{A},\hat{H}$ are essentially self-adjoint (for a proof see \cite{ro}). 
Let us remark that, under the assumption on $H_0$ in Theorem \ref{Propag1}, the classical flow $\Phi^t$ 
exists globally in $Z$. Indeed, the Hamiltonian vector field 
$(\partial_\xi H_0, -\partial_x H_0)$ has at most a linear growth 
at infinity and hence no classical trajectory can blow up in a finite time. Moreover, 
using usual methods in non linear O.D.E (variation equation), we can prove 
that for every $\gamma\in\N^{2n}$, $\vert \gamma\vert\geq 1$ 
$\partial^\gamma_z A(\Phi^t)\in {\cal O}(0)$ is uniformly bounded for 
$z\in Z$ and $t$ bounded. 
Now, from the Heisenberg equation and the classical equation of motion, we get 
\b\label{heis} 
\frac{d}{ds}U(-s)\widehat{A_0}(t-s)U(s) = 
U(-s)\left\{\frac{i}{\hbar} 
[\hat{H}, \widehat{A_0}(t-s)] - \widehat{\{{H}, A_0\}}(\Phi^{t-s})\right\}U(s), 
\e 
where $A_0(t) = A(\Phi^t)$. 
But, from the product rule formula (see Appendix), the principal symbol of 
$\frac{i}{\hbar} 
[\hat{H}, \widehat{A}_0(t-s)] - \widehat{\{{H}, A_0\}}(\Phi^{t-s})$ vanishes. So, 
in the first step, we get the error term 
\b 
U(-t)\hat{A}U(t) - \widehat{A_0}(t) = 
\int_0^t U(-s)\left(\frac{i}{\hbar} 
[\hat{H}, \widehat{A_0}(t-s)] - \widehat{\{{H}, A_0\}\Phi^{t-s}}\right)U(s)ds. 
\e 
Now, it is not difficult to obtain, by induction on $j$, the full asymptotics in $\hbar$ 
(see \cite{ro} for details). 
\QED 
\begin{remark} 
If $H=H_0$ is a polynomial function of degree $\leq 2$ on the 
phase space $Z$ then the propagation Theorem has a very simple form~: 
$A(t) = A\circ\Phi^t$ and the remainder term is null ($U(t)$ is a metaplectic 
transformation).\\ 
In \cite{pu2} a semi-classical version of Egorov Theorem on compact manifolds 
is given. 
\end{remark} 
Our first result is an improvement of Theorem \ref{Propag1} by giving estimates 
for large time, in the $C^\infty$-case. 
\begin{theorem}\label{Propag2} Let us assume that the Hamiltonian $H$ 
and the observable $A$ satisfy the assumptions 
of Theorem \ref{Propag1}. Let us introduce an upper bound 
of the stability exponents of the classical system 
$$ 
\Gamma := \sup_{z\in Z}\vert\!\vert\!\vert J\nabla_z^{(2)} H_0(z)\vert\!\vert\!\vert, 
$$ 
where for any observable $f$, $\nabla_z^{(2)}f$ is the corresponding Hessian matrix 
\footnote{Here the norm of a symmetric matrix $M$ 
is defined by $\vert\!\vert\!\vert M\vert\!\vert\!\vert = \sup_{\Vert x\Vert \leq 1}\vert\!<Mx,x>\!\vert$.}. 
Then, for every $j\in \N$ and every multi-index $\gamma$ such that $j+\vert \gamma\vert \geq 1$, 
there exists $C_{j, \gamma} > 0$ such that for every $z\in Z$ and every $t\in \R$ we have 
\b\label{reg1} 
\vert \partial^\gamma_z A_j(t, z)\vert \leq 
C_{j, \gamma}\exp(\Gamma(2j-1+\vert \gamma\vert)\vert t\vert). 
\e 
Furthermore we have the following estimates in $L^2$ operator-norm of the remainder 
term. For every $N \in \N$ there exists $C_N$ such that for every $t\in \R$ we have 
\b\label{res1} 
\Vert \hat{A}(t) - \sum_{0\leq j\leq N}\hbar^j\hat{A}_j(t)\Vert_{L^2} \leq C_N \hbar^{N+1} 
(1+\vert t\vert)^{N+\delta_n}\exp(\Gamma(2N+\delta_n)\vert t\vert), 
\e 
where $\delta_n$ is a universal constant ($\delta_n \leq 5n+3$). 
\end{theorem} 
This result entails the following corollary about the Ehrenfest time 
for the validity of the semi-classical approximation. 
\begin{corollary} Under the assumptions of Theorem \ref{Propag2}, for every $N\geq 1$ there exists $C_N>0$ 
such that for every $\varepsilon >0$ and for 
$\vert t\vert \leq \frac{1-\varepsilon}{2\Gamma}\log(\hbar^{-1})$, 
we have 
\b 
\Vert \hat{A}(t) - \sum_{0\leq j\leq N}\hbar^j\hat{A}_j(t)\Vert_{L^2} 
\leq C_N \hbar^{\varepsilon N+1}\hbar^{\frac{\varepsilon-1}{2}(5n+3)}. 
\e 
In particular the semi-classical asymptotic expansion is valid under the above 
condition on $t$. 
\end{corollary} 
\noindent 
{\bf Remarks in the case of ``classical" energy observables} 
{\em 
We shall see from the proof of Theorem \ref{Propag2} that if the expansion of $H$ in $\hbar$ is even 
(in particular if $H$ is ``classical"~: $H=H_0$) 
then the $\hbar$-expansion of $ A(t)$ is even and the exponential term in (\ref{reg1}) becomes 
$\exp({\Gamma(3j/2-1+\vert \gamma\vert)\vert t\vert})$. In the remainder 
estimate (\ref{res1}) the $2N$ becomes also $3N/2$ so that in this case the Ehrenfest time is not smaller 
than $\frac{2}{3\Gamma}\log\hbar^{-1}$. More precisely, 
for every $\varepsilon >0$ and for 
$\vert t\vert \leq \frac{2-\varepsilon}{3\Gamma}\log(\hbar^{-1})$, 
we have 
\b 
\Vert \hat{A}(t) - \sum_{0\leq j\leq N}\hbar^j\hat{A}_j(t)\Vert_{L^2} 
\leq C_N \hbar^{\frac{\varepsilon}{2}N+1}\hbar^{\frac{\varepsilon-2}{3}(5n+3)}.\label{classical} 
\e 
We remember that some times ago, S. De Bi\`evre suggested that the Ehrenfest time 
could be greater that $\frac{1}{2\Gamma}\log\hbar^{-1}$. The above results confirm 
this guess. In section 5 we shall make more comments about this. 
} 
\vskip5pt 
\begin{remark} 
In many cases the assumptions of Theorem \ref{Propag1} on the Hamiltonian $H$ 
are not satisfied. 
In particular for Schr\"odinger Hamiltonian with non constant magnetic field with vector-potential $a(q)$, 
$ H(q,p) = \Vert p-a(q)\Vert^2 + V(q)$ or for a quartic electric potential 
$ H(q,p) = \Vert p\Vert^2 + \Vert q\Vert^4$ the second derivatives are not always bounded. 
But if the observable $A$ is supported in a flow-invariant compact set $\bar{\Omega}$, 
of the phase space, then, using the functionnal calculus on $\hbar$-pseudodifferential operators (see \cite{ro}) 
we can get the same conclusion as in Theorem \ref{Propag2} with a constant $\Gamma$ defined as follows. 
$$ 
\Gamma := \sup_{\Omega}\vert\!\vert\!\vert J\nabla_z^{(2)} H_0(z)\vert\!\vert\!\vert 
$$ 
where $A$ is such that supp$(A)\subset \Omega$. 
A more precise statement will be given at the end of {\rm section 2} (see Proposition \ref{localization}). \\ 
As we can see from the proofs, the use of the numerical constant $\Gamma$ is to control 
the exponential growth of the flow in time. Of course it is also possible to make directly 
some growth assumptions on the flow or to make some geometrical assumptions on the classical Hamiltonian. 
This will be done in section 4 with periodic systems (see Proposition \ref{periodic}) and and stated in this 
section with integrable systems (see Theorem \ref{gevstab}). Some extensions to manifolds 
are also clearly possible. But in general (compact) manifolds no exact global quantization procedure exists 
so it seems difficult to compute $\hat{A}_j(t)$ for $j\geq 2$ for non classical symbols. 
\end{remark} 
As it is expected, the dependence in $j, \gamma, N$ of the constants in Theorem 
\ref{Propag2} can be specified under analyticity assumptions on $A$ and $H$. Let us first 
recall the following definition (see \cite{bk, sj}). 
\begin{definition} Let $A(\hbar,z)$ be a $C^\infty$-semi-classical observable, 
$A\in {\cal O}_{sc}(0)$. We say that $A(\hbar,z)$ is a semi-classical observable 
with a Gevrey index $\sigma\geq 1$, 
if the following conditions are satisfied, for some constant $C>0$, 
\ba 
\forall \gamma\in\N^{2n},\; \forall j\in\N,\;\forall z\in Z,\;{\rm we\;have}\; 
\vert\partial^\gamma_z A_j(z)\vert \leq C^{\vert \gamma\vert+j+1}(\vert\gamma\vert+j)!^\sigma ;\\ 
\left\vert \partial^\gamma_z\left(A(\hbar, z) - \sum_{0\leq j\leq N}\hbar^jA_j(z)\right)\right\vert 
\leq \hbar^{N+1}C^{\vert \gamma\vert+N+2}(\vert\gamma\vert+N+1)!^\sigma. 
\ea 
As usual, a semi-classical observable with a Gevrey index equal to 1 is said analytic. 
\end{definition} 
\begin{theorem}\label{Propag3} 
Let us assume that the energy Hamiltonian $H$ has an expansion in $\hbar$ 
such that $\hbar^{-2}(H-H_0-\hbar H_1)$ is an analytic, semi-classical observable 
and that it exists some $\delta >0$ such that $H_0, H_1$ 
and the initial observable $A$ are holomorphic in the 
complex domain $D(\delta):= \{z\in \C^{2n}, \Vert \Im z\Vert <\delta \}$. Suppose moreover that 
$\nabla_z^{(2)}H_0$, $\nabla_z H_1$, $\nabla_z A$ are bounded 
and continuous in $D(\delta)$. 
