\documentstyle[leqno]{article}
\newfont{\bb}{msbm10 scaled\magstep1}
\newfont{\frak}{eufm10 scaled\magstep1}
\newcommand{\noi}{\noindent}
\newcommand{\bra}[1]{{<}#1|}
\newcommand{\ket}[1]{|#1{>}}
\newcommand{\braket}[2]{{<}#1|#2{>}}
\newcommand{\slashD}{D\hspace{-.65em}\slash }
\newcommand{\isitequal}{\stackrel{?}{=}}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\h}{\hbar}
\newcommand{\Exp}{\mbox{Exp}}
\newcommand{\ad}{\mbox{Ad}}
\newcommand{\bbC}{\mbox{\bb C}}
\newcommand{\bbR}{\mbox{\bb R}}
\newcommand{\bbQ}{\mbox{\bb Q}}
\newcommand{\bbE}{\mbox{\bb E}}
\newcommand{\bbN}{\mbox{\bb N}}
\newcommand{\bbZ}{\mbox{\bb Z}}
\newcommand{\bbH}{\mbox{\bb H}}
\newcommand{\bbL}{\mbox{\bb L}}
\newcommand{\bbP}{\mbox{\bb P}}
\newcommand{\bbS}{\mbox{\bb S}}
\newcommand{\bbT}{\mbox{\bb T}}
\newcommand{\fraka}{\mbox{\frak a}}
\newcommand{\frakb}{\mbox{\frak b}}
\newcommand{\frakk}{\mbox{\frak k}}
\newcommand{\frakg}{\mbox{\frak g}}
\newcommand{\frakq}{\mbox{\frak q}}
\newcommand{\frakh}{\mbox{\frak h}}
\newcommand{\frakp}{\mbox{\frak p}}
\newcommand{\fraku}{\mbox{\frak u}}
\newcommand{\frakU}{\mbox{\frak U}}
\newcommand{\frakH}{\mbox{\frak H}}
\newcommand{\frakt}{\mbox{\frak t}}
\newcommand{\lig}{\mbox{\frak g}}
\newcommand{\liad}{{\frak g}_{\mbox{Ad}}}
\newcommand{\lt}{\mbox{\frak t}}
\newcommand{\dlg}{\mbox{\frak g}^*}
\newcommand{\zbar}{\overline{z}}
\newcommand{\wbar}{\overline{w}}
\newcommand{\dbar}{\overline{\partial}}
\newcommand{\tr}{\mbox{Tr}}
\newcommand{\wf}{\mbox{WF}}
\newcommand{\pdo}{\Psi\mbox{DO}}
\newcommand{\tildf}{\skew5\tilde f}
\newcommand{\tildh}{\skew5\tilde h}
\newcommand{\tildC}{\skew4\tilde C}
\newcommand{\tildN}{\skew4\tilde N}
\newcommand{\tildS}{\skew4\tilde S}
\newcommand{\tildM}{\skew4\tilde M}
\newcommand{\tildx}{\skew4\tilde x}
\newcommand{\dteta}{\partial_\theta}
\newcommand{\por}{{\cdot}}
\newcommand{\transverse}{\mbox{\bb \symbol{'164}}}
\newcommand{\grad}{\mbox{grad}\,}
\newcommand{\Isom}{\mbox{Isom}}
\newcommand{\GL}{\mbox{GL}}
\newcommand{\SL}{\mbox{SL}}
\newcommand{\Op}{\mbox{Op}}
\newcommand{\comp}{{\circ}}
\newcommand{\calW}{{\cal W}}
\newcommand{\calZ}{{\cal Z}}
\newcommand{\calY}{{\cal Y}}
\newcommand{\calP}{{\cal P}}
\newcommand{\calR}{{\cal R}}
\newcommand{\calB}{{\cal B}}
\newcommand{\calF}{{\cal F}}
\newcommand{\calG}{{\cal G}}
\newcommand{\calS}{{\cal S}}
\newcommand{\calO}{{\cal O}}
\newcommand{\calH}{{\cal H}}
\newcommand{\calL}{{\cal L}}
\newcommand{\calV}{{\cal V}}
\newcommand{\calC}{{\cal C}}
\newcommand{\calQ}{{\cal Q}}
\newcommand{\calD}{{\cal D}}
\newcommand{\calU}{{\cal U}}
\newcommand{\calA}{{\cal A}}
\newcommand{\calM}{{\cal M}}
\newcommand{\calK}{{\cal K}}
\newcommand{\calI}{{\cal I}}
\newcommand{\calE}{{\cal E}}
\newcommand{\calT}{{\cal T}}
\newcommand{\calN}{{\cal N}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{question}{Question}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumptions}[theorem]{Assumptions}
\newtheorem{example}[theorem]{Example}
\newcommand{\qdot}{\dot{q}}
\newcommand{\fmj}{f^{(m)}_j}
\newcommand{\lmj}{\lambda^{m}_j}
\newcommand{\vp}{\varphi}
\newcommand{\x}{\overline{x}}
\newcommand{\y}{\overline{y}}
\newcommand{\vs}{\varsigma}
\newcommand{\splus}{S^{+}}
\newcommand{\sminus}{S^{-}}
\newcommand{\hphi}{\hat{\varphi}}
\newcommand{\Msupp}{\mbox{MSupp}}
\newcommand{\supp}{\mbox{Supp}}
\newcommand{\st}{\sqrt\tau}
\newcommand{\e}{\epsilon}
\newenvironment{proof}{ {\sc Proof} \hskip
.1truein \rm}{\nobreak\par\hfill
$\Box$
\vskip .1truein \par}
\newcounter{kount}
\newenvironment{examples}{\vspace{.3in}
{\noindent \bf Examples }\hspace{.3in}\noindent}{\hspace{.3in}}
\def\today{\ifcase\month\or
January\or February\or March\or April\or May\or June\or
July\or August\or September\or October\or November\or December\fi
\space\number\day, \number\year}
\addtolength{\textwidth}{1.0in}
\addtolength{\textheight}{2.0in}
\voffset=-1.0in
\hoffset=-.45in
\addtolength{\baselineskip}{1.5pt}
\title{On the pointwise behaviour of semi-classical measures}
\author{T. Paul\thanks {CEREMADE, Universit\'e Paris-Dauphine,
Place de Lattre de Tassigny, 75775 Paris Cedex 16 et CNRS}
\ and A. Uribe\thanks{ Mathematics Department,
University of Michigan, Ann Arbor, Michigan 48109.
Research supported in part by NSF grant DMS-9303778.}}
\begin{document}
\setcounter{footnote}{2}
\date{}
\date{August 17, 1994}
\maketitle
\tableofcontents
\bigskip
\bigskip
\centerline{\bf Abstract}
\bigskip
In this paper we concern ourselves with
the small $\h$ asymptotics of the inner products of the eigenfunctions
of a Schr\"odinger-type operator with a coherent state.
More precisely,
let $\psi_j^\h$ and $E_j^\h$ denote the eigenfunctions and eigenvalues
of a Schr\"odinger-type operator $H_\h$ with discrete spectrum.
Let $\psi_{(x,\xi)}$ be a coherent state centered at the point
$(x,\xi)$ in phase space. We estimate as $\h\to 0$ the
averages of the squares of the inner products
$ \mid(\psi_{(x,\xi)}^a,\psi_j^\hbar)\mid ^2 $
over an energy interval of size $\h$ around a fixed energy, $E$.
This follows from asymptotic expansions of the form
\[
\sum_j\varphi\left( \frac{E_j(\hbar)-E}{\hbar}\right)
\mid(\psi_{(x,\xi)}^a,\psi_j^\hbar)\mid ^2\
\sim \ \sum_{k=0}^\infty\, c_k(a) \hbar^{-n+\frac{1}{2}+k}\,
\]
for certain test functions $\varphi$ and Schwartz amplitudes $a$ of the
coherent state. We compute the leading coefficient in the expansion, which
depends on whether the classical trajectory through $(x,\xi)$ is
periodic or not. In the periodic case the iterates of the trajectory
contribute to the leading coefficient.
We also discuss the case of the Laplacian on a compact Riemannian manifold.
\vfill\eject
\section{Introduction}
Let $H=-\hbar^2\Delta + V(x)$ be a Schr\"odinger operator with $V$
smooth, on $\bbR^n$ (in which case we assume $V$
tends to infinity at infinity and therefore $H$ has
discrete spectrum) or on a compact Riemannian manifold, $M$.
The trace formula, \cite{PU1},
describes the small $\h$ asymptotics of the average,
over a spectral interval of size $\hbar$, of the
matrix elements of a semi-classical
observable, $b(x,\hbar D_x)$, between
eigenvectors of $H$: Let $\varphi$ be a Schwartz function
whose Fourier transform is
compactly supported and let $E^\hbar _j$ and $\psi^\hbar _j$ the
eigenvalues and eigenvectors of $H$. Then, under certain conditions on
the Hamilton flow of the Hamiltonian
$\calH(x,\xi)= {1\over 2} | \xi | ^2 +
V(x)$ on $\Sigma_E\ =\ \{(\xi,x); {1\over 2} | \xi | ^2 + V(x)
\ =\ E\}$, we have an asymptotic expansion of the form
\begin{eqnarray}\label{1}
\sum_j\ \varphi\left( {E_j(\h) - E\over\h }\right)
(\psi _j^\hbar,b(x,\hbar D_x)\psi_j^\hbar) \sim \nonumber \\
\left(\hat{\varphi}(0)\frac {\int_{\Sigma_E}b(\xi,x)d\mu^L}{(2\pi)^{-n+1}} \hbar^{-(n-1)}\ + \
\sum_{k=-n+2}^{+\infty}c_k(\vp)\hbar^k\right) \ + \\
\sum_\gamma \left(\hat{\varphi}(T_\gamma)
\frac { e^{i(S_\gamma/ \hbar + \sigma_\gamma)}}{\sqrt{|\det(1-P_\gamma)|}}
\int_0^{T_\gamma^*}b(x(t),\xi(t))dt
\ + \sum_{j=1}^\infty d_j^{\gamma}(\vp)\hbar^j\right)\,. \nonumber
\end{eqnarray}
Here:
\begin{itemize}
\item d$\mu^L$ is the Liouville measure on $\Sigma_E$,
\item $\gamma$ runs over the
periodic trajectories of $ {1\over 2} | \xi | ^2 + V(x)$ on $\Sigma_E$
with periods $T_\gamma$
in the support of $\hat \varphi$ (the Fourier transform of $\varphi$),
\item $T^*_\gamma$ is the primitive period of $\gamma$,
\item $S_\gamma = \int _\gamma \xi dx$,
\item $\sigma_\gamma$ is the Maslov index of $\gamma$,
\item $P_\gamma$ is the Poincar\'e mapping of $\gamma$,
\item $c_k(\vp)$ are distributions with support in \{0\},
\item $d_j^{\gamma}(\vp)$ are distributions with supports in \{$T_\gamma$\}.
