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Egorov Theorem, semi-classical approximation, Ehrenfest time, 
Quantum and Classical dynamics
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\begin{center}


{\bf \Large Uniform Semi-classical Estimates for the propagation\\ of Heisenberg 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, 
Facult\'e des Sciences de Nantes,\\
2, rue de la Houssini\`ere, 44072 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 Heisenberg 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 Heisenberg observable is 
a semi-classical observable of index Gevrey 2 (3/2 if the Hamiltonian is purely classical, without
lower terms in $\hbar$).
\end{abstract}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% INTRODUCTION-ASSUMPTIONS- RESULTS (section)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Introduction-Assumptions-Results} 
According to the Bohr's correspondance principle, the quantum evolution of one observable
is close to its classical evolution as the Planck constant $\hbar$ is small. In the mathematical
litterature 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.


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 Heisenberg
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  Heisenberg
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 is
$ Z = X\times X$,   a symplectic linear space equipped with the symplectic form $\sigma$ defined 
through the restriction to $\R^n$ of the Hermitian scalar product $<\cdot,\cdot>$ on $\C^n$
\b
\sigma(p,q;p^\prime,q^\prime) = \langle p,q^\prime\rangle -  
\langle p^\prime,q\rangle,\;\;q,q^\prime\in X,\;p, p^\prime\in X^*.\label{sympform}
\e
A classical Hamiltonian is a smooth real function $H:Z\mapsto \R$. Our basic
example will be $H(q,p) = \frac{\Vert p\Vert^2}{2m} + V(q)$  ($m>0$)
where $\Vert p\Vert^2=<p,p>$.
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 
\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_p^\alpha\partial_q^\beta A_{\ell})
(\Phi^\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 (because $U(t)$ is a metaplectic
transformation).
\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 
$$
\Gamma := \sup_{z\in Z}\vert\!\vert\!\vert \nabla_z^{(2)} H_0(z)\vert\!\vert\!\vert,
$$  
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}
\exp(\Gamma(2N+5n+3)\vert t\vert).
\e
\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
$\exp({\Gamma(3j/2+\vert \gamma\vert)\vert t\vert})$. 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)}.
\e
We remember that some times ago, S. Debi\`evre suggested that the Ehrenfest time
could be   greater that $\frac{1}{2\Gamma}\log\hbar^{-1}$. The above  results confirm
this guess.
}
\vskip5pt


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 $\Omega(\delta):= \{z\in \C^{2n}, \vert \Im z\vert <\delta \}$. Suppose moreover that
$\nabla_z^{(2)}H_0$, $\nabla_z H_1$, $\nabla_z A$ are bounded 
and continuous in $\Omega(\delta)$.
Then, if we define 
$$
\Gamma := \sup_{z\in\Omega(\delta)}\vert\!\vert\!\vert \nabla_z^{(2)} H_0(z)\vert\!\vert\!\vert. 
$$
we have  the following improvement of Theorem \ref{Propag2}, saying
that $\hbar^{-1}(A(t,z)-A\circ\Phi^t)$ is a semi-classical observable with Gevrey
index 2, 
with   control in time.
More explicitely,  there
exist constants $M>0, K>0$ such that 
\ba
\vert \partial^\gamma_z A_j(t, z)\vert \leq 
M(j+\vert\gamma\vert)!^2K^{j+\vert \gamma\vert}\exp(\Gamma(2j+\vert \gamma\vert)\vert t\vert)\\
\left\vert \partial_z^\gamma A(t,z)[\hbar] - \sum_{0\leq j\leq N}\hbar^jA_j(t,z)\right\vert
\leq \hbar^{N+1}\vert t\vert MK^{N+1+\vert \gamma\vert}(N+1)!^2\gamma!\\ \nonumber
\times\exp(\vert t\vert\Gamma(2N+4n+1)).
\ea
Furthermore we have 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 M(N+1)!^2K^{N+1}\hbar^{N+1}
\exp(\Gamma(2N+5n+3)\vert t\vert).\label{res2}
\e
\end{theorem}
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
$\varepsilon+C <1/2$, there exists a constant $K$ such that
for $\vert t\vert \leq C\log(\hbar^{-1})$  we have
\b
\Vert \hat{A}(t) - \sum_{0\leq j\leq \hbar^{-\varepsilon}}\hbar^j\hat{A}_j(t)\Vert_{L^2} 
\leq K\exp\left(-\frac{1}{\hbar^{\varepsilon}}\right)
\e
\end{corollary}
\begin{remark}
Estimates (\ref{res1}) and (\ref{res2}) improve \cite{bgp} in two ways.
In the $C^\infty$ case we get similar results without analyticity assumptions.
In the analytic case we get better estimates with respect to $N$ (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} If the expansion of $H$ in $\hbar$  is even then the estimates given in Theorem \ref{Propag3}
hold with Gevrey index equal to $3/2$ and the $2N$ in the exponential term becomes $3N/2$.
\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}






 