Then, if we define 
$$ 
\Gamma := \sup_{z\in D(\delta)}\vert\!\vert\!\vert J\nabla_z^{(2)} H_0(z)\vert\!\vert\!\vert, 
$$ 
we have the following improvement of Theorem \ref{Propag2}, saying 
that $\hbar^{-1}(A(t,z)-A\circ\Phi^t)$ is an analytic semi-classical observable, 
with control in time. More explicitely, there exist constants $K_0>0, K_1>0$ such that for every 
$t\in\R$, $j\in\N$, $z\in Z$, $N\geq 1$ and $\hbar\in]0, 1]$, 
\b 
\vert \partial^\gamma_z A_j(t, z)\vert \leq 
K_0(j+\vert\gamma\vert)!K_1^{j+\vert \gamma\vert}(1+\vert t\vert)^j\exp(\Gamma(2j-1+\vert \gamma\vert)\vert t\vert); 
\label{eq:analytic1} 
\e 
and the following estimate in $L^2$ operator-norm of the remainder term~: 
\b 
\Vert \hat{A}(t) - \sum_{0\leq j\leq N}\hbar^j\hat{A}_j(t)\Vert_{L^2} \leq \hbar^{N+1} K_0(N+1)!K_1^{N+1} 
\vert t\vert(1+\vert t\vert)^{N+\delta_n}\exp(\Gamma(2N+\delta_n)\vert t\vert).\label{res2} 
\e 
where $\delta_n \leq 5n+3$ is a universal constant. 
\end{theorem} 
\begin{remark} 
As in the $C^\infty$-case, if the expansion of $H$ in $\hbar$ is even then in 
the estimates (\ref{eq:analytic1}) and (\ref{res2}), in the exponential term, the $2N$ becomes 
$3N/2$. This remark is an extension of the remark concerning the ``classical" observables. 
\end{remark} 
We can deduce from Theorem \ref{Propag3} a semi-classical approximation of $A(t)$ 
with an exponentially small error in $\hbar$, for long time intervals. 
\begin{corollary} Under the assumptions of Theorem \ref{Propag3}, for every 
$ \varepsilon > 0, C>0$, such that 
$\varepsilon+2C\Gamma <1$, there exist constant $K>0$, $a>0$, such that 
for $\vert t\vert \leq C\log(\hbar^{-1})$ and $N=[\frac{a}{\hbar^\varepsilon}]$ we have 
\b 
\Vert \hat{A}(t) - \sum_{0\leq j\leq N}\hbar^j\hat{A}_j(t)\Vert_{L^2} 
\leq K\exp\left(-\frac{a}{\hbar^{\varepsilon}}\right). 
\e 
\end{corollary} 
In applications it can be useful to consider observables with support outside some given non empty open set. 
Such observables can not be analytic but can be in some Gevrey class $s > 1$. 
Let us introduce the following technical assumptions~: 
\begin{itemize} 
\item{(C.0)} $H$ is analytic in $D(\delta)$ and there exist $C>0$, $M\geq 0$ such that 
\b 
\vert H(z)\vert \leq C(1+\vert z\vert)^M,\;\; \forall z\in D(\delta). 
\e 
Moreover $\hat{H}$ is supposed to be self-adjoint in $L^2(\R^n)$. 
\item{(C.1)} Let $\Omega$ be a bounded open subset of $Z$ such that $\bar{\Omega}$ is $\Phi^t$-invariant for every $t\in\R$ and $A$ 
a smooth observable such that supp$(A) \subset \Omega$. 
\item{(C.2)} $A$ is Gevrey $s>1$. 
\end{itemize} 
Let us define 
$$ 
\Gamma_\delta := \sup_{\Re z\in \Omega; \Im z <\delta}\vert\!\vert\!\vert J\nabla_z^{(2)} H(z)\vert\!\vert\!\vert. 
$$ 
\begin{theorem}\label{gevinst} Let us assume {\rm (C.0), (C.1), (C.2)}. Then we have the following estimates, 
for some constant $K_0$, $K_1$, 
\b 
\vert \partial^\gamma_z A_j(t, z)\vert \leq 
K_0(3j/2+\vert\gamma\vert)!^s (j/2)!^{-1}K_1^{j+\vert \gamma\vert}\vert t\vert(1+\vert t\vert)^j 
\exp(\Gamma_\delta(3j/2 +\vert \gamma\vert)\vert t\vert) ;\label{eq:gev1} 
\e 
and the following estimate in $L^2$ operator-norm of the remainder 
term~: 
\b 
\Vert \hat{A}(t) - \sum_{0\leq j\leq N}\hbar^j\hat{A}_j(t)\Vert_{L^2} \leq \hbar^{N+1} K_0(N+1)!^{(3s-1)/2}K_1^{N+1} 
\vert t\vert(1+\vert t\vert)^N\exp(\Gamma_\delta(3N/2+\delta_n)\vert t\vert).\label{eq:resgev} 
\e 
where $\delta_n$ is a universal constant. Moreover, as above, we have a long time approximation of $\hat{A}(t)$ 
with exponentially small error estimate~:\\ 
for every 
$ \varepsilon > 0, C>0$, such that 
$\varepsilon (3s-1)/2+3C\Gamma/2 <1$, there exist constants $K>0$, $a>0$, such that 
for $\vert t\vert \leq C\log(\hbar^{-1})$, $N=[\frac{a}{\hbar^\varepsilon}]$ and $\hbar\in]0, 1]$, we have 
\b 
\Vert \hat{A}(t) - \sum_{0\leq j\leq N}\hbar^j\hat{A}_j(t)\Vert_{L^2} 
\leq K\exp\left(-\frac{a}{\hbar^{\varepsilon}}\right). 
\e 
\end{theorem} 
\begin{remark} If in condition (C.1) $\bar\Omega$ in $\Phi^t$-invariant for every $t\geq 0$ (resp $t\leq 0$) then all 
the conclusions of the previous theorem hold for $t\geq 0$ (resp. $t\leq 0$). 
\end{remark} 
The above results are general and do not depend on specific properties of the classical dynamic. 
Nevertheless some important systems in classical and quantum mechanics are integrable and for them 
we can prove much better results~: the semiclassical expansions are valid for time less than some algebraic power 
$\hbar^{-1}$ instead of $O(\log(\hbar^{-1}))$ (which is optimal for instable systems). Let us first introduce the following analytic-integrability condition~: 
\begin{itemize} 
\item{(C.3)} There exists a symplectic map $\chi$ from $\Omega$ into ${\cal A}\times {\bf T}^n$ where ${\cal A}$ 
is an open set in $\R^n$, ${\bf T}^n$ is an n-dimensional torus, such that 
\b 
\chi(\Phi^t(z)) = (I(z), \varphi(z)+t\omega(I(z)),\;\; {\rm where}\; \chi(z) = (I(z), \varphi(z)), \forall z\in \Omega, \forall t\in \R. 
\e 
Moreover there exist complex open neighborhoods $\tilde{\Omega}$, $\tilde{\cal A}$, $\tilde{\bf T}^n$ of $\Omega, {\cal A}, {\bf T}^n$ 
such that $\chi$ is an analytic diffeomorphism from $\tilde{\Omega}$ onto $\tilde{\cal A}\times{\bf \tilde T}^n$. 
\end{itemize} 
\begin{theorem}\label{gevstab} 
Let us assume conditions {\rm (C.0), (C.1), (C.2), (C.3)}. Then we have the following Gevrey type estimates, 
for some constants $K_0, K_1$, 
\b 
\vert \partial^\gamma_z A_j(t, z)\vert \leq 
K_0(3j/2+\vert\gamma\vert)!^s (j/2)!^{-1}K_1^{j+\vert \gamma\vert} 
(1+\vert t\vert)^{\vert \gamma\vert + 2j};\label{eq:gevv} 
\e 
and the following estimate in $L^2$ operator-norm of the remainder term~: 
\b 
\Vert \hat{A}(t) - \sum_{0\leq j\leq N}\hbar^j\hat{A}_j(t)\Vert_{L^2} \leq \hbar^{N+1}K_0(N+1)!^{(3s-1)/2}K_1^{N+1} 
\vert t\vert(1+\vert t\vert)^{2N + \delta_n}\label{resgev} 
\e 
where $\delta_n \leq 5n+3 $ is a universal constant. Moreover as above we have a long time approximation of $\hat{A}(t)$ 
with exponentially small error estimate~:\\ 
for every 
$ \varepsilon > 0$, there exist $K > 0$, $ a > 0$, such that 
for $\vert t\vert \leq C\hbar^{-1/2+\varepsilon}$ and $N = [\frac{a}{\hbar^{3\varepsilon/(2s-1)}}]$ we have 
\b 
\Vert \hat{A}(t) - \sum_{0\leq j\leq N}\hbar^j\hat{A}_j(t)\Vert_{L^2} 
\leq K\exp\left(-\frac{a}{\hbar^{3\varepsilon/(2s-1)}}\right). 
\e 
\end{theorem} 
\begin{remark} 
Many integrable systems satisfy condition {\rm (C.3)}. Let us assume that the system is analytic-integrable in the Liouville sense. 
That means that there exist $n$ real-analytic Hamiltonians $F_1$, ..., $F_n$ in $\Omega$ with $F_1=H$, 
Poisson commuting, $\{F_j, F_k\} =0$. Assume that $F:=(F_1,\cdots, F_n)$ is a regular map 
from $\bar{\Omega}$ into $\R^n$ and that for every $\lambda\in F(\Omega)$, $F^{-1}(\lambda)\cap\bar{\Omega}$ is a 
compact and connected component of $F^{-1}(\lambda)$. 
Then revisiting the proof of Liouville Theorem (Arnold \cite{Ar}) we can prove that {\rm (C.3)} is satisfied. 
\end{remark} 
\begin{remark} Let us recall that only even terms $A_j$ appear in Theorem \ref{gevstab}. 
In the more general case with non null subprincipal 
terms 
($H = H_0 + \hbar H_1$) where $H_1$ is analytic and $H_0$ analytically integrable in $\Omega$ then odd terms contribute and 
in the above estimates we have to replace $(3s-1)/2$ by $2s-1$. In particular the exponential error estimate holds 
for a smaller time $\vert t\vert \leq \hbar^{-1/3 +\varepsilon}$. 
\end{remark} 
\begin{remark} 
Estimates (\ref{res1}) and (\ref{res2}) improve results given in \cite{iv} and in \cite{bgp} in different ways. 
In the $C^\infty$ case, we get results similar to those of \cite{bgp} without analyticity assumptions and 
with a better control for large time. 
In the analytic and Gevrey cases we get better estimates with respect to $N$ and $t$ (see (\ref{res2})). 
Note that in \cite{bgp} the $A_j(t)$'s are not defined through the algorithm (\ref{termj}).\\ 
Similar exponential estimates were recently proved in \cite{hj} for propagation of Gaussian 
coherent states. 
\end{remark} 
\begin{remark} 
The above results still hold true if the Hamiltonian $H$ has singularities (like Coulomb ptentials) 
outside of $\Omega$. This will be comment in section 4. 