\end{itemize}
Tauberian theorems allow to "pass to the limit" where $\varphi$ tends to the
characteristic function of $[E-c\hbar,E+c\hbar]$. This gives, assuming
that $b(x,\h D_x)$ is non-negative and that the set of periodic points
on $\Sigma_E$ has measure zero, that
\begin{eqnarray}\label{2}
\sum_{\mid E_j - E\mid \leq c\hbar} (\varphi _j^\hbar,b(x,\hbar D_x)\psi_j^\hbar) =\nonumber\\
\frac{2c}{{(2\pi)}^n}\int_{\Sigma_E}b(x,\xi)d\mu^L\hbar^{-(n-1)} +
o(\hbar^{-(n-1)})\,.
\end{eqnarray}
>From this one gets the following result on
"ergodicity of eigenfunctions": If the flow of
$ {1\over 2} | \xi | ^2 + V(x)$ is ergodic
on $\Sigma_E$, then, except for a subsequence of density $0$,
\begin{equation}\label{3}
\lim_{E_j^\hbar \rightarrow E, \hbar \rightarrow 0}
(\varphi _j^\hbar,b(x,\hbar D_x)\psi_j^\hbar) = \frac{\int _{\Sigma_E}
b(x,\xi)d\mu^L}{\int _{\Sigma_E}d\mu^L}\,.
\end{equation}
(For a precise statement see \cite{HMR}.)
Another way of writing (\ref{3}) in $\bbR^n$ is by using the so-called
anti-Wick or Toeplitz quantization.
Let ($\psi_{(x,\xi)}$), $(x,\xi)\in T^*\bbR^n$,
be the family of coherent states:
\begin{equation}\label{4a}
\psi_{(x,\xi)}(y)\ =\ 2^{-n/4}(2\pi\hbar)^{-\frac{3n}{4}} e^{-i\frac{\xi x}{2\hbar}}
e^{i\frac{\xi y}{\hbar}}
e^{-\frac{(y-x)^2}{2\hbar}}.
\end{equation}
The anti-Wick quantization of $b(x,\xi)$,
$b^{AW}(x,\hbar D_x)$, is the operator defined by the formula
\begin{equation}\label{5a}
b^{AW}(x,\hbar D_x) = \int b(x,\xi)(\psi_{(x,\xi)}, .)\psi_{(x,\xi)}dxd\xi.
\end{equation}
It is easy to check that, under very general assumptions on $V$,
\begin{equation}\label{6a}
(\psi _j^\hbar,b^{AW}(x,\hbar D_x)\psi^\hbar) =
(\psi _j^\hbar,b(x,\hbar D_x)\psi^\hbar) + O(\hbar)\,.
\end{equation}
So (\ref{3}) can be written as
\be\label{7}
\lim_{E_j^\hbar \rightarrow E, \hbar \rightarrow 0}
\int b(x,\xi)\mid(\psi_{(x,\xi)},\psi_j^\hbar)\mid ^2 dxd\xi =
\frac{\int_{\Sigma_E}b(x,\xi)d\mu^L}
{\int_{\Sigma_E}d\mu^L}.
\ee
In other words, ergodicity of the classical flow on $\Sigma_E$
implies that the measures
$\mid(\psi_{(x,\xi)},\psi_j^\hbar)\mid ^2 dxd\xi $
converge weakly to the normalized Liouville measure of $\Sigma_E$.
\medskip
Numerical computations, for the so-called billiard problem on the
stadium and for the hydrogen atom in a strong magnetic field, show
however that some concentration of eigenfunctions near unstable
periodic orbits may occur. This scar phenomenon seems to disappear in
the classical limit, contrary to the case of modes and quasi modes
associated to stable periodic orbits.
Our purpose in this paper is to show that, on the average,
the pointwise limit of
\[
\mid(\psi_{(x,\xi)},\psi_j^\hbar)\mid ^2
\]
depends strongly on whether
or not $(x,\xi)$ belongs to a periodic trajectory, and to analyze
its behavior in each case.
\medskip
Before we state the results precisely, we would like to present the
main ideas. The contribution of the periodic trajectories in the trace
formula disappears in formula (\ref{2}), since it appears in (\ref{1})
at a lower order in $\hbar$. On the other hand, the coefficient of the
contribution of $\gamma$ depends strongly on the support of $b(x,\xi)$,
so it is natural to think that if one takes
symbols whose supports concentrate near a
part of a periodic trajectory as $\hbar$ goes to zero,
the periodic orbit can make a contribution to the leading order term.
Using such symbols amounts, in effect, to observing the wave functions at a
smaller scale in phase space. Although this
type of symbols do not belong to classical pseudo-differential
classes, the 'anti-Wick' quantization allows to consider such singular
symbols. The simplest example is a symbol of the form
\be\label{8}
b_{(x_0,\xi_0)}(x,\xi) \ =\ \delta (x-x_0)\delta (\xi-\xi_0)\,,
\ee
a Dirac mass at $(x_0,\xi_0)$. Then (\ref{5a}) becomes
\be\label{9}
b^{AW}_{(x_0,\xi_0)}(x,\hbar D_x) \ =
\ ( \psi_{(x_0,\xi_0)},.) \psi_{(x_0,\xi_0)}\,.
\ee
Such an operator is related to the theory of Hermite Fourier integral
operators. Formally, it
can be viewed as a pseudo-differential operator with Weyl symbol
\be\label{10}
B_{(x_0,\xi_0)} (x,\xi)\ =
\ (4\pi\hbar)^{-n}e^{-\frac{(x-x_0)^2+(\xi -\xi_0)^2}{\hbar}}
\ee
which obviously is not in any standard symbol class.
If (\ref{3}) were still true for $b^{AW}$ given by (\ref{9}),
we would get that for almost all
eigenfunctions, the limit as $E_j\to E$ and
$\h\to 0$ of $\mid(\psi_{(x,\xi)},\psi_j^\hbar)\mid ^2$
would be the result of
applying the normalized Liouville measure to $B_{(x_0,\xi_0)}$, which is
\be\label{11}
\int B_{(x_0,\xi_0)} d\mu^L \ = \
\frac{(4\pi\hbar)^{-\frac{1}{2}}}{\mid\nabla H(x_0,\xi_0)\mid}
+ O(1)\,.
\ee
(Invariantly, the gradient is with respect to the quadratic form
appearing in the Gaussian.) Our main result shows that, on average,
there are extra contributions to
$\mid(\psi_{(x,\xi)},\psi_j^\hbar)\mid ^2$.
\medskip
It is useful to express our main result for a more general class of coherent
states (see the next section for details).
\par\bigskip\noindent
{\bf Preliminary Definition.}
{\it
Let $a\in S (R^n)$, and $(x,\xi) \in R^n$ or $T^*M$. A generalized coherent
state centered at $(x,\xi)$ and symbol $a$ is defined locally around x as:
\be\label{16}
\psi_{(x,\xi)}^a(y)\ = \ \rho(y-x)(2\pi\h)^{-\frac{3n}{4}}2^{-n/4}e^{-ix\xi/2\h}e^{i\xi y/\h}
\hat a(\frac{y-x}{\sqrt\h})\,.
\ee
Here $\rho$ is a $C_0^\infty$ cutoff function equal to $1$ near $0$, and
in the manifold case the formula above is in a given coordinate system.}
\bigskip
\noindent
{\bf Remarks:}
- The formal definition agrees with this one to leading order in $\h$,
but it allows for higher order terms which are needed to make
the definition coordinate-independent. The Schwartz function $a$
is invariantly a symplectic spinor, which is the symbol of the generalized
coherent state, see \S 2.
- The Proposition 2.4 below shows that the cutoff $\rho$ is semiclassically
inessential: modulo $O(\h^\infty)$ the state above is independent of
it.
- In the case where $a(\eta) \ = \ (4\pi)^{-n/4}e^{-(\eta^2/2)}$, this is the
usual coherent states definition (up to the inessential cut-off and
normalization).
- The normalization in (\ref{16}) is such that the $L^2$ norm of
$\psi_{(x,\xi)}^a$ is $O(\h^{-n/2})$. It is chosen so that the Wigner
function of the coherent state (see below) converges, as $\h\to 0$,
to a Dirac mass at $(x,\xi)$.
\medskip
We will now state the main results.
Let $H_\hbar\,=\,\sum_{l=0}^{L}\hbar^lP_l(x,D_x)$ where $P_l$ is a
differential operator of order l on $\bbR^n$ (or $M$) of principal
symbol $P_l^0$ and smooth coefficients. Let
$\calH(x,\xi)\ =\ \sum_{l=0}^{L}P_l^0(x,\xi)$ be the principal symbol of
$H_\hbar$. We assume that $\calH$ is positive, and in case $M=\bbR^n$,
that it tends to infinity at infinity.
Let $E^\hbar _j$ and $\psi^\hbar _j$ denote the
eigenvalues and eigenvectors of $H_\h$.
Throughout we will use the following normalization of the Fourier transform:
\[
\hat f(\xi)\ =\ (2\pi)^{-\frac{n}{2}}\int e^{i\xi x}f(x)dx\,.
\]
\begin{theorem}
Let $(x,\xi$) be a point in $T^*(\bbR^n)$ (or $T^*(M)$) not in classical
equilibrium, and let $E=\calH(x,\xi$).
Assume that $(x,\xi)$ is not periodic with respect to the Hamilton flow of
$\calH$.
Let $\varphi\in C^\infty(\bbR)$ with compactly supported Fourier transform,
and let $\psi_{(x,\xi)}^a$ be defined by (\ref{16}).
Then, as $\h\to 0$,
\be\label{13}
\sum_j\varphi\left( \frac{E_j(\hbar)-E}{\hbar}\right)
\mid(\psi_{(x,\xi)}^a,\psi_j^\hbar)\mid ^2\
\sim \ \sum_{k=0}^\infty\, c_k(a) \hbar^{-n+\frac{1}{2}+k}\,,
\ee
with
\be
c_0(a)\ =\ \frac{2^{-n/2}}{\sqrt{2\pi}}(2\pi)^{-3n/2}
\widehat\varphi(0)\int e^{-it^2\dot{x}\dot{\xi}/2}e^{it\eta\dot{x}}
\overline{a(\eta)}a(\eta -t\dot{\xi})d\eta dt\,.
\ee
Moreover, $\forall\;c>0$
\begin{eqnarray}
\label{tauber1}
\sum_{|E_j(\h) - E| \leq c\h}\mid(\psi_{(x,\xi)},\psi_j^\hbar)\mid ^2 &\ = \ &
\frac{c }{\pi}2^{-n/2}(2\pi)^{-3n/2}
\hbar^{-n+\frac{1}{2}}\int e^{-it^2\dot{x}\dot{\xi}/2}e^{it\eta\dot{x}}
\overline{a(\eta)}a(\eta -t\dot{\xi})d\eta dt \nonumber \\
& & \mbox{}+o(\hbar^{-n+\frac{1}{2}})\,.
\end{eqnarray}
\end{theorem}
\medskip
We will next state the result in the periodic case, in coordinates.
