$. 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_{\gammaĦj},\;{\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\!