\end{remark} 
\begin{remark} 
The above results are still valid for Hamiltonians $H$ which admit singularities (e.g. Coulomb potential) outside 
$\Omega$ (see comments in section 4). 
\end{remark} 
\begin{example} 
Our results apply in particular to the following examples~: 
\begin{enumerate} 
\item $H(q,p)=\Vert p\Vert^2+V(q)$, $A(q,p)=q_1$ with $V$ holomorphic in 
$\{q\in\C^n,\;\Vert\Im q\Vert <\delta\}$, for some $\delta>0$ and such that 
$$ 
\sup_{\{q\in\C^n,\;\Vert\Im q\Vert <\delta\}}\vert\!\vert\!\vert\nabla_q^{(2)}V(q)\vert\!\vert\!\vert <+\infty. 
$$ 
\item $H(q,p)=\sqrt{1+\Vert p\Vert^2}+V(q)$ with $V$ and $A$ as above. 
\end{enumerate} 
\end{example} 
A natural question in the study of remainders in Egorov's Theorem is to know in which 
way the error bounds are optimal. More precisely, this raises the question whether 
remainders actually do have exponential growth in generic cases. We treat this problem 
in section 6 through two examples of Schr\"odinger Hamiltonians for which exponential 
remainders are highlighted. The sections 2, 3 and 4 are respectively devoted 
to the proof of Theorems \ref{Propag2}, \ref{Propag3} and \ref{gevstab}. 
Another related problem is to study the evolution of the quantum 
observables $\hat{A}(t)$ along coherent states $\phi_z$ ($z\in Z$), defined by the mean value 
quantity $\langle \hat{A}U(t)\phi_z,U(t)\phi_z \rangle$, when $\hbar$ goes to zero. It's a 
kind of ``weak" version of our main results. It is well known that as long as $t$ (possibly 
depending on $\hbar$) is reasonably big then $U(t)\phi_z$ stays concentrated around the classical 
trajectory $\Phi^t(z)$. We give in section 5 estimates of the maximum duration of this 
phenomenon (Ehrenfest time) and discuss our results with similar previous ones. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%% $C^\infty$ case (section 2) %%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{ The $C^\infty$ case} 
The goal of this section is to prove Theorem \ref{Propag2}. 
In order to control derivatives of observables moving along 
the classical flow, we shall use the Fa\`a de Bruno formula 
\cite{co} : 
\begin{lemma} Let $g: \R^d \rightarrow \R^d$ and $f: \R^d \rightarrow \R$ 
smooth enough mappings defined in suitable neighborhoods. Then for $\alpha\in \N^d$, 
we have 
\ba\label{fdb} 
\partial^\alpha(f\circ g) &=& \sum_{\gamma\neq 0, \gamma\leq \alpha} 
(\partial^\gamma f)\circ g\cdot{\cal B}_{\alpha,\gamma}[\partial^\beta g],\; 
{\rm where}\; \\ 
{\cal B}_{\alpha,\gamma}[\partial^\beta g] &=& 
\alpha!\sum_{\sum_\beta \beta=\gamma \atop \sum_\beta \beta\vert a_\beta\vert=\alpha} 
\prod_{0\neq\beta\in\N^d}\frac{1}{a_\beta!} 
\left(\frac{\partial^\beta g}{\beta!}\right)^{a_\beta}. 
\nonumber 
\ea 
\end{lemma} 
In formula (\ref{fdb}), $\beta$ and $a_\beta$ are multi-index and we have used the usual 
rules for multi-index. Let us explain one term. 
We have $g=(g_1, \cdots g_d)\in \R^d$, $a_\beta=(a_{\beta,1}, \cdots, a_{\beta,d})$, 
$\beta!=\beta_1!\cdots\beta_d!$, then 
\b 
\left(\frac{\partial^\beta g}{\beta!}\right)^{a_\beta} = 
\left(\frac{\partial^\beta g_1}{\beta!}\right)^{a_{\beta,1}} 
\cdots \left(\frac{\partial^\beta g_d}{\beta!}\right)^{a_{\beta,d}}. 
\e 
Our first step is to estimate the classical flow. 
\begin{lemma} For every $\gamma\in \N^{2n}$, there exists $C_\gamma >0$ such that 
\b\label{estflow} 
\forall t\in\R,\;\forall z\in Z,\;\; 
\Vert \partial^\gamma_z\Phi^tz\Vert \leq C_\gamma\exp(\vert \gamma\vert\vert t\vert \Gamma) 
\e 
\end{lemma} 
{\bf Proof} : We proceed by induction on $\vert \gamma\vert$. We start with the Jacobi 
stability equation 
\b\label{ja} 
\frac{d}{dt}\nabla_z\Phi^t(z) = J\nabla_z^{(2)}H_0(\Phi^tz)\nabla_z\Phi^tz. 
\e 
Using the definition of $\Gamma$ and the Gronwall inequality, 
we get (\ref{estflow}) for $\vert \gamma\vert = 1$ with $C_\gamma=1$.\\ 
For $\vert \gamma\vert=k \geq 2$, let us assume that (\ref{estflow}) holds for 
$\vert \gamma\vert <k$. Computing derivatives in $z$ of (\ref{ja}) and applying 
Fa\`a de Bruno formula, we get 
\b 
\frac{d}{dt}\partial^\gamma_z(\Phi^tz) = 
\gamma!\sum_{\vert\alpha\vert=1}\partial^\alpha(J\nabla_zH_0)(\Phi^tz) 
(\partial^\gamma_z\Phi^tz)^\alpha + Y(t). 
\e 
Using induction assumption, 
there exists $C>0$ such that 
\b 
\forall t\in\R,\,\; \Vert Y(t)\Vert \leq C\exp(\Gamma\vert \gamma\vert\vert t\vert). 
\e 
To complete the proof of the lemma, we need some standard properties of linear differential equations 
which are recalled in the following lemma. 
\begin{lemma}\label{lindifeq} 
Let us consider the linear differential equation 
\b\label{lin} 
\frac{d}{dt}X(t) = M(t)X(t) +Y(t), 
\e 
where $M(t)$ is a smooth family of $d\times d$ matrices defined on $\R$ such 
that \\ 
$\Gamma= \sup_{t\in\R}\vert\!\vert\!\vert M(t)\vert\!\vert\!\vert <+\infty$. 
Let $R(t,s)$ be the resolvent of the homogeneous system i.e such that 
\b\label{homsyst} 
\frac{\partial R(t,s)}{\partial t} = M(t)R(t,s),\;\; R(s,s) = \Id. 
\e 
Then we have 
\ba 
X(t) = R(t,0)X(0) + \int_0^tR(t,s)Y(s)ds,\;\;{\rm and}\\ 
\Vert R(t,s)\Vert \leq \exp(\Gamma\vert t-s\vert),\;\: \forall t, s\in\R. 
\ea 
In particular, if $\Vert Y(t)\Vert \leq C{\rm e}^{K\vert t\vert} $ with $K\geq\Gamma$, 
then there exists $C^\prime>0$ such that 
\b 
\Vert X(t)\Vert \leq C^\prime{\rm e}^{K\vert t\vert}\;\; \forall t\in\R. 
\e 
\end{lemma} 
Using the inequality (\ref{estflow}) and the Fa\`a de Bruno formula again, 
we get easily the next lemma. 
\begin{lemma} \label{1erterme}For every multi-index $\gamma\neq 0$, there exists $C_{0,\gamma}>0$ such 
that 
\b 
\forall t\in\R, \forall z\in Z,\;\; \vert \partial^\gamma_z A_0(t, z)\vert \leq 
C_{0, \gamma}\exp(\Gamma\vert \gamma \vert\vert t\vert) 
\e 
\end{lemma} 
Now using the induction formula (\ref{termj}), the Leibniz formula and again the Fa\`a de Bruno 
formula, we get estimates for $A_j(t)$. 
\begin{lemma} 
For every $j\geq 1$ and every multi-index $\gamma$, 
there exists $C_{j, \gamma} > 0$ such that for every $z\in Z$ and every $t\in \R$, we have 
\b 
\vert \partial^\gamma_z A_j(t, z)\vert \leq 
C_{j, \gamma}\exp(\Gamma(2j-1+\vert \gamma\vert)\vert t\vert). 
\e 
\end{lemma} 
Now we want to estimate the error term in the propagation of observables. Let us 
define 
\b 
A^{(N)}(t) := \sum_{0\leq j\leq N}\hbar^jA_j(t). 
\e 
Recall that the Moyal bracket, $\{K,B\}^*$ of two observables $K, B$ is defined 
as the Weyl symbol of $\frac{i}{\hbar}[\hat K, \hat B]$ which admits 
the following formal $\hbar$-expansion 
$$ 
\{K, B\}^*\asymp\sum_{j\geq 0}\hbar^j\{K, B\}_{j+1}. 
$$ 
Using the rule product (see Appendix), we can expand $\{H, B\}^*$ in a power serie in $\hbar$, 
\ba 
\{H, B\}^* = \{H_0, B\}_1 + \hbar(\{H_ 0, B\}_2 +\{H_1, B\}_1) 
+\cdots + \nonumber\\ \hbar^{k-1}(\{H_0, B\}_k +\{H_1, B\}_{k-1}) + 
\hbar^k(\delta^k_{H_0,B}+\delta^{k-1}_{H_1,B}) + \cdots 
\ea 
where $\{H_0, B\}_1$ is the usual Poisson bracket (see (\ref{poisson})). The remainder term is given by 
the remainder term in the product rule, with notations defined in Appendix, 
\b 
\delta^k_{H,B} = i\hbar^{-k-1}(R_k(H,B) - R_k(B,H)). 
\e 
The algorithm used to construct the $A_j$'s is such that we have 
\b 
\frac{d}{dt}A^{(N)}(t) = \{H,A^{(N)}(t)\}^* + \hbar^{N+1} R^{(N+1)}(t), 
\e 
where 
\b\label{remform} 
R^{(N+1)}(t) = \delta^{N+1}_{H_0,A_0} + \cdots + \delta^1_{H_0,A_N} + 
\delta^{N}_{H_1,A_0} + \cdots + \delta^0_{H_1,A_N}. 
\e 
The following lemma gives the error in $L^2$-operator norm. 
\begin{lemma}\label{remest} 
For every $N\in\N$ and every $t\in\R$, we have 
\b 
\Vert \hat{A}(t) - \hat{A}^{(N)}(t)\Vert_{L^2} \leq 
\hbar^{N+1}\vert t\vert\sup_{\vert s\vert \leq\vert t\vert}\Vert \hat{R}^{(N+1)}(s)\Vert_{L^2}. 