If $(x,\xi$) belongs to a periodic trajectory $\gamma$ of action $S_\gamma$
and primitive period $T_\gamma >0$, let $S(t)$ be the matrix solution of
\be
\dot S(t) \ = \ J \mbox{Hess(\calH)}(x(t),\xi(t))\cdot S(t)\ ,\quad
S(0)\,=\,\mbox{Identity}\,,
\ee
where J is the matrix
$ \left( \begin{array}{clcr} 0 & -Id \\ Id & \ 0 \end{array} \right)$,
$\{(x(t),\xi(t)\}$ is the trajectory of the Hamilton flow generated by $\calH$
starting at $(x,\xi)$
and Hess($\calH$) is the Hessian of $\calH$ (see section 3). Invariantly,
the mapping
defined by $S(t)$ is the differential of the Hamilton flow of $\calH$,
and $S(t)$ determines, by continuity in $t$, an element
of $Mp(R^n)$ (starting with the identity element at $t=0$). Therefore one can
associate to it a unitary operator, $M(S(t))$, on $L^2(\bbR^n)$ through
the metaplectic representation. A key role will be played by
the metaplectic quantization of $S(T_\gamma)$,
\[
U:=M(S(T_\gamma ))\,.
\]
\begin{theorem}
With the above notations, if $(x,\xi$) belongs to the periodic trajectory
$\gamma$,
\be\label{lopopo}
\sum_j\varphi\left( \frac{E_j(\hbar)-E}{\hbar}\right) \mid(\psi_{(x,\xi)}^a,\psi_j^\hbar)\mid ^2\
\sim \ \sum_k d_k(a) \hbar^{-n+\frac{1}{2}+k}
\ee
with
\begin{equation}\label{lapapa}
d_0\,=\,
\frac{2^{-n/2}}{\sqrt{2\pi}}(2\pi)^{-3n/2}\sum _{l\in\bbZ } \hat\varphi
(lT_\gamma) e^{i(\frac{lS_\gamma}{\h})}
\int e^{-it^2\dot{x}\dot{\xi}/2}
e^{it\eta\dot{x}}\overline{a(\eta)}(U^l a)(\eta -t\dot{\xi})d\eta dt
\end{equation}
(the term $l=0$ is precisely the previous coefficient $c_0$).
Moreover if
$\gamma$ has an infinitesimal
Poincar\'e section invariant by the linearized flow,
and if the Poincar\'e mapping of $\gamma$
is diagonalizable over
$\bbC$ and it has at least one hyperbolic summand, then
as $\h \rightarrow 0$ along
the Bohr-Sommerfeld values $\h=\frac{S_\gamma} {2\pi m}$,
$m=1,2,\cdots $
\begin{eqnarray}
\label{tauber2}
\lefteqn{\sum_{|E_j(\h) - E| \leq c\h}\mid(\psi_{(x,\xi)},\psi_j^\hbar)
\mid ^2 \ = \hbar^{-n+\frac{1}{2}}
\frac{c }{\pi}2^{-n/2}(2\pi)^{-3n/2}
\int e^{-it^2\dot{x}\dot{\xi}/2}e^{it\eta\dot{x}}
\overline{a(\eta)}a(\eta -t\dot{\xi})d\eta dt}\\
& &
\nonumber
+ \hbar^{-n+\frac{1}{2}}\,
\frac{1}{\pi}2^{-n/2}(2\pi)^{-3n/2}\,
\sum _{l\neq 0} \frac{\sin(clT_\gamma)}{(lT_\gamma)}
\int e^{-it^2\dot{x}\dot{\xi}/2}e^{it\eta\dot{x}}\overline{a(\eta)}(U^l a)
(\eta -t\dot{\xi})d\eta dt \\
& &\mbox{}+o(\hbar^{-n+\frac{1}{2}})\nonumber
\end{eqnarray}
where the series above converges absolutely.
\end{theorem}
\medskip
We will now give an interpretation of the coefficient $c_0$. Let
\[
\Omega_\h :=\{ j\,;\, E_j\in[E-c\h,E+c\h]\}\, .
\]
Assuming $E$ is a regular value of $\calH$ and that almost all points
on the energy shell $\calH^{-1}(E)$ are not periodic, the differentiated
Weyl law of \cite{PU1} says that the cardinality of this set is
\[
\sharp\Omega_\h\,=\,\h^{-(n-1)}\,
{\int_{\Sigma_E} d\mu^L\over (2\pi)^n}
+ o(\h^{-(n-1)})\,.
\]
\begin{proposition}
Assume that almost all points
on the energy shell $\calH^{-1}(E)$ are not periodic.
Let $W^a_{(x,\xi)}$ be the Weyl symbol of the orthogonal projector on
$\psi_{(x,\xi)}^a(y)$ (on a manifold this is in a given coordinate system),
that is the Wigner function of $\psi_{(x,\xi)}^a(y)$. Then,
\be
\int_{\Sigma_E} W^a_{(x,\xi)}d\mu^L \ =\
\frac{\h^{-\frac{1}{2}}}{2\pi}(4\pi)^{-n/2}
\int e^{-it^2\dot{x}\dot{\xi}/2}e^{it\eta\dot{x}}
\overline{a(\eta)}a(\eta -t\dot{\xi})d\eta dt + O(\h^{-1/4})\,,
\ee
therefore the first term on the right-hand side of (\ref{13}) is
\be
\left(\int\varphi\right)\,\h\,(2\pi\h)^{-n}\,
\int_{\Sigma_E}W^a_{(x,\xi)}d\mu^L + O(\h^{-n+3/4})\,.
\ee
In particular:
\begin{itemize}
\item If $(x,\xi)$ is not periodic,
\be
\sum_{j\in\Omega_\h}\,
\mid(\psi_{(x,\xi)},\psi_j^\hbar)\mid ^2 \ = \
\sharp\Omega_\h\times\left( \frac{\int W^a_{(x,\xi)}d\mu^L}
{\int d\mu^L}+o(\hbar^{-\frac{1}{2}})\right) \,.
\ee
\item
If $(x,\xi)\in\gamma$ and $\gamma$ satisfies the assumptions of Theorem 1.2,
$\exists \nu\in\bbR$ generically non-zero such that
\be
\sum_{j\in\Omega_\h}\,
\mid(\psi_{(x,\xi)},\psi_j^\hbar)\mid ^2 \ =\
\sharp\Omega\gamma\times\left( \frac{\int W^a_{(x,\xi)}d\mu^L}
{\int d\mu^L}+\nu \hbar^{-\frac{1}{2}}+ o(\hbar^{-\frac{1}{2}})\right)
\ee
for $\h$ of the form $\h= \frac{S_\gamma} {2\pi m}$, as before.
\end{itemize}
\end{proposition}
The next result is an immediate consequence of the preceding and show a little
more precisely the role of periodic trajectories on the pointwise behavior of
semiclassical measures.
\begin{corollary}
\begin{itemize}
\item Assume $(x,\xi)$ is not periodic. Then
$\forall\epsilon>0$ there exists a subsequence
$\{E_{j_k}\}\subset\Omega_\h$ of positive density such that,
for $\h$ small enough,
\be
\mid(\psi_{(x,\xi)},\psi_j^\hbar)\mid ^2 \ \leq\ \h^{-\frac{1}{2}}(
\frac{\int W^a_{(x,\xi)}d\mu^L}
{\int d\mu^L}+\epsilon)\,.
\ee
\item Assume $(x,\xi)\in\gamma$ with $\gamma$ an unstable trajectory,
and suppose moreover that
\be
\frac{1}{\pi}2^{-n/2}(2\pi)^{-3n/2}\,
\sum _{l\neq 0} \frac{\sin(clT_\gamma)}{(lT_\gamma)}
\int e^{-it^2\dot{x}\dot{\xi}/2}e^{it\eta\dot{x}}
\overline{a(\eta)}(U^l a)(\eta -t\dot{\xi})d\eta dt =:\ b > 0
\ee
(this is true in the some of the Gaussian examples of \S 6). Then
$\forall\epsilon>0$ there exists a subsequence
$\{ E_{j_k}\} \subset\Omega_\h$ of positive density such that, for $\h$
small enough and of Bohr-Sommerfeld type,
\be
\mid(\psi_{(x,\xi)},\psi_j^\hbar)\mid ^2 \ \geq\ \h^{-\frac{1}{2}}(
\frac{\int W^a_{(x,\xi)}d\mu^L}
{\int d\mu^L}+\frac{b}{2c\int d\mu^L}-\epsilon)\,.
\ee
\end{itemize}
\end{corollary}
\medskip
We will now give a coordinate-free interpretation of the integral appearing in
the $l$-th term in (\ref{lapapa}), $\forall l\in\bbZ$.
\renewcommand{\mp}{\mbox{Mp }}
\smallskip
Consider the symplectic vector space $V = T_{(x,\xi)}(T^*M)$; let
$\mp (V)$ denote the metaplectic group of $V$, double cover of the
group of linear symplectic transformations of $V$.
Choosing a coordinate system near $x$, we get naturally-induced coordinates
on $T^*M$ and therefore linear
coordinates $(\delta x,\eta)$ on $V$.
Given a coherent state $\psi^a_{(x,\xi)}$ we claim that, intrinsically,
the function $a(\eta)$ should be thought of as
a smooth vector in the
metaplectic representation of $\mp (V)$ (see \S 2).
Let $\rho$ be the Weyl representation of the Heisenberg
group of $V$. Explicitly, acting on functions of $\eta$,
$d\rho$ is the representation
\be\label{waltz}
\left\{
\begin{array}{rcl}
d\rho(\delta x_j)(a)(\eta)&=& \eta_j a(\eta)\\
d\rho(\eta_j)(a)(\eta) &=& -i{\partial \over\partial\eta_j}a(\eta)\,.
\end{array}
\right.
\ee
Then the integral
\be\label{ave}
\int e^{-it^2\dot{x}\dot{\xi}/2}e^{it\eta\dot{x}}
U^l (a)(\eta -t \dot{\xi}) dt
\ee
is the projection of $U^l(a)$ onto the space of tempered distributions
$f$ satisfying
\[
d\rho (\Xi) (f)\,=\,0\,,
\]
where
\be\label{Xi}
\Xi\,=\,(\dot{x},\dot{\xi})
\ee
is the vector tangent to the trajectory at $(x,\xi)$. Indeed the operator
\[
\calG\,:\,\calS (\bbR^n) \ni b\mapsto
\bigl(\,\eta\mapsto \int_{-\infty}^\infty
e^{-it^2\dot{x}\dot{\xi}/2}e^{it\eta\dot{x}}
b(\eta -t \dot{\xi})dt\,\bigr)
\in\calS'(\bbR^n)
\]
is precisely $\int \rho(t\exp(\Xi))dt$ (the integral should be understood
in the weak sense). With this notation, the integral
appearing in the $l$-th term in (\ref{lapapa}) is
\[
(\overline{a}, \calG(U^l(a)))\,.
\]
where the outer parenthesis denote the pairing between $\calS (\bbR^n)$
and $\calS' (\bbR^n)$.