 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%% $C^\infty$ case (section) %%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\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_z^{(2)}H_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 $K>\Gamma$,
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} 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}),
\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
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}
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)}(t)\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
\noindent{\bf End of the proof of Theorem \ref{Propag2}}


Using Appendix, we can estimate $L^\infty$-norms of derivatives for 
$\delta^{N+1-j}_{H_i,A_j(t)}$ and then we 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




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%  Analytic case (section)  %%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\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
were considered in \cite{bgp}). That is under the assumptions of Theorem \ref{Propag3}, we have
\begin{lemma}
$\Phi^t$ 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 $F(z) := J\nabla_zH_0(z)$. $F(z)$ 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
>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}


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 $M>0, a>0$ 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}
There exists a constant $C_1$ large enough, depending only on $n, A, H$, 
such that for all $j\geq 1$, $t\in\R$, we have
\b\label{anestj}
\sup_{z\in\Omega(\tau\delta{\rm e}^{-\vert t\vert\Gamma})}\vert A_j(t,z)\vert \leq 
Mj!^2C_1^j\left(\frac{\rm e}{1-\tau}\right)^{2j-1}{\rm e}^{(2j-1)\vert t\vert\Gamma}.
\e
\end{proposition}
{\bf Proof}: 
Let us assume for simplicity that $t>0$.  We shall prove (\ref{anestj}) by induction
on $j$. From Lemma \ref{anest0}, the estimate is clearly satisfied for $j=1$.
Let us assume that (\ref{anestj}) is satisfied for $\ell \leq j-1$. Then using
Lemma \ref{anal}, we get, for $\Vert\Im z\Vert\leq \tau\delta{\rm e}^{-t\Gamma}$,
\ba
\vert\partial^\gamma_z A_\ell(s,\Phi^{t-s}z\vert \leq 
M\ell!^2\left(\frac{\rm e}{1-\tau}\right)^{2\ell-1+
\vert\gamma\vert}\delta^{-\vert\gamma\vert}
{\rm e}^{(2\ell-1+\vert\gamma\vert)s\Gamma}\nonumber\\
\times 2\ell(2\ell+1) \cdots (2\ell-1+\vert\gamma\vert).
\ea
In the induction  formula (\ref{termj}), we have $\vert\gamma\vert+\ell=j+1$
and $\ell\leq j-1$. So we shall use 
the two following elementary inequalities
\ba
  2\ell(2\ell+1) \cdots (2j-1+) \leq 4^{j-\ell}\frac{j!^2}{\ell!^2},\\
\sum_{\vert\gamma\vert=j-\ell+1}1 = (2n)^{j-\ell+1}.
\ea
>From the assumptions on $H$, taking a smaller $\delta>0$ if necessary,
  there exists $C_0>0$ such that for
$\vert\gamma\vert+k\geq 2$, we have 
\b
\sup_{\Omega(\delta)}\vert\partial^\gamma_z H_k\vert \leq \gamma!C_0^{\vert\gamma\vert+1}.
\e
Then, it follows from (\ref{termj})  that the following estimate holds
\ba
\sup_{z\in\Omega(\tau\delta{\rm e}^{-\vert t\vert\Gamma})}\vert A_j(t,z)\vert \leq 
Mj!^2C_1^j\left(\frac{\rm e}{1-\tau}\right)^{2j-1}{\rm e}^{(2j-1)\vert t\vert\Gamma}\nonumber\\
\times \frac{4nMC_0^2}{\delta\Gamma}\sum_{0\leq\ell\leq j-1}\left(\frac{4nC_0}{\delta C_1}\right)^{j-\ell}.
\ea
Taking now  $C_1$ large enough, we get the conclusion. \QED


Using Cauchy inequalities, we deduce the following result on the $A_j$'s
\begin{corollary} For every $\gamma\in N^{2n}$ and $j\in\N$ such that
$\vert\gamma\vert+j\geq 1$, we have
\b
\sup_{z\in Z}\vert \partial^\gamma_z A_j(t,z)\vert \leq 
Mj!^2\gamma!C_1^j\delta^{-\vert\gamma\vert}{\rm e}^{(2j-1+\vert\gamma\vert)\vert t\vert\Gamma}.
\e
\end{corollary}
{\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 (\ref{remest}). We left the details
to the reader.\QED
\vskip10pt
\noindent{\bf Acknowledgement}:  The authors thank S.Graffi,  for discussions about long  time semi-classical approximation, 
  P. Bolley, B. Gr\'ebert and JP. Guillement for discussions
concerning  the Fa\`a de Bruno formula.


\newpage


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%  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{remest}
\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{remest}) 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{remest}) 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}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% REFERENCES %%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{thebibliography}{abcd} \baselineskip=10pt


\bibitem[BGP]{bgp} D. Bambusi, S. Graffi, T. Paul, 
{\em Long time semi-classical approximation of quantum flows: 
a proof of the Ehrenfest time}, (1998), Asymptotic Analysis (to appear).



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