\e 
\end{lemma} 
{\bf Proof} : Let us denote $E(t)=\hat{A}(t) - \hat{A}^{(N)}(t)$ and compute (\ref{heis}) 
\b 
\frac{d}{ds}U(-s)E(t-s)U(s) = 
U(-s)\left\{\frac{i}{\hbar} 
[\hat{H}, E(t-s)] - \frac{d}{dt}E(t-s)\right\}U(s). 
\e 
After integration in $s$ and using $E(0)=0$, we get the lemma. \QED 
\vskip5pt 
\noindent{\bf End of the proof of Theorem \ref{Propag2} :} 
Using Appendix, we can estimate $L^\infty$-norms of the derivatives of 
$\delta^{N+1-j}_{H_i,A_j(t)}$ and then conclude using the 
Calderon-Vaillancourt Theorem, with the recent improvement 
by A. Boulkhemair (\cite{bo}). The statement is the following. 
There exists $\gamma_n$ such that for all $B\in{\cal O}(0)$, 
we have~: 
\b\label{cv} 
\Vert\hat{B}\Vert_{L^2} \leq \gamma_n \sup_{\vert\alpha\vert, \vert\beta\vert \leq[n/2]+1\atop z\in Z} 
\vert \partial^{\alpha,\beta}_z B(z)\vert. 
\e 
\QED 
\vskip5pt 
By revisiting the proof, we can easily see that the conclusions of Theorem \ref{Propag2} are still valid if the following conditions 
on the Hamiltonian $H$ and the observable $A$ are fulfilled. For simplicity we assume that $H$ is classical. 
We relax growth assumptions on $H$ but we reinforce assumptions on the observable $A$. 
\begin{itemize} 
\item{(C.4)} there exists $ M > 0$ such that for every $\gamma\in\N^{2n}$ there exists $C_\gamma$ 
such that 
\b 
\vert \partial^\gamma_z H(z)\vert \leq C_\gamma (1+\vert z\vert)^M,\;\;\forall 
z\in\R^{2n}, \; 
\e 
\item{(C.5)} Let $A$ a smooth 
observable such that supp$(A) \subset \Omega$ where $\bar{\Omega}$ is a compact, $\Phi^t$-invariant set of $Z$, $\forall t\geq 0$. 
Let us introduce the real number 
$$ 
\Gamma := \sup_{z\in \Omega}\vert\!\vert\!\vert J\nabla_z^{(2)} H(z)\vert\!\vert\!\vert. 
$$ 
\end{itemize} 
\begin{proposition} \label{localization} 
Let us assume that $H$ and $A$ satisfy the conditions {\rm (C.4)}, {\rm (C.5)}. Then the following estimates 
holds, for every $t\geq 0$ and $j\geq 0$, 
\b\label{reg3} 
\vert \partial^\gamma_z A_j(t, z)\vert \leq 
C_{j, \gamma}\exp(\Gamma(3j/2+\vert \gamma\vert)\vert t\vert). 
\e 
Furthermore we have the following estimates in $L^2$ operator-norm of the remainder 
term. For every $N \in \N$ there exists $C_N$ such that we have 
\b\label{res3} 
\Vert \hat{A}(t) - \sum_{0\leq j\leq N}\hbar^j\hat{A}_j(t)\Vert_{L^2} \leq C_N \hbar^{N+1} 
\exp(\Gamma(3N/2+\delta_n)\vert t\vert). 
\e 
\end{proposition} 
As we shall see later in more details for integrable sytems, 
when the classical system is stable it is expected that the quantum motion stays localized in the phase space 
for much larger time interval. 
This can be easily proved for periodic systems as we shall see now. 
Let us consider a classical Hamiltonian $H$ and an observable $A$ satisfying the assumptions (C.4), (C.5) 
with $\bar{\Omega}$ is a connected component of the compact set $H^{-1}([E_1, E_2])$ 
where $E_1 <E_2$. Let us introduce the following periodicity property~: 
\begin{itemize} 
\item{(Per)} (i) For every $\lambda\in [E_1, E_2]$, $H^{-1}(\lambda)\cap \bar\Omega$ is a compact connected component of $H^{-1}(\lambda)$.\\ 
(ii) $\nabla H(z)\not= 0$ for every $z\in\bar\Omega$. \\ 
(iii) The Hamiltonian flow $\Phi_H^t$ defined by $H$ is periodic with period $T(z)$ in $\Omega $ 
where $T$ is a smooth function in $\Omega$. 
\end{itemize} 
\vskip5pt 
\begin{proposition}\label{periodic} 
Let us assume that $A$ and $H$ satisfy the above assumptions, in particular {\em (Per)}. 
Then for any $N$ there exists $C_N>0$ such 
that for any time $t\in\R$, 
\b 
\Vert \hat{A}(t) - \sum_{0\leq j\leq N}\hbar^j\hat{A}_j(t)\Vert_{L^2} 
\leq C_N \hbar^{N+1}\vert t\vert(1+\vert t\vert)^{2N+\delta_n}. 
\e 
where $\delta_n$ is a universal constant. 
In particular the semiclassical regime is still valid in 
time interval $[-\hbar^{\epsilon-1/2}, \hbar^{\epsilon-1/2}]$ for every $\epsilon >0$, 
and the quantum particle stays 
localized close to the classical trajectory in this time interval. 
\end{proposition} 
{\bf Proof} : 
We have to revisit the proof of Theorem \ref{Propag2} in this particular case. 
The proof of Proposition \ref{periodic} is clear if we can choose the period $T(z)$ 
constant in $\Omega$. 
Indeed, under this stronger assumption, for any multi-index $\gamma\in\N^{2n}$, 
there exists $C_\gamma>0$, such that for all $z\in \Omega$ and all $t\in \R$, we have 
the uniform control in time : $\vert\partial^\gamma_zA(\Phi^tz)\vert\leq C_\gamma$. 
This is easily proved 
using conservation of the energy and the periodicity of the flow. 
Using the induction formula (\ref{termj}), the derivatives of the $A_j$'s are also 
uniformly controled in time for the same reasons. To get the conclusion, 
we follow again the same method as in the $C^\infty$-case (see proof of Theorem \ref{Propag2}).\\ 
When the period is not constant but is smoothly varying, we can replace the Hamiltonian $H$ 
by a new one, $\widetilde{H} = f(H)$, such that the Hamiltonian flow for $ \widetilde{H}$ 
has a constant period in $\Omega$. For a detailed construction of $f$ we refer for exemple to \cite{ds}. 
In particular we have $f^\prime(E) \neq 0$ for $E\in [E_1, E_2]$ 
and 
$$ 
\Phi_H^t(z) = \Phi_{\tilde H}^{t/f^\prime(H(z))}(z), \;\; \forall z\in \Omega. 
$$ 
We get from that $\vert\partial^\gamma_zA(\Phi^tz)\vert\leq C_\gamma(1+\vert t\vert)^{\vert \gamma\vert}$. 
We then finish the proof using the same method as for Theorem \ref{Propag2}. 
\QED 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%% Analytic case (section3) %%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{The analytic case} 
To prove Theorem \ref{Propag3}, we shall use the same strategy 
as for Theorem \ref{Propag2}. For simplicity, 
we assume that $H=H_0+\hbar H_1$. The general case is not more difficult. Let us point out that we 
will use in this section the same notation to define the Hermitian norm on $\C^n$ and on $\C^{2n}$; 
that is $\Vert\cdot\Vert$. 
First of all, we estimate 
the classical flow in complex domains (similar estimates 
are considered in \cite{bgp}). That is under the assumptions of Theorem \ref{Propag3}, we have 
\begin{lemma}\label{estiflot} 
$\Phi^t$ is holomorphic in $\Omega(\delta{\rm e}^{-\vert t\vert\ \Gamma})$ 
and the following estimate holds 
\b\label{fa} 
\Vert \partial^\gamma_z\Phi^tz\Vert \leq \gamma!\delta^{1-\vert\gamma\vert} 
\exp(\vert\gamma\vert\vert t\vert\ \Gamma),\;\;\forall \gamma\in\N^{2n},\; 
\vert\gamma\vert\geq 1,\;\forall z\in Z. 
\e 
Moreover if $0\leq s\leq t$ or if $t\leq s\leq 0$, for $\tau\in [0,1]$, we have 
\b\label{I0} 
\Vert\Im z\Vert \leq \tau\delta{\rm e}^{-\vert t\vert\Gamma} \Rightarrow \Vert\Im(\Phi^{t-s}z)\Vert 
\leq\tau\delta{\rm e}^{-\vert s\vert\Gamma}. 
\e 
\end{lemma} 
{\bf Proof} : 
We can assume that $t>0$. 
Let us consider the differential equation 
\b 
\frac{d}{dt}\Phi^tz = F(\Phi^tz),\; \Phi^0z=z 
\e 
where $F(z) := J\nabla_zH_0(z)$. $F(\cdot)$ is then holomorphic in $\Omega(\delta)$. 
Using Lemma \ref{lindifeq} on linear differential, we get easily 
\b\label{I1} 
\Vert\Im\Phi^sz\Vert<\delta, \;\forall s\in [0,t^\prime] \Rightarrow 
\Vert\nabla_z\Phi^sz\Vert < {\rm e}^{s\Gamma},\;\;\forall s\in [0,t^\prime]. 
\e 
We also have 
\b\label{I2} 
\Vert\Im\Phi^sz\Vert \leq \Vert\Phi^sz-\Phi^s(\Re z)\Vert \leq \Vert\Im z\Vert 
\sup_{\Vert y\Vert\leq\Vert\Im z\Vert}\Vert\nabla_z\Phi^s(\Re z+iy)\Vert. 
\e 
Let us assume now that $\Vert\Im z\Vert < \delta{\rm e}^{-t\Gamma}$ and define 
$$ 
t^\prime = \sup\{t^{\prime\prime}>0,\; \Vert\Im \Phi^s \zeta\Vert <\delta, \forall \zeta,\; 
\Vert\Im\zeta\Vert\leq \Vert\Im z\Vert,\; \forall s\in [0,t^{\prime\prime}]\}. 
$$ 
Using (\ref{I1}) and (\ref{I2}), we get 
\b 
\Vert\Im \Phi^{t^\prime} \zeta\Vert \leq {\rm e}^{(t^\prime-t)\Gamma},\; 
\forall \zeta, \;\Vert \Im\zeta\Vert \leq \Vert\Im z\Vert. 
\e 
So that for $\varepsilon >0$, we have $ \Vert\Im \Phi^{t^\prime+\varepsilon} \zeta\Vert < \delta$ 
which contradicts the assumption on $t^\prime$ and proves that $t^\prime = t$. 
This proves (\ref{I0}) for $s=0$. 