\medskip
\noindent
\underline{The Riemannian case.}
\smallskip
\par\noindent
We finish this introduction by observing that the previous results apply
in particular to the large eigenvalue asymptotics of the eigenfunctions
of the Laplacian on a compact Riemannian manifold.
Let $M$ be a Riemannian manifold, $\Delta$ the (negative)
Laplacian on $M$ and $\lambda_j,\psi_j$ the eigenvalues
and eigenfunctions of $\Delta$. Instead of working with $\h^2\lambda_j$
it is customary in the Riemannian context
to work with the square roots of the eigenvalues
\[
\mu_j\,=\,\sqrt{-\lambda_j}\,,
\]
which can be done with trivial modifications to the proof.
Pick $(x, \xi)\in S^*M$, the unit cotangent bundle of $M$,
periodic with respect to geodesic flow.
Let $(r,s)$, $r=(r_1,\ldots r_{n-1})$ be Fermi coordinates
in a neighborhood of $x$, adapted to the geodesic $\gamma$
through $(x,\xi)$. Thus if $(r,s,\varrho,\sigma)$ are the
coordinates induced on $T^*M$, locally the geodesic $\gamma$ is
the parametrized curve
$\{ r = 0\,, \varrho = 0\,, s = t\,, \sigma = 1\}$.
Let $V$ be the tangent space to $T^*M$ at $(x,\xi)$.
The coordinates $(r,s,\varrho,\sigma)$ induce linear coordinates
$(\delta r, \delta s, \eta_r, \eta_s)$ on $V$,
and in these coordinates the vector $\Xi_{(x,\xi)}$ is
$(0,\ldots ,1,0,\ldots 0)$ ($1$ in the $n$-th entry).
Recall the interpretation given above
of the coefficient (\ref{lapapa}).
We were led to consider the operator
$ \calG \,=\,\int_{-\infty}^\infty\,\rho(\exp (t\Xi))\,dt$
mapping Schwartz functions of the variable $\eta = (\eta_r, \eta_s)$
to tempered distributional functions of $\eta$, where $\rho$ is the Weyl
representation of the Heisenberg group of $V$, (\ref{waltz}).
Since $d\rho (\Xi) = i\eta_s$
in the present case, the operator $\calG$ is basically
multiplication by the Dirac delta of $\eta_s$
\[
\calG (a) (\eta)\,=\,2\pi\delta(\eta_s)\,a(\eta)\,.
\]
Therefore, our main theorem (in the periodic case) takes on the following
form:
\begin{theorem}
Assume the geodesic through $(x,\xi)$ is periodic, of primitive
length $L$.
Let $\chi$ be a smooth cut-off function in the Fermi coordinate chart.
Then, for every test function $\varphi$ with Fourier transform
in $C_0^\infty(\bbR)$, and every Schwartz function $a\in\calS (\bbR^n)$
\be\label{waw}
\sum_j\,\varphi(\mu_j-\tau)\,
\left| \int \overline{\psi_j(r,s)}\,e^{i\tau s}\,
\hat{a}(\sqrt{\tau}r,\sqrt{\tau}s)\,\chi(r,s)drds \right|^2
\,=
\ee
\[
=\,\tau^{\frac{n}{4}-\frac{1}{2}}\,
2^{1-\frac{n}{4}}(2\pi)^{-3n/4}\sum _{l\in\bbZ} \hat\varphi
(lL)\, e^{-i\tau lL }
\int \overline{a(\eta_r,0)}\,(U^l a)(\eta_r,0)\,d\eta_r +
O(\tau^{\frac{n}{4}-1})
\]
as $\tau\to\infty$ (one has in fact a full asymptotic expansion
in powers of $\sqrt{\tau}$).
\end{theorem}
We won't bother to formally state the formula regarding
\[
\sum_{j; |\mu_j -\tau|\leq c}
\left| \int \overline{\psi_j(r,s)}\,e^{i\tau s}\,
\hat{a}(\sqrt{\tau}r,\sqrt{\tau}s)\,\chi(r,s)drds \right|^2
\]
in case $\gamma$ is unstable.
\newcommand{\calJ}{{\cal J}}
\medskip
Formula (\ref{waw}) simplifies for certain choices of test functions $a$,
as we will now see.
Recall that the operator $U$ is the metaplectic quantization of the
differential of geodesic flow at $(x,\xi)$, at time $L$.
Such a differential leaves invariant both
$\Xi$ and the radial direction in $T^*M$.
Those two directions span a symplectic subspace $V_1$ of $V$.
Let $V_2$ be the symplectic orthogonal to $V_1$. Then
the differential of the flow preserves this decomposition of $V$; it
is the identity on $V_1$ and the linearized Poincar\'e map on $V_2$.
Accordingly, it is natural to consider Schwartz functions $a$ of the form
\be\label{fac}
a(\eta)\,=\, e^{-\eta_s^2/2}\,a_1(\eta_r)\,.
\ee
On such an $a$, the operator $U$ has the form
\be\label{UU}
U(a)(\eta)\,=\, e^{-\eta_s^2/2}\,U_P(a_1)(\eta_r)
\ee
where $U_P$ is the metaplectic quantization of the linearized Poincar\'e
map of $\gamma$. On such amplitudes, our formula becomes
\be\label{waw2}
\sum_j\,\varphi(\mu_j-\tau)\,
\left| \int \overline{\psi_j(r,s)}\,e^{\tau( is-s^2/2)}\,
\hat{a_1}(\sqrt{\tau}r)\,\chi(r,s)drds \right|^2
\,=
\ee
\[
=\,\tau^{\frac{n}{4}-\frac{1}{2}}\,
2^{1-\frac{n}{4}}(2\pi)^{-3n/4}
\sum _{l\in\bbZ} \hat\varphi
(lL)\, e^{-i\tau lL }
\int \overline{a_1(\eta_r)}\,(U_P^l a_1)(\eta_r)\,d\eta_r +
O(\tau^{\frac{n}{4}-1})\,.
\]
This can be simplified further. Recall that the Fourier transform
$\calJ$
is the metaplectic quantization of a certain element $J$ of the
metaplectic group such that $J^2=-I$. Let
\be\label{bab}
\calM\,:=\,\calJ\,U_P\,\calJ^{-1}\,.
\ee
This is nothing but the metaplectic quantization of the Poincar\'e map
of $\gamma$ on $L^2$ functions of the variable $\delta r$.
Since $\calJ$ is unitary, we have:
\begin{corollary}
In the Riemannian context described above, for every Schwartz function
$b\in\calS(\bbR^{n-1})$ and every test function $\varphi$ with Fourier
transform in $C_0^\infty(\bbR)$, we have
\[
\sum_j\,\varphi(\mu_j-\tau)\,
\left| \int \overline{\psi_j(r,s)}\,e^{\tau( is-s^2/2)}\,
b(\sqrt{\tau}r)\,\chi(r,s)drds \right|^2
\,=
\]
\[
=\,\tau^{\frac{n}{4}-\frac{1}{2}}\,
2^{1-\frac{n}{4}}(2\pi)^{-3n/4}
\sum _{l\in\bbZ} \hat\varphi (lL)\, e^{-i\tau lL }
\int \overline{b(r)}\,\calM^l(b)(r)\,dr + O(\tau^{\frac{n}{4}-1})\,.
\]
\end{corollary}
Although we won't go into details here, we mention that
the operator $\calM$ can be computed in
terms of the transverse Jacobi fields of $\gamma$.
\medskip
The paper is organized as follows: sections 2 and 3 deal with
propagation of coherent states, section 4 contains the proof of the
Theorems 1.1 and 1.2 and section 5 the proof of additional results. In section
6 we treat the case of Gaussian symbols and show that the elliptic case
gives rise to "Poisson formulae". We conclude in section 7 by a
discussion of the results.
\bigskip\noindent
\section{Coherent states and Hermite distributions}
Let $S$ a Riemannian manifold. In \cite{BG} (see also \cite{G2}) Boutet
de Monvel and Guillemin associate to any conic isotropic manifold
$\Gamma$ in $T^*(S)$ a family of distributions on $S$ whose wave-front
sets are included in $\Gamma$. These distributions have symbols that
are symplectic spinors on $\Gamma$. We will concentrate in this paper
in the case where $S=M\times\bbR$, with $M$ an $n$-dimensional
Riemannian manifold ($M$ might be $\bbR^n$) and $\Gamma$ is one
dimensional. We will work on a local system of coordinates, but, by the
theory of Hermite distributions, the main results are independent of
any choice of coordinates.
\medskip
We begin by briefly recalling the definition of Hermite
distributions as it applies to the present setting.
Let $a(x,\tau,\eta)\in
C^{\infty}(M\times\bbR^+\times\bbR^n)$
compactly supported in $x$ and rapidly decreasing in $\eta$
admitting, as $\tau\to\infty$, an asymptotic expansion of the following form:
\be\label{2.1}
a(x,\tau,\eta) \ \sim \ \sum_{j=0}^{\infty} \tau^{-j/2}a_j(x,\eta)\,,
\ee
where $\forall j$ the function $a_j$ is in the class
$C_0\calS(\bbR^n\times\bbR^n)$ defined as follows:
\begin{definition}
We'll denote by $C_0\calS(\bbR^n\times\bbR^n)$ the set of all smooth functions
$a(x,\eta)$ that are compactly supported in $x$ and satisfy:
$\forall K,M,N$ non-negative integers $\exists C_{KMN}>0$ such that
\be\label{c0s}
\forall (x,\eta)\in\bbR^n\times\bbR^n\qquad
\mid \eta^K\partial^M_x\partial^N_\eta a\mid \leq C_{KMN}\,.
\ee
\end{definition}
(For the precise meaning of (\ref{2.1}) see \cite{BG} \S 3.)
\medskip
For $(x,\xi,\alpha)$$\in$$ T^*(M)\times\bbR$, let $I_{x,\xi,\alpha}^a$
be the distribution defined locally by the oscillatory integral
\be
I_{x,\xi,\alpha}^a(y,\theta)\ = \ \int_{R^n\times R^+} (2\pi)^{\frac{3n}{2}}
\tau^{\frac{n}{4}}
e^{-i\tau(\theta+\alpha-\xi
y)}e^{i\eta(y-x)}a(y,\tau,\frac{\eta}{\sqrt\tau})d\eta d\tau\,.
\ee
By definition this is an Hermite distribution associated to
\be\label{Gamma}
\Gamma \ = \
\{(\theta,y;\tau,p)\; ;\; \theta = -\alpha+y\cdot\xi, y=x, p=\tau \xi\}\,.
\ee
In particular:
\begin{proposition}
The wave front of $I_{x,\xi,\alpha}^a$ is contained in $\Gamma$.
\end{proposition}
To $a,x,\xi$ we associate as in \cite{PU1} the following family of
functions on $M$ (actually the inverse
Fourier transform of $I_{x,\xi,\alpha =\frac{x\xi}{2}}^a$)
\be
\psi^a_{(x,\xi)}(y) \ = \ (2\pi)^{-n}\tau^{\frac{n}{4}}
e^{-i\tau(\frac{x\xi}{2}-\xi y)}
\int_{R^n }
e^{i\eta(y-x)}a(y,\tau,\frac{\eta}{\sqrt\tau})d\eta \,.
\ee
\begin{definition}
The family
$\{ \psi^a_{(x,\xi)}\}_\tau$ is called a coherent state or wave packet
centered at $(x,\xi)$ and of symbol $a$.
\end{definition}
Each
$\psi^a_{(x,\xi)}$ is a compactly supported $C^\infty$ function.
As we showed in \cite{PU1}, the previous proposition implies
the following:
\begin{proposition}
The frequency set (or micro-support) of $\psi^a_{(x,\xi)}$ is $\{(x,\xi)\}$.
\end{proposition}
Coherent states are localized in space around $x$ to the
extent that, to leading order, the $y$ dependence of the
amplitude $a$ can be suppressed. This fact will be very
useful in what follows, and it reconciles the definition above with
the preliminary one introduced in \S 1:
\begin{proposition}
Let $a(y,\tau,\eta)$ be an Hermite amplitude satisfying
(\ref{2.1}). Let $\rho$ be any $C_0^\infty$ function identically
one near the origin in $\bbR^n$. Then
\be\label{2.2}
\psi^a_{(x,\xi)}(y) \,=\, \rho(y-x)\, (2\pi)^{-n}\tau^{\frac{n}{4}}\,
e^{-i\tau(\frac{x\xi}{2}-\xi y)}
\int_{\bbR^n}\,
e^{i(y-x)\eta}\,
a_0(x,\eta/\sqrt{\tau})\,d\eta\; +\; O(\tau^{n/4-1/2})\,,
\ee
uniformly on compact sets.
\end{proposition}
\begin{proof}
By the estimates (\ref{2.1}), it is easy to see that one has
(\ref{2.2}) with the right-hand side replaced by
\be
\tau^{-n/4}\,\rho(y-x)\,e^{-i\tau(\frac{x\xi}{2}-\xi y)}
\int_{\bbR^n}\,e^{i(y-x)\eta}\,
a_0(y,\eta/\sqrt{\tau})\,d\eta\,.
\ee
To go from here to (\ref{2.2}), we do a Taylor expansion of
$a$ in $y$ near $y=x$ and we integrate by parts.
Specifically, write
\[
a_0(y,\eta)\,=\,a_0(x,\eta) + \sum_{j=1}^n\,(y_j-x_j)g_j(y,\eta)
\]
where the $g_j$ are smooth functions. We must estimate, $\forall j$,
\[
I_j(y,\tau)\,=\,
\int e^{i(y-x)\eta}\,(y_j-x_j)g_j(y,\eta/\sqrt{\tau})\,d\eta\,.
\]
Evidently
\[
I_j\,=\, i \int e^{i(y-x)\eta}\,
{\partial\over\eta_j}(g_j(y,\eta/\sqrt{\tau}))\,d\eta
\]
\[
= {i\over \sqrt{\tau}} \int e^{i(y-x)\eta}\,
{\partial g_j\over\eta_j}(y,\eta/\sqrt{\tau})\,d\eta\,,
\]
which is clearly $O(\tau^{(n-1)/2})$ uniformly on compacts.
\end{proof}
\noindent
{\bf Observation:} If $b$ is the inverse Fourier transform of the
Schwartz function $\eta\mapsto a_0(x,\eta)$, then in a
neighborhood of $x$ the coherent state above equals
\be
\psi^a_{(x,\xi)}(y)\,=\, (2\pi)^{n}\,\tau^{\frac{n}{4}}\,\rho(y-x)\,
e^{-i\tau( {x\xi\over 2}-\xi y)}\,
b(\sqrt{\tau} (y-x))\quad \mbox{mod }O(\tau^{n/4-1/2})\,.
\ee
\begin{corollary}
\be
||\psi^a_{(x,\xi)}||^2_{L^2(M)} \ = \ (2\pi\h)^{-n}||a_0(x,.)||^2_{L^2(\bbR^n)}\ +\
\mbox{O}(\hbar^{-n+1}).
\ee
Moreover,
if $(x,\xi)\neq(x',\xi')$,
\be
(\psi^a_{(x,\xi)},\psi^{a'}_{x',\xi'})_{L^2(M)}\ =\ \mbox{O}(\hbar^\infty),
\ee
and for any compact $\Omega$ containing $x$
\be
(\psi^a_{(x,\xi)},\psi^{a'}_{x,\xi})_{L^2(M)}\ - \int_\Omega \overline
{\psi^a_{(x,\xi)}(y)}\psi^{a'}_{x,\xi}(y)dy\ =\ \mbox{O}(\hbar^\infty)\,.
\ee
\end{corollary}
\bigskip\noindent
\renewcommand{\L}{\ell}
\section{Semi-classical propagation of coherent states}
Let
\be
\psi_{(x,\xi)}^a(y)\ =\ (2\pi)^{-\frac{3n}{4}}(2\pi\h)^{-\frac{n}{4}}\rho(y-x)
e^{-ix\xi/2\h}e^{i\xi y/\h}\int a(\sqrt\h \eta)e^{i\eta (y-x)}d\eta
\ee
be a coherent state at $(x,\xi)$.
The next theorem shows how such a state evolves under the Schr\"odinger
equation. Let
\be
H_\h\, =\, \sum_{l=0}^{L}\hbar^lP_l(x,D_x)\, ,\qquad
\calH(x,\xi)\,=\, \sum_{l=0}^{L} P^0_l(x,\xi)\,,
\ee
and $S(t)$ be as in \S 1.
$S(t)$ is the matrix of the differential of the Hamilton flow in
coordinates. The associated linear transformation is symplectic and
maps the tangent vector to the trajectory at $(x(0),\xi(0))$ to the one at
$(x(t),\xi(t))$. Since $S(0)$ is the identity,
one can naturally lift the $S(t)$ to the metaplectic group, $Mp(\bbR^n)$
in a continuous way, starting at the identity.
We will continue to denote the lift by $S(t)$. Let $M(S(t))$ be
the family of unitary operators image of $S(t)$ by the metaplectic
representation.
\medskip
The following result shows that after evolution a coherent state remains
a coherent state and gives the leading term of the symbol.
\begin{theorem}
Under the previous assumptions, $\forall t\in\bbR$ there exists a symbol
$a(t)$ of the form (\ref{2.1}) depending smoothly in $t$ and such that
\be
e^{-i\frac{tH_\h}{\h}}\left(\psi_{(x,\xi)}^a\right)
\ = \ e^{i\L(t)/\h} \psi_{(x(t),\xi(t))}^{a(t)}
\quad \mbox{mod}(\h^\infty)
\ee
uniformly on each compact in $(t,x)$-space. Here
\[
\L(t) \ = \ \int_0^t\left(\frac{\xi\dot x -x\dot\xi}{2}-\calH(x,\xi)\right)dt\,.
\]
Moreover, the leading term of $a$ evolves according to
\be\label{3.1b}
a_0(t) \ = \ M(S(t))\,\left( a_0|_{t=0}\right)\,.
\ee
\end{theorem}
\medskip
The proof of this theorem will be based on the theorem of
propagation of Hermite distributions by Fourier integral operators,
namely Theorem 7.5 in \cite{BG}.
We will consider the distribution on
M$\times S^1$ whose Fourier coefficients are precisely the l.h.s of
Theorem 3.1, with $\h = 1/(m+c)$.
This distribution satisfies a certain equation, which we analyze.
Then we will show that the solution of this equation is
microlocally equal, in the region of interest,
to an Hermite distribution whose Fourier
coefficients are given by the right-hand side
of the Theorem. Finally we will
identify the symbol of this Hermite distribution. These ideas have been
used in the compact case in \cite{PU1}, but we give an independent
proof.
\medskip
To $H_\h$ we associate the following family of operators on $M\times S^1$
\be
A\,=\,\sum_{l=0}^{L}D_\theta^{-l+L}P_l(x,D_x)
\ee
where $D = D_\theta + c$, $c\in[0,1]$ a parameter.
$A$ is a differential operator of order $L$.
\medskip
We break the proof in a series of lemmas. The first one is straightforward:
\begin{lemma}
Let
\be
\psi_\tau^t(y)\ =\ e^{-it\tau H_{\tau^{-1}}}\psi_{(x,\xi)}^a(y)
\ee
and let
\be
u(t,y,\theta)\ =\ \sum_{\tau=1}^\infty
e^{i\tau\theta}\psi_{\tau + c}^t(y)\,.
\ee
Then $u(t)$ is a distribution that satisfies
\be\label{3.1}
D^{L-1}D_t u\ =\ Au
\ee
with initial condition
\be\label{3.2}
u|_{t=0}\ =\ I_{x,\xi,\frac{x\xi}{2}}^a\,.
\ee
\end{lemma}
Next, we use (\ref{3.1}, \ref{3.2}) to control the wave-front
set of $u$.
\begin{lemma}
Let $u$ be a distribution on $\bbR\times M\times S^1$ satisfying
(\ref{3.1}) and (\ref{3.2}). Then
\begin{eqnarray}
\nonumber\lefteqn{\wf (u)\,=\, }\\
& & \{\,(t,y,\theta;\epsilon,\eta,\tau)\;;\;
\tau\not= 0\,,\; (y,\eta/\tau)=
\phi_t(x,\xi/\tau)\,,\;\theta=-{x\xi\over 2}+\ell(t)\,,\;\epsilon = \tau
\calH(x,\xi/\tau)\,\}
\label{front}
\end{eqnarray}
\end{lemma}
\begin{proof}
Observing that $u$ satisfies the differential equation
$(D^{L-1}D_t - A)(u)=0$, one knows that the wave-front set
of $u$ is contained in the characteristic variety of the operator
\[
Q\,:=\, D^{L-1}D_t - A\,,
\]
which is the set
\be
\mbox{Char }(Q)\ =\
\{(t,y,\theta;\epsilon,\eta,\tau)\;;\;
\tau^{L-1}\epsilon = \sum_{l=0}^L\tau^{L-l}P_l^0(y,\eta)\,\}\,.
\ee
Observe that
\be\label{cosa}
\mbox{Char }(Q)\cap\{\,\tau\not=0\,\}\ =\
\{(t,y,\theta;\epsilon,\eta,\tau)\;;\;
\epsilon/\tau = \calH(y,\eta/\tau)\,\}\,.
\ee
Since the principal symbol $\sigma_Q$
of $Q$ and the function $\tau$ Poisson commute, it follows from
(\ref{cosa}) that the null-bicharacteristic strips of $Q$
in the region $\{\tau\not= 0\}$ are the same as the
trajectories of the Hamilton flow of the function
$ \calH(y,\eta/\tau)-\epsilon/\tau$.
We also know that the wave-front set of $u$ is invariant under the
Hamilton flow of the principal symbol of $Q$ on $T^*(M\times S^1)$.