To prove it for all $s\in [0,t]$, we use the same method but taking 
$\tau\delta{\rm e}^{-s\Gamma}$ as a new $\delta$ and $t-s$ as a new $t$. 
Finally, using smoothness for solutions of differential equations, we get that 
$\Phi^t$ is holomorphic in $\Omega(\delta{\rm e}^{-t\Gamma})$. Then using 
Cauchy inequalities, we get (\ref{fa}).\QED 
\vskip5pt 
From the above lemma and Cauchy inequalities, we get also the following estimates. 
\begin{lemma}\label{anest0} 
There exists a constant $M>0$ such that for every $\gamma\in\N^{2n}$, 
$\vert \gamma\vert \geq 1$, $z\in\Omega(\delta{\rm e}^{-\Gamma\vert t\vert})$, we have 
\b 
\vert \partial^\gamma_z[A(\Phi^t(z)]\vert \leq 
M\gamma!\delta^{1-\vert\gamma\vert}{\rm e}^{\Gamma\vert \gamma\vert\vert t\vert}. 
\e 
\end{lemma} 
To prove Theorem \ref{Propag3}, the analyticity assumptions on $H$ and $A$ will also 
be used through the following lemma (see also \cite{tr}). 
\begin{lemma}\label{anal} Let $f: \Omega(\delta)\rightarrow \C$ be an holomorphic function, 
where $\Omega(\delta)=\{z\in\C^d, \Vert\Im z\Vert<\delta\}$. 
For $\tau\in]0,1[$, let us define 
$$ 
\Vert f\Vert_\tau = \sup_{z\in \Omega(\tau\delta)}\vert f(z)\vert. 
$$ 
And assume that there exist $M>0, a\geq 0$ such that 
\b\label{inq} 
\Vert f\Vert_\tau \leq M\left(\frac{e}{1-\tau}\right)^a,\; \forall\tau\in]0,1[. 
\e 
Then $\forall \gamma\in\N^d$, $\forall \tau\in]0,1[$, we have 
\b 
\Vert \partial^\gamma_zf\Vert_\tau \leq M\delta^{-\vert\gamma\vert}(a+1)\cdots(a+\vert\gamma\vert) 
\left(\frac{e}{1-\tau}\right)^{a+\vert\gamma\vert}. 
\e 
\end{lemma} 
{\bf Proof} : Let us start with $\vert\gamma\vert=1$. Using the Cauchy inequality, we have that 
$\forall \tau^\prime\in]0,\tau[$, 
\b 
\Vert\frac{\partial f}{\partial z_j}\Vert_{\tau^\prime} \leq \frac{\delta^{-1}}{\tau-\tau^\prime} 
\Vert f\Vert_\tau. 
\e 
Using now (\ref{inq}) with $\tau=\tau^\prime+\frac{1-\tau^\prime}{1+a}$ and the elementary 
inequality $\left(\frac{1+a}{a}\right)^a\leq e$, we get 
\b 
\Vert\frac{\partial f}{\partial z_j}\Vert_{\tau^\prime}\leq M\delta^{-1}(a+1) 
\left(\frac{e}{1-\tau^\prime}\right)^{a+1}. 
\e 
Then coming back to the notation $\tau^\prime=\tau$, we prove easily the lemma by induction on 
$\vert \gamma\vert$. 
\QED 
The proof of Theorem \ref{Propag3} is based on the following estimate. 
\begin{proposition}\label{moyenne} 
Let $A$ and $H$ satisfying the assumptions of Theorem \ref{Propag3}. Then 
there exist two constant $K_0$ and $K_1$ such that for all $j\geq 1$, $t\in\R$, $\tau\in ]0, 1[$ we have 
\b\label{anestj} 
\sup_{z\in\Omega(\tau\delta{\rm e}^{-\vert t\vert\Gamma})}\vert A_j(t,z)\vert 
\leq 
K_0\left\{\left(\frac{\rm e}{1-\tau}\right)^{j-1}{\rm e}^{(2j-1)\vert t\vert\Gamma}\right\} 
\sum_{1\leq m\leq j}\frac{(j+m)!}{m!}\vert t\vert^m \pmatrix{j-1\cr m-1}K_1^{j+m} 
\e 
where $\bd\pmatrix{j-1\cr m-1}=\frac{(j-1)!}{(m-1)!(j-m)!}\ed$. 
\end{proposition} 
{\bf Proof :} 
We shall prove (\ref{anestj}) by induction on $j$ using the three previous lemmas. 
The estimate is clearly satisfied for $j=1$. Indeed, according to Lemma \ref{estiflot} 
and formula (\ref{eq:term1}), $A_1(t,\cdot)$ is holomorphic in the domain 
$\Omega(\tau\delta{\rm e}^{-\vert t\vert\Gamma})$. And then, from Lemma \ref{anest0}, 
there exists $K_0>0$ such that for any $z\in \Omega(\tau\delta{\rm e}^{-\vert t\vert\Gamma})$, 
\b 
\vert A_1(t,z)\vert 
\leq 
K_0\left(\frac{\rm e}{1-\tau}\right)^{0}{\rm e}^{\vert t\vert\Gamma}\vert t\vert. 
\e 
Note that by application of Lemma \ref{anal}, 
\b 
\sup_{z\in\Omega(\tau\delta{\rm e}^{-\vert t\vert\Gamma})} 
\vert \partial_z^\gamma A_1(t,z)\vert 
\leq 
K_0\gamma!\left(\frac{\rm e}{1-\tau}\right)^{\vert\gamma\vert}{\rm e}^{(1+\vert\gamma\vert)\vert t\vert\Gamma}\vert t\vert. 
\e 
Assume now that (\ref{anestj}) is satisfied for $0\leq\ell\leq j-1$. So that according to 
Lemmas \ref{estiflot} and \ref{anal}, for $\Vert\Im z\Vert\leq \tau\delta{\rm e}^{-\vert t\vert\Gamma}$, 
\b 
\vert\partial^\gamma_z A_{\ell}(t,z)\vert \leq 
M\left(\frac{\rm e}{1-\tau}\right)^{\ell-1+\vert\gamma\vert} 
{\rm e}^{(2\ell-1+\vert\gamma\vert)\vert t\vert\Gamma}\delta^{-\vert \gamma\vert}\gamma! 
\sum_{1\leq m\leq \ell}\frac{(\ell+m)!}{m!}\vert t\vert^m \pmatrix{\ell -1\cr m-1}K^{\ell+m}. 
\label{eq:recur} 
\e 
We apply now (\ref{termj}) and (\ref{eq:recur}) to estimate each term in the sum given $A_j$ . 
In particular we use the identity~: 
\b 
\pmatrix{j-1\cr m} = \sum_{m\leq k\leq j-1}\pmatrix{k-1\cr m-1} 
\e 
and the elementary inequality 
\b 
\gamma! \leq \vert \gamma\vert! \leq 2^{\vert \gamma\vert}\gamma!.\label{eq:inegalite} 
\e 
So we can prove easily, using standard estimates, that (\ref{anestj}) is satisfied. 
\QED 
\vskip5pt 
We then deduce the following result on the $A_j$'s. 
\begin{corollary} There exist $K_1>0$ and $K_2>0$ such that for every $\gamma\in \N^{2n}$, $j\in\N$ 
satisfying $\vert\gamma\vert+j\geq 1$, $t\in\R$ we have 
\b 
\sup_{z\in Z}\vert \partial^\gamma_z A_j(t,z)\vert \leq 
K_1(j+\vert\gamma\vert)!K_2^{j+\vert\gamma\vert} 
(1+\vert t\vert)^j{\rm e}^{(2j-1+\vert\gamma\vert) 
\vert t\vert\Gamma}. 
\e 
\end{corollary} 
{\bf Proof :} 
According to Lemma \ref{anal} and Proposition \ref{moyenne}, the formula (\ref{eq:recur}) is valid 
for any $j\geq 1$ in the domain $\Omega(\tau\delta{\rm e}^{-\vert t\vert\Gamma})$ where $\tau$ is arbitrary small and 
$\delta>1$. 
Note first that using (\ref{eq:inegalite}), for $0\leq m\leq j$, 
$$ 
\frac{(j+m)!}{m!}\leq \frac{(2j)!}{j!}\leq 4^jj!. 
$$ 
Thus for $\Vert\Im z\Vert\leq \tau\delta{\rm e}^{-\vert t\vert\Gamma}$, 
\baa 
\vert\partial^\gamma_z A_{j}(t,z)\vert & \leq & 
C_0\left(\frac{\rm e}{1-\tau}\right)^{j-1+\vert\gamma\vert} 
{\rm e}^{(2j-1+\vert\gamma\vert)\vert t\vert\Gamma}(j+\vert\gamma\vert)!(4C_1)^j 
\sum_{0\leq m\leq j}\pmatrix{j \cr m}(\vert t\vert C_1)^m\\ 
& \leq & C_0\left(\frac{\rm e}{1-\tau}\right)^{j-1+\vert\gamma\vert} 
{\rm e}^{(2j-1+\vert\gamma\vert)\vert t\vert\Gamma}(j+\vert\gamma\vert)!(4K)^j(1+C_1\vert t\vert)^j. 
\eaa 
The results follows by restriction to the real Euclidean space $Z$ ( taking $\tau$ small enough ). 
\QED 
\vskip5pt 
\noindent{\bf End of the proof of Theorem \ref{Propag3} :}\\ 
We follow the same method as in the $C^\infty$-case, with the above estimates 
on $A_j$ by using the Appendix and Lemma \ref{remest}. We left the details 
to the reader. 
\QED
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%$$ The Gevrey case (sect 4 ) %%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{The Gevrey case} 
The proofs of the Gevrey estimates in Theorem \ref{gevinst} and Theorem \ref{gevstab} use the 
following more or less well known lemma~: 
\begin{lemma}\label{gevest} 
Let us consider a real analytic map $f$ from $\Omega_1$ to $\Omega_2$ where $\Omega_j$ is an open set in $\R^{n_j}$ 
and $u$ a Gevrey $s$ function from $\Omega_2$ into the complex plane $\C$. Then $u\circ f$ is Gevrey $s$ in $\Omega_1$. 
More precisely there exist universal positive constants $\delta, \delta_1$ such that if there exist constants $C_0, C_1, K_0, K_1$ 
and 
\b 
\vert \partial _x^\alpha f(x) \vert \leq \alpha!K_0K_1^{\vert \alpha\vert},\;\;\;\;\; 
\vert\partial_z^\beta u(z)\vert \leq 
C_0C_1^{\vert\beta\vert}\beta!^s, 
\e 
then we have, for every real number $r$ such that $0<r\leq \frac{1}{2K_1}$ and $r\leq \frac{1}{2\delta K_0K_1C_1}$, 
\b 
\vert\partial_x^\alpha (u\circ f)(x)\vert \leq \delta_1C_0r^{-\vert\alpha\vert}\alpha!^s. 
\e 
\end{lemma} 
{\bf Proof} : As in L. H\"ormander, Vol.1, \cite{ho}, we fix $x\in\R^{n_1}$ and consider the Taylor expansion for 
$u$ in $f(x)$, at order $k = \vert \alpha\vert$, 
\b 
F(y) = \sum_{\vert \beta\vert \leq k}\frac{\partial_z^\beta u(f(x))}{\beta!}(f(y)-f(x))^\beta. 
\e 
Note that $\partial_x^\alpha u\circ f(x) = \partial_y^\alpha F(y)\vert_{y=x}$. 
The lemma is then proved by applying the Cauchy integral formula to $F$.\QED 
\vskip5pt 
Theorem 1.11 is proved by induction on $j$, using several times Lemma \ref{gevest}. 
The main step to prove Theorem 1.13 is the following result~: 
\begin{proposition}\label{coordest} 
There exist $K_0>0$, $K_1>0$ such that for all $t\in\R$, $j\in \N$, $z\in Z$, $\gamma\in\N^{2n}$ 
we have 
\b 
\vert\partial_z^\gamma A_j(t,z)\vert \leq 
K_0K_1^{\vert \gamma\vert+3j/2}(1+\vert t\vert)^{\vert \gamma\vert+2j}(\vert\gamma\vert+3j/2)!^s(j/2)!^{-1} 
\e 
\end{proposition} 
Let us recall that $A_j = 0$ for $j$ odd because here $H$ has no lower order terms.\\ 
It is convenient to prove estimates first in action-angle variables and afterwards to 
use Lemma \ref{gevest} to get estimates in position-momentum coordinates. 
So let us consider the new classical observables $\tilde{A}_j(t,I, \varphi) = A_j(t, \chi(I,\varphi+t\omega(I))$. 
\begin{lemma} 
There exist $K_0>0$, $K_1>0$, $K_2>0$, such that for all $t\in\R$, $j\in \N$, $z\in Z$, $\gamma\in\N^{2n}$, 
we have 
\b\label{AA0} 
\vert\partial_I^\alpha\partial_\varphi^\beta\tilde{A}_0(t,I,\varphi)\vert \leq K_0K_1^{\vert\alpha\vert+\vert\beta\vert } 
(\vert\alpha\vert+\vert\beta\vert)!^s (1+\vert t\vert)^{\vert\alpha\vert} 
\e and for $j \geq 2$, 
\ba\label{AAj} 
\vert\partial_I^\alpha\partial_\varphi^\beta\tilde{A}_j(t,I,\varphi)\vert \leq K_0K_1^{\vert\alpha\vert+\vert\beta\vert +3j/2} 
(\vert\alpha\vert+\vert\beta\vert+3j/2)!^{s-1} (1+\vert t\vert)^{\vert\alpha\vert+3j/2}\nonumber\\ 
\sum_{1\leq m\leq j/2} 
(\vert \alpha\vert+\vert\beta\vert+j+m)!\frac{\vert t\vert^m}{m!}\bd\pmatrix{j/2-1\cr m-1}\ed K_2^m 
\ea 
\end{lemma} 
{\bf Proof} : For $j=0$ we have $\tilde{A_0}(t,I,\varphi)=\tilde{A}(I,\varphi+t\omega(I))$ where $\tilde{A}:=A\circ \chi^{-1}$. 
We know that $\tilde{A}$ is Gevrey $s$ and $\omega$ is analytic so we get easily (\ref{AA0}) from Lemma \ref{gevest}.\\ 
For $j\geq 2$, with some tedious but not very difficult computations, we can prove (\ref{AAj}) by induction 
on $j$ as we have done in the analytic case (see section 3). 
\QED 
\vskip5pt 
\noindent{\bf End of proof of Proposition \ref{coordest} :} It results easily using estimates (\ref{AAj}) 
and systematic use of Lemma \ref{gevest}. 
\QED 
\vskip5pt 
\noindent{\bf End of proof of Theorem \ref{gevstab} :} We get control of the remainder terms with the same method as in section 2, using Lemma \ref{remest}, the Appendix and the Calderon-Vaillancourt Theorem. \QED 
\vskip10pt 
\noindent{\bf Remarks concerning Hamiltonians with singularities}\\ 
For the statement of Theorems 1.11 and 1.13, smoothness of the Hamiltonian $H$ is required only on a flow invariant domain 
of $Z$ containing the support of the observable $A$. These results can then be extended, as briefly explained below, in the case of local Hamiltonians of the form~: 
$$ 
H(x,\xi) = \sum_{\vert \alpha\vert\leq m}a_\alpha(x)\xi^\alpha 
$$ 
satisfying for every $\alpha$, $\vert\alpha\vert \leq m$ : 
\begin{itemize} 
\item{(H.1)} $\partial_x^\beta a_\alpha\in L_{\ell oc}^1(\R^n)$, $\forall\;\vert\beta\vert\leq\vert\alpha\vert$.
\item{(H.2)} There exists a compact $K \subset \R^n $ and $\rho >0$ such that $a_\alpha$ is analytic in 
$I_K:=$\hbox{$\{ \Vert\Im x\Vert <\rho,\; \Re x \notin K \} $} and satisfies in $I_K$, for some $C>0$ and $M>0$, 
$$ 
\vert a_\alpha(x)\vert \leq C (1+\vert x\vert)^M. 
$$ 
Moreover $\hat{H}$ is supposed to be self-adjoint in $L^2(\R^n)$. 
\end{itemize} 
As an example of such $H$, we can consider the following Coulomb like Hamiltonian 
on $\R^3$: 
$$ 
H(x,\xi) = \Vert\xi\Vert^2+\sum_{j=1}^L\frac{c_j}{\Vert x-x^{(j)}\Vert}+W(x) 
$$ 
where $c_j\in\R$, $x^{(j)}\in\R^3$ and $W$ satisfies (H.2) with $ K=\emptyset$. Of course, one needs to 
adapt to this case the previous assumptions 
(C.1) and (C.2) concerning the observable $A$~: 
\begin{itemize} 
\item{(C'.1)} Let $\Omega$ be a bounded open subset of $Z$ such that $\bar{\Omega}$ is 
$\Phi^t$-invariant for every $t\in\R$ and $\bar{\Omega}\subset\R^{2n}\setminus(K\times\R^n)$. 
The observable $A$ is such that $supp(A)\subset\Omega$. 
\item{(C'.2)} $A$ is Gevrey $s>1$. 
\end{itemize} 
Under the assumptions (H.1), (H.2), (C'.1), (C'.2), we can prove that the singular region contributes in the 
Egorov Theorem by an error term which is of the same order as in the smooth case. This fact can be seen by standard 
commutator arguments and integrations by parts. We give here the main steps of the proof without 
going to the details.\\ 
Let us introduce two $C^\infty$ cutoff functions $\chi$ and $\zeta$ where $\zeta$ is supported in 
a small neighborood of $K$ such that $\zeta=1$ on $K$ and $\chi=1$ on $supp(\zeta)$. The 
distance between the supports of $\zeta$ and $1-\chi$ is then positive. 
Write the commutator $[\hat{H}, \hat{A^{(N)}(t)}]$ as follows~: 
\b\label{decomp} 
[\hat{H}, \hat{A^{(N)}(t)}] = [\hat{H}, \hat{A^{(N)}(t)}]\chi + \hat{H}\zeta\hat{A^{(N)}(t)}(1-\chi) + 
[\hat{H}(1-\zeta), \hat{A^{(N)}(t)}](1-\chi) 
\e 
In the last term of the r.h.s of (\ref{decomp}) each operator has a smooth symbol so that 
we can use the same computations as in the $C^{\infty}$ case with $\hbar$-pseudodifferential operators. 
We explain now how the first two terms give negligible contributions. 
We suppose here, to fix ideas, that the system is integrable in $\Omega$ 
so that the Gevrey estimates on $A_j(t)$ established in Theorem \ref{gevstab} 
are still valid under the above assumptions. Using now integration by parts, taking into account of the support 
of the involved symbols, we can estimate the integral kernel of operators $[\hat{H}, \hat{A_j(t)}]\chi$ 
and $\hat{H}\zeta\hat{A_j(t)}(1-\chi)$ to get the $L^2$ estimate~: 
\ba\label{sremest} 
\Vert [\hat{H}, \hat{B}]\chi\Vert_{L^2} + \Vert\hat{H}\zeta\hat{B}(1-\chi)\Vert_{L^2} \leq 
K_0K_1^NN!^{(3s-1)/2}(1+\vert t\vert)^{2N}\hbar^N. 
\ea 
By combination of (\ref{decomp}) and (\ref{sremest}), we then obtain the same $L^2$ estimate 
for $\hat{A(t)} - \hat{A^{(N)}}(t)$ as in Theorem \ref{gevstab}. 
\QED

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%% Etats Coherents (SECTION 5) %%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{Propagation of coherent states and Ehrenfest time} 
Let us consider a usual Gaussian coherent state $\phi_{z_0}$ concentrated around the point $z_0=(q_0,p_0)$ 
of the phase space, defined as follows~: 
$$ 
\phi_{z_0}:=W(z_0)\eta 
$$ 
where $ 
\eta(x)=(\pi\hbar)^{-\frac{n}{4}} 
e^{-\frac{\vert x\vert^2}{2\hbar}} 
$ 
is the ground state of the $n$-dimensionnal harmonic oscillator $K_0=\frac{1}{2}(P^2+Q^2)$ 
and 
$ 
W(z_0)=e^{-\frac{i}{\hbar}(q_0P-p_0Q)} 
$ 
is the unitary translation operator in the phase space. 
Here $P=-i\hbar\partial_x$ and $Q\psi(x)=x\psi(x)$. 
Note that 
$W(0,0)=Id$ so that $\eta$ is concentrated at the origine (see \cite{cr} for a more complete 
description of the Gaussian coherent states and \cite{pu} for a more general class). 
Here, we study the semiclassical limit of a quantum observable $\hat{A}(t)$ 
along the coherent state $\phi_{z_0}$ through the quantity 
$\langle\hat{A}(t)\phi_{z_0},\phi_{z_0}\rangle$. In other words, 
we study the average on $A$ of the quantum evolution $U(t)\phi_{z_0}$ of a well localized wave paquet $\phi_{z_0}$. 