In the region $\{\tau\not=0\}$ that Hamilton flow is, up
to a rescaling, the Hamilton flow of $\calH$. From this,
using the fact that the initial condition has wave-front in the set
$\Gamma$ of (\ref{Gamma}) and the calculus of wave-front sets,
one can show that the wave-front set of $u$
is in fact (\ref{front}).
\end{proof}
\begin{lemma}
Let
\[
\widehat{\Sigma_E} \ =\
\{(t,y,\theta;\epsilon, \eta,\tau)\in T^*(M\times S^1)\;;\;
\tau\not=0\,,\;
\epsilon/\tau = \calH(y,\eta/\tau)\,\}\,.
\]
Then there exists a conic neighborhood of $\widehat{\Sigma_E}$, $\Omega$,
contained in $T^*(\bbR\times M\times S^1)\setminus\{\tau\not= 0\}$,
and a classical, first-order pseudodifferential
operator on $M\times S^1$, $B$,
such that $\hat{Q}:=D^{L-1}D_t - D^{L-1} B$
and $Q$ are microlocally equal on $\Omega$. Moreover
$[B,D_\theta]= 0$.
\end{lemma}
\begin{proof}
It suffices to construct a first order pseudodifferential
operator on $M\times S^1$ commuting with $D_\theta$ and such
that $A$ and $D^{L-1}B$ are microlocally equal in a neighborhood
of the set
\[
\{(y,\theta; \eta,\tau)\in T^*(M\times S^1)\;;\;
\tau\not=0\,,\;
\calH(y,\eta/\tau)= E \,\}\,.
\]
\end{proof}
The following lemma is truly the heart of the proof:
\begin{lemma}
Let $u(t)$ a solution of (\ref{3.1}) and (\ref{3.2}). Then there exists
an Hermite distribution $I_{x(t),\xi(t),\L(t)}^a(t)$ such that:
\be
\left( u(t)-I_{x(t),\xi(t), \L(t)}^a(t)\right) \ \in C^{\infty}\,.
\ee
Here $(x(t),\xi(t))=\phi_t(x,\xi)$ is the trajectory of $(x,\xi)$ under the
Hamilton flow of $\calH(x,\xi)$ and $\L(t)$ is as in Theorem 3.1 .
\end{lemma}
\begin{proof} On the one hand, by the previous lemmas,
\[
g\,:=\, \hat{Q} (u) \in C^\infty\,.
\]
Therefore, the distribution $u$ can also be described as the
solution of the pseudodifferential equation
\begin{eqnarray}
\hat{Q} (u) & = & g,\\
u|_{t=0} & = & I_{x,\xi, {x \xi\over 2}}^a\,.
\end{eqnarray}
On the other hand, one can construct a
Fourier integral operator $U(t)$ solving:
\begin{eqnarray}
D_tU(t)\ & =& \ B U(t)\ \mbox{mod } C^{\infty},\nonumber \\
U(0)\ & =& \ \mbox{identity}
\end{eqnarray}
even in case $M$ is non-compact, if in the equations above
we restrict $|t|$ to be bounded and we restrict $U(t)$
to act on functions supported in a compact set
(see for example \cite{TR}). In terms of such $U$,
\be\label{duham}
u(t)\,=\, \int_0^t\,U(t-s)(f)\,ds + U(t)(I_{x,\xi, {x \xi\over 2}}^a)\,,
\ee
where $f$ is a smooth function satisfying
\be\label{effe}
D^{L-1}f\,=\, g\,.
\ee
Such an $f$ indeed exists; observe that therefore the first summand on
the right-hand side of (\ref{duham}) is smooth.
\smallskip
To conclude the proof of the lemma we will use the theorem
of propagation of
Hermite distributions trough Fourier integral operators.
\medskip
$I^a_{x,\xi,\frac{x\xi}{2}}$ is an Hermite distribution associated to
$\Gamma =\{(\theta = -\frac{x\xi}{2},y=x;\tau,
p=\tau\xi)\}\subset\widehat{\Sigma_E}$. $U(t)$ is a Fourier
integral operator associated to the flow generated by the principal symbol
of $B$, which equals $\calH(x,\xi/\tau)$ in a neighborhood of $\Gamma$.
An easy calculation of the Hamilton equations
(see \cite{PU1} for details) shows that this flow maps $\Gamma$ into
$\Gamma(t)=\{ (\theta = -\frac{x\xi}{2}+\L(t),y=x(t);\tau, p=\tau\xi(t))\}$.
By the already cited theorem of propagation of \cite{BG},
$U(t)(I_{x,\xi,\frac{x\xi}{2}}^a)$ is an Hermite distribution associated
to $\Gamma(t)$. Since by (\ref{duham}) $u(t)$ is equal to it modulo
a smooth function, the proof is finished.
\end{proof}
\bigskip
To finish the proof of the first part of the Theorem just note that by the
previous Lemma the differences
\[
\psi_\tau^t - e^{i\tau\L(t)}
\psi_{(x(t),\xi(t))}^{a(t)}
\]
(with $\tau=1,2,\ldots$ are the Fourier coefficients in $\theta$
of a smooth function on $M\times S^1$; therefore they are rapidly
decreasing in $\tau$ uniformly on $x$ in compacts.
This proves the first part of the lemma for values of
$\h$ along the values $\h=1/(m+c)\,,\;m=1,2,\ldots$. It is
clear however that the estimates must be uniform in $c$, and
therefore we get the desired conclusion as $\h \to 0$ continuously.
\bigskip
Finally, we sketch the computation of the principal symbol.
A straightforward computation shows that
\be
H(\psi_{(x,\xi)}^{a})\ =\ \psi_{(x,\xi)}^{a'} + O(\tau^{-3/2})
\ee
with
\be
a'\ =\ \calH(x,\xi)a\ +\ \tau^{-1/2}(\nabla_xh.D_\eta -\nabla_\xi h .\eta)a\ +\
\tau^{-1}\frac{i}{2}
(D_\eta,\eta)\mbox{Hess }\calH(x,\xi)(D_\eta,\eta)a\,,
\ee
where
\be\label{hesse}
(D_\eta,\eta)\mbox{Hess }\calH(x,\xi)(D_\eta,\eta)\,=\qquad { }
\ee
\[
{ }
=\quad \sum_{j,k=0}^n{\partial^2 \calH\over\partial{x_jx_k}} D^2_{\eta_j\eta_k} +
{\partial^2 \calH\over\partial{x_j\xi_k}}(\eta_jD\eta_k+\eta_kD\eta_j) +
{\partial^2 \calH\over\partial{\xi_j\xi_k}}\eta_j\eta_k
\]
is the Weyl quantification of the Hessian of $\calH$.
Moreover,
\be
\tau^{-1}D_te^{i\tau\L(t)}\psi_{(x(t),\xi(t))}^{a(t)}\ =\ e^{i\tau\L(t)}
\psi_{(x(t),\xi(t))}^{a'(t)} + O(\tau^{-3/2})
\ee
with\
\be
a'(t)\ =\ \dot\L(t)\ -\ \tau^{-1/2}(\dot\xi D_\eta -\dot x\eta)a\ +\
\tau^{-1}\dot a\,.
\ee
Identifying term by term gives:
\be
\dot a_0\ =\ \frac{i}{2}(D_\eta,\eta)\mbox{Hess }\calH(x,\xi)(D_\eta,\eta)a_0\,.
\ee
We want to prove that a solution of this equation is given by
\be
a_0(t)\ =\ M(S(t))a_0(0)\,.
\ee
We can easily compute $\frac{d}{dt}{M(S(t))}$, \cite {LJ}:
\[
M(S(t+\delta t))=M(S(t)+\delta t\dot S(t) +O(\delta t^2)=M((1+\delta t\dot SS^{-1})S)
=M(1+\delta t\dot SS^{-1})M(S(t))\,.
\]
Now remark that $1+\delta t\dot SS^{-1}$ is the flow of Hamiltonian
$(x,\xi)^tJ\dot SS^{-1}(x,\xi)$ modulo $\delta t^2$. This implies that
\be
M(1+\delta t \dot SS^{-1})\ =\ e^{i\delta t(D_\eta,\eta)J\dot SS^{-1}
(D_\eta,\eta)}+O(\delta t^2) \,,
\ee
so
\be
\frac{d}{dt}{M(S(t))}\ =\ i(D_\eta,\eta)J\dot SS^{-1}(D_\eta,\eta)M(S(t))\,.
\ee
Identifying once more gives the equation for $S$.
\medskip
{\bf Remark:}
It is possible to get the same result directly from the symbolic properties of
Hermite distributions. Indeed the family of Hermite distributions constructed
before can be considered, taking $t$ as a variable,
as an Hermite distribution on $\bbR\times M\times S^1$ associated with the
flow-out of $\Gamma$ by the principal symbol of
the operator $(D_\theta^{L-1}D_t\ -\ A)$. This new
Hermite distribution is in the kernel of the operator
$(D_\theta^{L-1}D_t\ -\ A)$ whose symbol gives rise to a flow
tangent to $\Lambda$. The transport equation, see \cite{BG}, gives
exactly the result above. We leave this proof to another paper where the case
of a general Hermite distribution will be considered.
\newcommand{\ip}{{1\over 2\pi}}
\bigskip
\noindent
\underline{The Riemannian Case}
\nobreak
\par\smallskip
\noindent
For the Riemannian case discussed at the end of \S1 one also needs
a Theorem of propagation of wave packets, which is actually easier
to prove; we will be sketchy. Let $M$ be a compact Riemannian manifold,
and $\Delta$ the (negative) Laplacian on $M$.
The definition of coherent states in the Riemannian case is the
same as that of \S 2. Consider the operator
on $M\times S^1$
\[
P\,:=\,\sqrt{-\Delta + D^2}\,.
\]
This is of course a standard first-order $\Psi$DO, and the required
Theorem of propagation of wave packets is a consequence of the result that
the Fourier integral operator,
$\exp(itP)$, maps Hermite distributions to Hermite distributions.
\section{Proof of Theorems 1.1 and 1.2.}
Fix $(x,\xi)\in T^*M$, not in classical equilibrium, and
$a\in\calS(\bbR^n)$.
Denoting $\h^{-1}$ by $\tau$, we have:
\begin{eqnarray}
\sum_j\varphi\left(\frac{E_j(\hbar)-E}{\hbar}\right)
\mid(\psi_{(x,\xi)}^a,\psi_j^\hbar)\mid ^2\
= \ {1\over {\sqrt{2\pi}}}\,
\int \hat \varphi(-t)\,(\psi^a_{(x,\xi)},e^{-it\tau(H-E)}\psi^a_{(x,\xi)})\,dt
\nonumber \\
= \ {1\over {\sqrt{2\pi}}}\,\int \hat \varphi(-t)\,e^{i\tau\int^t_0(\frac{\xi\dot x
-x\dot\xi}{2}-\calH(x,\xi)+E)dt}\,
(\psi^a_{(x,\xi)},\psi^{a(t)}_{x(t),\xi(t)})\, dt + O(\h^\infty)\,.