It is well known that $U(t)\phi_{z_0}$ stays concentrated around the classical trajectory 
$\Phi^t(z_0)$ (see Remark~\ref{rmqpropa} below). Here we give an estimate 
of the duration $T_\hbar$, such that for $\vert t\vert \leq T_\hbar$, 
$\langle\hat{A}(t)\phi_{z_0},\phi_{z_0}\rangle$ goes to its classical limit $A(\Phi^t(z_0))$ 
when $\hbar$ goes to zero. The following result can be interpreted as a weak version of 
Theorem \ref{Propag2}. 
\begin{proposition}\label{weakpro} 
Under the hypotheses of the Theorem \ref{Propag2}, for any $\epsilon>0$, we have the following uniform (in time) limit. 
\b 
\lim_{\hbar\searrow 0 \atop \vert t\vert\leq \frac{1-\epsilon}{2\Gamma}\ln({\frac{1}{\hbar}})} 
\left(\langle\hat{A}(t)\phi_{z_0},\phi_{z_0}\rangle - A(\Phi^t(z_0))\right) = 0. 
\label{weak} 
\e 
\end{proposition} 
{\bf Proof} : 
Using 
the Wigner function of $\phi_{z_0}$ and the Weyl $\hbar$-symbol $A_\hbar(t,z)$ of $\hat{A}(t)$, 
we get the following well known integral~: 
\b 
\langle\hat{A}(t)\phi_{z_0},\phi_{z_0}\rangle 
=(\pi\hbar)^{-n}\int_{\R^{2n}} A_\hbar(t,z)e^{-\frac{\vert z-z_0\vert^2}{\hbar}}dz.\label{wigner} 
\e 
Then since we are interested in the limit as $\hbar$ goes to zero, 
according to Theorem~\ref{Propag2}, one can substitute $\hat{A_0}(t)$ to $\hat{A}(t)$ 
in (\ref{wigner}). So that we have now to estimate the following expression~: 
\baa 
\rho(t,\hbar)&=& 
(\pi\hbar)^{-n}\int_{\R^{2n}} A(\Phi^t(z))e^{-\frac{\vert z-z_0\vert^2}{\hbar}}dz 
-A(\Phi^t(z_0))\\ 
&=&(\pi\hbar)^{-n}\int_{\R^{2n}} (A(\Phi^t(z))-A(\Phi^t(z_0))) 
e^{-\frac{\vert z-z_0\vert^2}{\hbar}}dz. 
\eaa 
Now using the estimate $\vert\nabla_zA(\Phi^t(z))\vert\leq C e^{\Gamma\vert t\vert}$ 
for any $t\in\R$ (see Lemma \ref{1erterme}) and the Taylor formula, we get 
\baa 
\vert\rho(t,\hbar)\vert&\leq & 
C'\hbar^{-n} 
\left(\int_{\R^{2n}} \vert z-z_0\vert e^{-\frac{\vert z-z_0\vert^2}{\hbar}}dz\right) 
e^{\Gamma\vert t\vert}\\ 
&\leq&C''\hbar^{\frac{1}{2}}e^{\Gamma\vert t\vert}, 
\eaa 
where $C', C"$ are constant independent on $t$ and $\hbar$. 
Hence for any $t$ satisfying 
$\vert t\vert\leq \frac{1-\epsilon}{2\Gamma}\ln({\frac{1}{\hbar}})$ with $\epsilon>0$, we have 
$\vert\rho(t,\hbar)\vert\leq C''\hbar^{\epsilon}$. 
\QED 
\vskip5pt 
\begin{Remark}\label{rmqpropa} 
\noindent {\em 
(i) Since for any $A\in{\cal O}(0)$, we have that 
$$ 
\Vert\hat{A}-\hat{A}^{aw}\Vert_{L^2}={\cal O}(\hbar) 
$$ 
where $\hat{A}^{aw}$ is the anti-Wick quantization of $A$, we can replace $\hat{A}$ 
by $\hat{A}^{aw}$ in (\ref{weak}) (see \cite{hmr} for definition and properties of anti-Wick quantization) 
and consider the semiclassical Husimi measures~: 
$$ 
\mu_{t,\hbar,z_0}:A\longmapsto \langle\hat{A}^{aw}U(t)\phi_{z_0},U(t)\phi_{z_0}\rangle. 
$$ 
Then, if we denote by $\delta_\zeta$ the Dirac measure in $\zeta\in Z$, 
the Proposition \ref{weakpro} means that we have the following uniform (in time) weak limit of measures. 
For any $\epsilon >0$ we have 
\b\label{wl} 
\lim_{\hbar\searrow 0 \atop \vert t\vert\leq \frac{1-\epsilon}{2\Gamma}\ln({\frac{1}{\hbar}})} 
\left(\mu_{t,\hbar,z_0} - \delta_{\Phi^t(z_0)}\right) = 0. 
\e 
\noindent 
(ii) A result similar to (\ref{wl}) has been already obtained for some specific quantized hyperbolic maps on the torus 
(hyperbolic automorphisms and Baker map) \cite{bdb}.\\ 
\noindent 
(iii) In \cite{cr} a result similar to 
Proposition \ref{weakpro} and (\ref{wl}) was obtained as a consequence of propagation results 
for coherent states but for smaller times~: $\vert t\vert \leq \frac{1-\epsilon}{6\Gamma}\ln({\frac{1}{\hbar}})$. 
So by the method of this paper we have improved 1/6 by 1/2. \\ 
\noindent 
(iv) Now, a natural question is what happens for times $\vert t \vert \geq \frac{1}{2\Gamma}\ln({\frac{1}{\hbar}}) $? 
An answer is given in \cite{bdb} for quantized hyperbolic maps on the torus. After the critical time $T_\hbar := \frac{1}{2\Gamma}\ln({\frac{1}{\hbar}})$, 
appears a new regime, called ``mixing regime", where the Dirac measure at $\Phi^t(z_0)$ 
is replaced by some absolutely continuous measure on some flow invariant set (delocalization phenomenon).\\ 
For Schr\"odinger operators, 
and more generally for classical Hamiltonians $H$, we have already seen that the semiclassical expansion 
for the quantum observable $\hat{A}(t)$ is valid for $\vert t \vert \leq \frac{2-\epsilon}{3\Gamma}\ln({\frac{1}{\hbar}})$ 
for every $0<\epsilon$. This result suggests that the mixing regime is also accessible for Schr\"odinger operators, 
at least for some examples. In \cite{dbr} we shall consider this problem, in particular when the starting point $z_0$ is 
an hyperbolic fixed point of the flow.} 
\end{Remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%% LOWER BOUNDS (section 6) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{Examples} 
The goal of this section is to check through two examples that the error bounds in Theorems 
\ref{Propag2} and \ref{Propag3} are in some sense optimal. 
To this end, we establish, for these examples, exponential lower bounds of the remainder term. We 
have already seen in previous sections that for some particular (but important) dynamical sytems that this phenomenon does 
not hold. We suppose in our examples that $t>0$. For the complementary case $t<0$ we have the same results. 
\vskip5pt 
\noindent\underline{\bf Example 1} :\\ 
Let us consider the semi-classical Hamiltonian 
$$ 
H=H_0+\hbar H_1 
$$ 
where $H_0,H_1:\R^2\longrightarrow \R$ are defined as follows 
$$ 
H_0(q,p)=\frac{1}{2}(p^2-q^2),\;\;\;\;H_1(q,p)=\varepsilon q. 
$$ 
The linear Hamiltonian flow $\Phi^t:\R^2\longrightarrow\R^2$ associated to $H_0$ is given by 
$$ 
\Phi^t(q,p)=(q\cosh t+p\sinh t,q\sinh t+p\cosh t). 
$$ 
If we consider the observable $A(q,p)=q$, according to Theorem \ref{Propag2}, 
we have that for any coherent states $\phi_z$ concentrated around the point $z=(q,p)$, 
$$ 
\lim_{\hbar\rightarrow 0} 
\frac{\langle \hat{A}(t)\phi_z,\phi_z\rangle-\langle \hat{A}_0(t)\phi_z,\phi_z\rangle} 
{\hbar}=A_1(t,z). 
$$ 
Using the formula (\ref{termj}), for any $z\in\R^2$ we have 
$ 
A_1(t,z)=\varepsilon(\cosh t -1)$. 
So that 
\b 
\liminf_{\hbar\rightarrow 0} 
\frac{\Vert\hat{A}(t)- \hat{A}_0(t)\Vert_{L^2}} 
{\hbar} 
\geq\lim_{\hbar\rightarrow 0} 
\frac{\langle \hat{A}(t)\phi_z,\phi_z\rangle-\langle \hat{A}_0(t)\phi_z,\phi_z\rangle} 
{\hbar}=\varepsilon(\cosh t -1).\label{eq:cosh} 
\e 
Then, according to (\ref{eq:cosh}), we can assert that the remainder in Egorov's Theorem is exponentially large in time. 
Indeed, suppose that $H$ and $A$ are as above. More precisely there exists a function $M:\R^*\longrightarrow\R$ 
such that 
\b 
\Vert\hat{A}(t)-\hat{A}_0(t)\Vert_{L^2}\leq \hbar M(t), \forall \hbar\in]0, 1],\;\forall t>0, 
\e 
then there exists $C_\varepsilon>0$ such that $M(t)\geq C_{\varepsilon}e^t$, $\forall t>0$. 
\QED 
\vskip0.5cm 
\noindent\underline{\bf Example 2} : (Non symmetric double well. Local phenomenon around an instable fixed point)\\ 
Let us consider the classical Hamiltonian $H$ defined on $\R^2$ as follows~: 
$$ 
H(q,p)=\frac{p^2}{2}+V(q) 
$$ 
where $V'(0)=0$, $V''(0)=-1$ and $V^{(3)}(0)\neq 0$. 
The condition on the third derivative of $V$ implies that the double well is non symmetric. It 
will ensure that the coefficient $A_2(t,0)$ in the formal expansion of the Heisenberg observable is not 
zero. Consider, for example, the following potential 
$$ 
V(q)=\frac{1}{4}(q^2-1)^2-\frac{1}{4}+\frac{q^3}{3}\chi(q)\;\; 
\mbox{where}\;\;\chi\in C^{\infty}_0(\R)\;\;\mbox{and}\;\;\chi(0)\neq 0. 
$$ 
Note that the associated linearized flow at the equilibrium point is 
$$ 
\nabla_z\Phi^t(0,0)=\pmatrix{\cosh t & \sinh t \cr \sinh t &\cosh t}. 
$$ 
According to the Theorem \ref{Propag2}, if $\phi_0$ is a coherent state concentrated around $(0,0)$, we have 
as in the previous example, 
\b 
\lim_{\hbar\rightarrow 0} 
\frac{\langle \hat{A}(t)\phi_0,\phi_0\rangle-\langle \hat{A}_0(t)\phi_0,\phi_0\rangle} 
{\hbar^2}=A_2(t,0). 