\label{basic}
\end{eqnarray}
For simplicity we will take
$a(t)= a(t,y,\eta)$ to be the leading term in the
expansion (\ref{2.1}) of the symbol of Theorem 3.1, and therefore
it evolves according to the equation (\ref{3.1b});
higher order terms are treated identically.
\smallskip
By Corollary 2.5 we know that
$(\psi^a_{(x,\xi)},\psi^{a(t)}_{x(t),\xi(t)})$
is $O(\hbar^\infty)$ if $(x,\xi)\neq(x(t),\xi(t))$, so in (\ref{basic})
it is enough to
integrate over intervals around the periods $lT_\gamma$ of $\gamma$ if
$(x,\xi)$ is periodic and only around zero if it is not.
The analysis is therefore localized to a neighborhood of $x$, which
enables us to work in a fixed local coordinate system.
Let $\beta(t)$ be a cut-off function around zero , and let
$f_l(t)=\beta(t-lT_\gamma)\hat{\varphi}(t)$ (only the term $l=0$ arises
in the non-periodic case).
Since $\calH(x,\xi)=E$, we must estimate the integrals
\be\label{sarabande}
I_l(\tau)\, =\, \int f_l(t)\, e^{i\tau\int^t_0\frac{\xi\dot x -x\dot\xi}{2}}\,
(\psi^a_{(x,\xi)},\psi^{a(t)}_{x(t),\xi(t)})\,dt\, .
\ee
\begin{lemma}
Let
\[
g_l(t-T_\gamma)\, :=\,
f_l(t)\, e^{i\tau\int^t_0\frac{\xi\dot x -x\dot\xi}{2}}\,
(\psi^a_{(x,\xi)},\psi^{a(t)}_{x(t),\xi(t)})
\]
be the integrand in (\ref{sarabande}). Then:
\be
g_l(t)\ =\
2^{-n/2}(2\pi)^{-\frac{3n}{2}}\tau^n e^{il\tau S_\gamma}e^{-i\tau\int^t_0 x\dot\xi ds +i\tau x(\xi(t)-\xi)}\,
\alpha_l(t,\sqrt\tau t)+O(\tau^{-\infty})
\ee
with
$\alpha_l(t,u) \in C_0\calS(\bbR\times\bbR)$, the class of functions defined in
(\ref{c0s}).
Moreover :
\be\label{sas}
\alpha_l(0,u)\ =\
\hat\varphi(lT_\gamma)\int e^{iu y\dot\xi(0)}\overline{\hat a(y)}
{\hat a}^{lT_\gamma}(y-u\dot x(0))dy\,.
\ee
\end{lemma}
\begin{proof}
By Proposition 2.4 one can get rid of the cutoffs $\rho$ in
$(\psi^a_{(x,\xi)},\psi^{a^t}_{x(t),\xi(t)})$, since we compute $\alpha$
mod $\tau^{-\infty}$. This gives, after some manipulations:
\be
\alpha_l(t,\sqrt\tau t)\ =\
(2\pi)^{-n/2}\beta(t)\hat\varphi(lT_\gamma+t)\int
e^{i\sqrt\tau[y(\xi(t)-\xi)-\eta(x(t)-x)]}e^{i\eta y}
\overline{\hat a(y)}a^{lT_\gamma+t}(\eta)dyd\eta
\ee
So
\be
\alpha_l(t,u)\ =\
(2\pi)^{-n/2}\beta(t)\hat\varphi(lT_\gamma+t)\int
e^{iu[y\frac{\xi(t)-\xi}{t}-\eta\frac{x(t)-x}{t}]}e^{i\eta y}
\overline{\hat a(y)}a^{lT_\gamma+t}(\eta)dyd\eta\,.
\ee
By the stationary phase lemma, since $(\dot x,\dot\xi)\neq 0$ one can see that
$\alpha$ decreases rapidly with $u$.
The same argument gives the result for the derivatives of $\alpha$, and
the desired uniformity as well.
\end{proof}
We have:
\be
I_l \sim 2^{-n/2}(2\pi)^{-\frac{3n}{2}}\tau^n \int e^{-i\tau\int^t_0 x\dot\xi ds +i\tau x(\xi(t)-\xi)}\,
\alpha(t,\sqrt\tau t)dt\,.
\ee
This integral will be estimated thanks to the following Proposition:
\begin{proposition}\label{fase}
Let $\alpha \in C_0\calS(\bbR\times\bbR)$ and
$\Phi \in C^\infty(\bbR)$ satisfying:
\be
\Phi(0)\, =\, 0\,=\, \Phi'(0)\,.
\ee
Let
\be
I(\tau)\, :=\, \int e^{i\tau\Phi(t)}\alpha(t,\sqrt\tau t)dt\,.
\ee
Then:
\be\label{maya}
I(\tau)\sim \tau^{-1/2}\sum_{j=0}^{\infty}c_j\tau^{-j/2}\,.
\ee
Moreover,
\be
c_0\ =\ \int e^{i\frac{\Phi''(0)}{2}t^2}\alpha(0,t)dt\,.
\ee
\end{proposition}
\begin{proof}
By the assumptions on $\Phi$ there exists a $\Psi\in C_0^\infty(\bbR)$
such that
\[
\Phi (t)\,=\, t^2\,\Psi(t)
\]
in the support of $t\mapsto\alpha(t,u)$, $\forall u\in\bbR$.
Substituting into $I(\tau)$ and
making a dilation $s:=\sqrt{\tau}t$ we get:
\be\label{ulm}
I(\tau)\ =\ \tau^{-1/2}\int e^{is^2 \Phi(s/\sqrt{\tau})}
\alpha\left(\frac{s}{\sqrt\tau}\, ,\,s\right)\,ds\,.
\ee
Since $\alpha$ is rapidly decreasing in the second variable,
uniformly with respect to the first, the integrand above is
bounded by, say, the integrable function $(s^2+1)^{-1}$ uniformly in $\tau$.
Therefore
\[
\lim_{\tau\to\infty}\,\tau^{1/2}I(\tau)\,=\,\int\,e^{is^2\Psi(0)}\,
\alpha(0,s)\,ds\,=\,c_0\,.
\]
To obtain the asymptotic expansion observe that in fact the function
of $\sigma$
\be\label{ixtla}
g(\sigma)\,=\, \int\,e^{is^2\Psi(s \sigma)}\,
\alpha(s \sigma ,s)\,ds
\ee
is smooth in a neighborhood of $\sigma = 0$: since
$\alpha$ is Schwartz in the second variable uniformly with respect to
the first every derivative with respect to $\sigma$ of the integrand in
(\ref{ixtla}) is bounded by an integrable function uniformly in $\sigma$.
The expansion (\ref{maya}) is nothing but the Taylor expansion of
$g(\sigma)$ around $\sigma =0$.
\end{proof}
\noindent
{\bf Remark:} If $0$ is not a critical point of $\Phi$, then one
can easily show that $I(\tau)$ above is $O(\tau^{-\infty})$.
\medskip
We now return to $I_l$. Thanks to Lemma 4.1, $I_l$ can rewritten as:
\be\label{rose}
I_l\ =\ 2^{-n/2}(2\pi)^{-\frac{3n}{2}}\tau^n e^{il\frac{S_\gamma}{\h}}
\int e^{i\tau\Phi(t)}\alpha_l(t,\sqrt\tau t)\, dt\, +\,O(\tau^{-\infty})
\ee
with
\be
\Phi(t)=-\int_0^tx(s)\cdot\dot\xi(s)\,ds \ +\ x\cdot(\xi(t)-\xi)\,.
\ee
We obviously have $\Phi'(t)=(x-x(t))\cdot \dot{\xi}$.
If $(x,\xi)$ is not a periodic point, then as mentioned we only
need to consider the term $l=0$, and therefore the asymptotic expansion
of Theorem 1.1 follows from Proposition 4.2.
\smallskip
If $(x,\xi)$ is periodic,
\[
\forall\ l\in\bbZ\quad\quad \Phi(lT_\gamma)\,=\,0\,=\,\Phi'(lT_\gamma)\,.
\]
and
\[
\Phi''(lT_\gamma)\ =\ -\dot{x} \cdot\dot{\xi}\,.
\]
This means that each integral $I_l(\tau)$ has, by Proposition 4.2, an
asymptotic expansion of the form:
\be
I_l\ \sim \ 2^{-n/2}(2\pi)^{-\frac{3n}{2}} e^{il\frac{S_\gamma}{\h}}
\tau^{n-\frac{1}{2}}\sum_{j=0}^\infty c_j^l\tau^{-j/2}
\ee with
\be\label{trs}
c_0^l\ =\ \int e^{-i\frac{\dot x\dot\xi}{2} t^2}\alpha_l(0,t)dt\,.
\ee
Plugging (\ref{sas}) in (\ref{trs}) and summing over l gives the existence of
(\ref{lopopo}) and the leading term (\ref{lapapa}).
\bigskip
\noindent
\underline{Tauberian Arguments.}
\nobreak
\smallskip\noindent
To prove (15) and (19) we will use a the following Tauberian lemma proved
in \cite{BPU} (see also \cite{BU}).
Consider an expression of the following form:
\be \label{5.1}
\Upsilon ^w_{E, \hbar }(\varphi ) \, = \,
\sum _j \, w_j(\hbar ) \, \varphi \left(
{{E_j(\hbar ) - E} \over \hbar} \right) \ .
\ee
It will be useful to introduce the following
\par\smallskip\noindent
{\bf Notation:} We will denote by $\calR$ the set of all Schwartz
functions on the line with compactly supported Fourier transform.
\smallskip
The Tauberian lemma in question is:
\begin{theorem} (See \cite{BPU}.)
Suppose
$w_j(\h )$, $E_j(\h )$, $E$ and $\Upsilon^w_{\h }$ itself satisfy
all of the following:
\medskip \noindent
\begin{enumerate}
\item There exists a positive function $\omega(\h)$, defined on an interval
$(0, \h _0)$, and a functional $\calF_0 $ on $\calR$,
such that for all $\vp\in\calR$
\be \label{5.3}
\Upsilon _{E, \h }^w (\vp ) = \calF_0(\vp ) \omega(\h) + o(\omega(\h)),
\quad \h \to 0
\ee
(both $\calF_0 $ and $\omega $ depending on $E$, in general).
\item If $f\in\calR$ is non-negative, identically one near
the origin and of mass one, and if one defines $\forall \mu>0$
\[
f_\mu(r)\,=\, \mu^{-1}f(r/\mu)\,
\]
(so that $\{f_\mu\}$ an approximate identity
i.e. each $f_\mu$ is positive, of mass one, and $f_\mu (r)\to\delta(r)$
as $\mu\to 0$),
then $\forall c>0$ the limit
\[
\calL(c) \,=\, \lim_{\mu\to 0} \calF_0(f_\mu *\chi_{[-c,c]})
\]
exists. Here $\chi_{[-c,c]}$ is the characteristic function of
the interval $[-c,c]$.