\e 
Suppose that $A$ is a classical observable satisfying $A(q,p):=f(q)$ with $f^\prime(0)=f^{\prime\prime}(0)=0$ and $f^{(3)}(0)\neq 0$. 
According to the formula (\ref{termj}), 
$$ 
A_2(t,0)=\Gamma(0,3)V^{(3)}(0)f^{(3)}(0)\int_0^t(\sinh s)^3ds. 
$$ 
Then, as in the previous example, we get the following inequality, for some $C>0$ 
$$ 
\liminf_{\hbar\rightarrow 0} 
\frac{\Vert\hat{A}(t)- \hat{A}_0(t)\Vert_{L^2}} 
{\hbar^2} 
\geq Ce^{3t},\;\forall t>0. 
$$ 
Therefore, one concludes that if $M:\R^*\longrightarrow\R$ is a function 
satisfying 
\b 
\forall t\geq 0, \forall \hbar \in]0, 1],\;\; \Vert\hat{A}(t)-\hat{A}_0(t)\Vert_{L^2}\leq \hbar^2 M(t), 
\e 
then $M(t)\geq Ce^{3t}$, $\forall t>0$. 
\QED 
\vskip1cm 
\noindent{\bf Acknowledgments~:} {\tt The authors are grateful to the referees for their stimulating questions which have helped 
to improve this paper.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%% Composition of Observables (appendice) %%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\appendix 
\section{APPENDIX~: Product of observables} 
Let us first recall the formal product rule for quantum observables with Weyl 
quantization. Let $A, B \in {\cal S}(Z)$. We look for a semi-classical observable 
$C$ such that $\hat{A}\cdot\hat{B} = \hat{C}$. Some computations with the Fourier 
transform give the following formula \cite{ho} 
\b 
C(q,p) = 
\exp\left(\frac{i\hbar}{2}\sigma(D_q,D_p;D_{q^\prime},D_{p^\prime})\right)A(q,p) 
B(q^\prime,p^\prime)\vert_{(q,p)=(q^\prime,p^\prime)}, 
\e 
where $\sigma$ is the symplectic bilinear form (\ref{sympform}) and $D=i^{-1}\nabla$. 
By expanding the exponential term, we get 
\b\label{prod2} 
C(q,p) = 
\sum_{j\geq 0}\frac{\hbar^j}{j!}(\frac{i}{2}\sigma(D_q,D_p;D_{q^\prime},D_{p^\prime}))^j 
A(q,p)B(q^\prime,p^\prime)\vert_{(q,p)=(q^\prime,p^\prime)}. 
\e 
So that $C(q,p)$ is a formal power serie in $\hbar$ with coefficients given by 
\b 
C_j(q,p) = \frac{1}{2^j}\sum_{\vert\alpha+\beta\vert=j} 
\frac{(-1)^{\vert \beta\vert}}{\alpha!\beta!} 
(D^\beta_q\partial^\alpha_p A).( D^\alpha_q\partial^\beta_p B)(q,p). 
\e 
It is well known that if $A, B$ are observables with polynomial growth then $C$ 
is a $C^\infty$ semi-classical observable (see for example \cite{ho, ro}). 
Here we need 
more accurate remainder estimates. Let us denote $A\#B = C$, $z=(q,p)\in Z$ and 
for every $N \geq 1$, 
\b 
A\#B(z) - \sum_{0\leq j \leq N}\hbar^jC_j(z) =: R_N(A,B;z;\hbar). 
\e 
The main result of this appendix is the following 
\begin{theorem} There exists $K_n>0$ and for every $m\in \N$, $m\geq 4n$, for 
every $s > 4n$ there exists a constant $\rho_{n,m,s}$ such that 
for every $A, B \in {\cal S}(Z)$, for every $N\geq 1$, for every multi-index $\gamma$, 
the following estimate holds, for every $z\in Z$, 
\ba\label{remest1} 
\vert \partial_z^\gamma\left(A\#B(z) - \sum_{0\leq j \leq N}\hbar^jC_j(z)\right)\vert 
\leq \hbar^{N+1}\rho_{n,m,s}K_n^{N+\vert \gamma\vert}(N!)^{-1}\cdots \nonumber\\ 
\times \sup_{(*)}\left[(1+u^2+v^2)^{(s-m)/2}\vert\partial_u^{(\alpha,\beta)+\mu}A(u+z)\vert 
\vert\partial_v^{(\beta,\alpha)+\nu}B(v+z)\vert\right] 
\ea 
where $\displaystyle{\sup_{(*)}}$ means that the supremum holds 
under the conditions\\ $u, v \in Z$, $\vert \mu\vert +\vert \nu\vert\leq m+\vert\gamma \vert$, 
$\vert\alpha\vert+\vert\beta\vert = N+1$ ($\mu, \nu\in\N^{2n}$, $\alpha, \beta\in\N^n$). 
\end{theorem} 
{\bf Proof}~: We follow \cite{dr}. By Fourier transform computations and 
Taylor formula, we get the following formula 
\ba\label{reprem} 
R_N(A,B;z;\hbar) = \frac{1}{N!}\left(\frac{i\hbar}{2}\right)^{N+1} 
\int_0^1(1-t)^NR_{N,t}(z;\hbar)dt \;\; 
{\rm where}\;\;\; R_{N,t}(z;\hbar) = \nonumber \\ (2\pi\hbar t)^{-2n} 
\int\int_{Z\times Z} \exp(-\frac{i}{2t\hbar}\sigma(u,v))\sigma^{N+1}(D_u,D_v) 
A(u+z)B(v+z)dudv. 
\ea 
We shall use the following lemma to estimate $R_{N,t}(z;\hbar) $. 
\begin{lemma}\label{foi} 
Let us consider $F\in{\cal S}(Z\times Z)$ and the integral 
\b 
I(\lambda) = \lambda^{2n}\int \int_{Z\times Z}\exp[-i\lambda \sigma(u, v)]F(u,v)dudv. 
\e 
Then for every real number $s>4n$ and every integer $m\geq 4n$ there 
exists $\kappa(n,s,m) >0$ depending only on $n, s, m$ (but independent of $F$) 
such that the following estimate holds 
\b 
\vert I(\lambda)\vert \leq \kappa(n,s,m) 
\sup_{u, v\in Z, \atop \vert \mu\vert+\vert \nu\vert\leq m}(1+u^2+v^2 )^{s-m/2} 
\vert \partial^\mu_u\partial^\nu_v F(u,v)\vert. 
\e 
\end{lemma} 
{\bf Proof} : This lemma is more or less standard (see \cite{ho}). For completeness, 
we give here a direct proof. Let us introduce a cut-off $\chi_0$, $C^\infty$ on $\R$, 
$\chi_0(x) = 1$ for $\vert x\vert\leq 1/2$ and 
$\chi_0(x) = 0$ for $\vert x\vert\>\geq1$. We split $I(\lambda)$ into three pieces 
\ba 
I_0(\lambda) = \lambda^{2n}\int\!\!\int_{Z\times Z}\exp[-i\lambda \sigma(u, v), 
\chi_0(\lambda(u^2+v^2))F(u,v)dudv, \\ 
I_1(\lambda) = \lambda^{2n}\int\!\!\int_{Z\times Z}\exp[-i\lambda \sigma(u, v)] 
(1-\chi_0(u^2+v^2))\chi_0(\lambda(u^2+v^2))F(u,v)dudv,\\ 
I_2(\lambda) = \lambda^{2n}\int\!\!\int_{Z\times Z}\exp[-i\lambda \sigma(u, v)] 
(1-\chi_0(u^2+v^2))F(u,v)dudv. 
\ea 
For $I_0(\lambda)$, we easily have 
\b\label{AI0} 
\vert I_0(\lambda)\vert \leq \omega_{4n}\sup_{u^2+v^2\leq 1} \vert F(u,v)\vert, 
\e 
where $\omega_{4n}$ is the volume of the unit ball in $Z^2$. 
For $I_1(\lambda)$ and $I_2(\lambda)$, we integrate by parts with the differential 
operator 
\b 
L = \frac{i}{u^2+v^2}\left(Ju\frac{\partial}{\partial v}-Jv\frac{\partial}{\partial u}\right), 
\e 
where $J$ is the matrix associated to the symplectic form ($\sigma(u,v)=<Ju,v>$). 
Performing $4n$ integrations by parts, we can see that it exists a constant $c_n$ 
such that 
\b\label{AI1} 
\vert I_1(\lambda)\vert \leq c_n 
\sup_{u^2+v^2\leq 1\atop \vert \mu\vert+\vert \nu\vert\leq 4n } 
\vert\partial^\mu_u \partial^\nu_vF(u,v)\vert. 
\e 
Similarly, performing $m$ integrations by parts, we get for a constant $c(n,s,m)$, 
\b\label{AI2} 
\vert I_2(\lambda)\vert \leq c_n 
\sup_{u,v \in Z\atop \vert \mu\vert+\vert \nu\vert\leq m} 
(1+u^2+v^2 )^{(s-m)/2}\vert\partial^\mu_u \partial^\nu_vF(u,v)\vert. 
\e 
\QED 
Now we can complete the proof of the theorem by using Lemma \ref{foi}, the Leibniz formula 
and the following elementary estimate, using in $Z$ 
the coordinates $u=(x,\xi)$, $v=(y,\eta)$, 
\ba 
\vert \sigma^N(\partial_x,\partial_\xi; \partial_y,\partial_\eta))F(x,\xi)G(y,\eta)\vert \leq 
\nonumber\\ 
(2n)^N\sup_{\vert \alpha\vert+\vert \beta\vert=N} 
\vert\partial_x^\alpha\partial_\xi^\beta A(x,\xi)\partial_y^\beta\partial_\eta^\alpha B(y,\eta)\vert. 
\ea 
\QED 
\begin{remark} 
We can easily extend the estimate (\ref{remest1}) for observables $A, B$ with polynomial 
growth at infinity, by choosing $m$ large enough to get a finite r.h.s. 
Let us assume that $A\in{\cal O}(\mu_A)$, $B\in{\cal O}(\mu_B)$, 
where $\mu_A, \mu_B \in\R$. Then we can apply (\ref{remest1}) to 
$A_\varepsilon(u) = {\rm e}^{-\varepsilon u^2}A(u)$ and 
$B_\varepsilon(v) = {\rm e}^{-\varepsilon v^2}B(v)$ for $\varepsilon >0$ and 
pass to the limit $\varepsilon \rightarrow 0$ with $m-s\geq \mu_A+\mu_B$. 
\end{remark} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
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