\item There exists a $k \in Z$ such that $\h ^k = \calO (\omega(\h )), \
\h \to 0$.
\item There exists an $\varepsilon > 0$ such that for every $\vp $
there is a constant $C_{\vp }$ such that for all
$E' \in [E - \varepsilon , E + \varepsilon ]$ :
\be \label{4}
|\Upsilon ^w_{E', \h}(\vp )| \leq C_{\vp } \omega (\h )
\ee
(rough uniformity in $E$).
\item The $w_j(\h )$ are non-negative and bounded: there exists a constant
$C \geq 0$ such that for all $j$ and all $\h, 0 < \h < \h _0$:
\be \label{5}
0 \leq w_j(\h ) \leq C\,.
\ee
\item The eigenvalues $E_j(\h )$ satisfy the following rough estimate:
for each $C_1$ there
exist constants $C_2, N_0$ such that $\forall k$
\be \label{6}
\# \{j: E_j(\h) \leq C_1 + k \h \} \leq C_2 (\h ^{-1} k)^{N_0} \,.
\ee
\end{enumerate}
\medskip \noindent
Define the weighted counting function by
\be
N_{E,c}^w(\h ) = \sum _{|x_j(\h)| \leq c} w_j(\h )
\ee
where
\be
x_j(\h)\,:=\, {E_j(\hbar ) - E \over \hbar}\,.
\ee
{\bf Then the conclusion is:}
$\forall c>0$
\be
N_{E,c}^w(\h ) = \calL (c) \omega (\h) + o(\omega (\h)), \
\h \to 0\,.
\ee
\end{theorem}
\medskip
In the present context, we wish to take
\be\label{wish1}
\omega_j = \sqrt{h}\,
\mid(\psi_{(x,\xi)}^a,\psi_j^\hbar)\mid ^2\,.
\ee
With the shown normalization, property (\ref{5}) is ensured. The function
$\omega$ is
\be\label{whish2}
\omega(\h)\,=\,\h^{-n+1}\,,
\ee
so that hypothesis 3 is automatically satisfied,
while the functional $\calF_0$ is given by Theorems 1.1 and 1.2 so that
property (\ref{5.3}) is true. The rough estimate on the eigenvalues
(\ref{6}) is certainly true in this case, see e.g. \cite{BPU}.
We need to verify the remaining assumptions 2 and 4 of the
Tauberian lemma above.
\smallskip\noindent
Assumption 2:
- if $(x,\xi)$ is not periodic, then
\be
\calF_0(\varphi)\,=\,
\int_{-\infty}^{+\infty}\varphi(t)\,dt
\int e^{-it^2\dot{x}\dot{\xi}}e^{it\eta\dot{x}}
\overline{a(\eta)}a(\eta -t\dot{\xi})d\eta dt\,.
\ee
that is $\calF_0$ is
proportional to Lebgesgue measure on $\bbR$. Thus assumption
2 is trivially satisfied, with $\calL (c) = 2c$.
-if $(x,\xi)$ is periodic, let us only consider values of $\h$ of the
form $\h =(\frac{S_\gamma}{2\pi m})$.
Then we must prove that the functional
\be
\calF_0 (\varphi)\, =\, \sum _{l} \hat\varphi(lT_\gamma)
\int e^{-it^2\dot{x}\dot{\xi}}e^{it\eta\dot{x}}
\overline{a(\eta)}(U^l a)(\eta -t\dot{\xi})d\eta dt
\ee
has the required property.
This is true under a hypothesis of instability; this is an easy
consequence of the following:
\begin{lemma}
Let us suppose that the differential $d(\phi_T)_{(x,\xi)}$ of the
classical flow at time $T_\gamma$ and at $(x,\xi)$
is diagonalizable over $\bbC$,
and has $r$ elliptic directions of angles $\theta_j$,
$j=1,\ldots, r$ and $n-1-r$ hyperbolic directions of Liapunov
angles $\mu_k$, $k=1,\ldots ,n-1-r$, with $rFrom this we get that
\begin{eqnarray}
(H_\h-E)\Phi^a_\gamma &=&i\h\int_0^{T_\gamma}
\partial_t(e^{i\frac{l(t)-Et}{\h}}\psi_{(x(t),\xi(t))}^{M(S(t))})+
O(\h^{3/2})\nonumber \\
&=&i\h
\left( e^{i\frac{S_\gamma}{\h}}
\psi_{(x,\xi)}^{M(S(T_\gamma))a}-\psi_{(x,\xi)}^a \right)
+O(\h^{3/2})\,.
\end{eqnarray}
Therefore $(H_\h-E)\Phi^a_\gamma\ =\ O(\h^{3/2})$ provided that:
i) $M(S(T_\gamma))a_0\ =\ e^{i\lambda}a_0$, and
ii) $\lambda\ =\ \frac{S_\gamma}{\h}+2\pi k$ for some integer $k$.
\smallskip\noindent
Here $a_0$ is the leading term of $a$ and $S_\gamma$ the action of $\gamma$.
A solution to this problem is precisely
given by $a=a^m_R$, $m\in\bbZ^n$, defined in the preceding section.
Moreover such an $a^m_R$ gives
$\lambda=\sigma_\gamma+\sum m_j\theta_j+\frac{1}{2}\sum\theta_j$, so (ii) is
nothing but the Bohr-Sommerfeld condition (see \cite{V}):
\be\label{polka}
S_\gamma\,=\,\left(2\pi k+\sum m_j\theta_j+\sigma_\gamma+\frac{1}{2}\sum\theta_j
\right)\h\,.
\ee
The computation of the constant $C$ shows that if $\Phi^a_\gamma$ is
normalized one has:
\be\label{kron}
\mid <\psi^{a^m_R}_{x,\xi},\Phi^{a^{m'}_R}_\gamma>\mid^2\ =\
\frac{(4\pi\h)^{-\frac{1}{2}}}{T_\gamma\mid\nabla H(x,\xi)\mid}\delta_{mm'}+
O(\h^{\frac{1}{2}})\,.
\ee
\medskip
Let us suppose now that $\gamma$ is non degenerate on $\Sigma_E$, so it belongs
to a family $\{\gamma_s\}$ of elliptic trajectories indexed by their action, $s$,
where $s$ ranges in a neighborhood
of $S_\gamma$. Let $E(s)$ and $T(s)$ denote
the energy and the period of $\gamma_s$. Then the energies of the
quasi-modes associated with this family are
\be\label{exacto}
\calE^{k,m}\,=\,
E\left((2\pi k+\sum (m_j+\frac{1}{2})\theta_j+\sigma_\gamma)\h\right)\,.
\ee
Moreover it is well-known that
\be
\frac{d E(s)}{ds}\ =\ \frac{1}{T(s)}\,.
\ee
Therefore, there is a smooth function $\nu(s)$ such that
\be\label{timo}
\calE^{k,m} - E\,=\,
\frac{1}{T_\gamma}\left((2\pi k+c_m)\h- S_\gamma
\right) + ((2\pi k+c_m)\h -S_\gamma)^2\,\nu((2\pi k+c_m)\h -S_\gamma)
\ee
where $c_m := \sum (m_j+\frac{1}{2})\theta_j+\sigma_\gamma$.
Let us {\em define} the numbers $E^{k,m}_{QM}$ by the equation
\be
\label{EQM}
\frac {E^{k,m}_{QM}(\h)-E}{\h}\ =\
\frac{1}{T_\gamma}\left( 2\pi k+\sum m_j\theta_j+\sigma_\gamma+
\frac{1}{2}\sum\theta_j-\frac{S_\gamma}{\h}\right)\,.
\ee
Then (\ref{timo}) shows that, $\forall c>0$,
as $\h\to 0$ and for $k$'s such that $|2\pi k\h - S_\gamma|\mid^2 +T
\ee
where T is $O(\h^{\frac{1}{2}})$ in the weak topology.
Moreover, the $E^{k,m}_{QM}(\h)$ are asymptotic to the energies of the
quasi-modes $\Phi^{a^{m}_R}$ in the sense of (\ref{timo2}).
\end{proposition}
\noindent
{\bf Remark:} One also has a similar result for the weighted spectral measure for an
arbitrary symbol $a$, by decomposing $a$ on the Hermite basis, in which case
the summation over $m'$ would be non-trivial.
\bigskip
\noindent
\underline{The unstable case.}
Let us turn now to the case where $\gamma$ is unstable.
\begin{proposition}
If $\gamma$ is fully hyperbolic then on the class $\calR$
the weighted spectral measure $\rho_{x,\xi}^{a_R}$ (where $a_R$ is the
Gaussian (\ref{goose})) is,
modulo $\h^{\frac{1}{2}}$ (also in the weak topology),
Lebesgue-continuous of the form $g(\lambda)d\lambda$ where
\be\label{plo}
g(\lambda)\ =\ \frac{(4\pi)^{-\frac{1}{2}}}{T_\gamma\mid\nabla H(x,\xi)\mid}\sum_{-\infty}^{+\infty}
h(\frac{1}{T_\gamma}(2\pi k-\frac{S_\gamma}{\h}+\sigma_\gamma)-\lambda)
\ee
with
\be
\label{ouf}
h(\lambda)\ =\ \frac{1}{\sqrt{2\pi}}\int \frac{e^{i\lambda t}}
{\sqrt{\Pi_0^{n-1}\mbox{\em cosh}(\mu_k t)}}dt\,.
\ee
\end{proposition}
\begin{proof}
By the results of \S 6 in this case the coefficient $d_0$ has the form:
\be
\frac{(4\pi)^{-\frac{1}{2}}}{\mid\nabla H(x,\xi)\mid}\left(
\sum_l \frac{1}{\sqrt{\Pi_{k=1}^{n-1}\mbox{cosh}(l\mu_k)}}
\widehat\varphi (lT_\gamma)\right)\,.
\ee
By the Poisson summation formula, and using the fact that the Fourier transform of a
product is a convolution, one gets the result.
\end{proof}
\medskip
\noindent
{\bf Remark:}
The formula above is, for small Liapunov exponents, a kind of
``smeared-out Poisson formula".
As shown by (\ref{ouf}), $h(\lambda)\to\delta(\lambda)$ as all the
Liapunov exponents $\mu_k$ tend to $0$.
This shows that $g(\lambda)$ has peaks around the lattice
$\frac{1}{T_\gamma}(2\pi k-\frac{S_\gamma}{\h}+\sigma_\gamma)$
if $\gamma$ is not too unstable.
\bigskip
\noindent
{\it We would like to finish with two informal remarks:}
\smallskip\noi
{\bf 1.}
No construction of quasi-modes is available in the unstable case;
nevertheless numerical computations (see \cite{He},\cite{D},\cite{Gu})
show that some phenomenon of localization of eigenfunctions near
unstable periodic orbits are visible. Among the main properties of this
controversial ``scarring" phenomenon we point out:
- The apparent localization doesn't occur more strongly as $\hbar$ goes to 0.
- This localization is more visible if the Liapunov exponents of $\gamma$ are
small.
Our results show that the average over a band of energy of the Husimi
functions (namely $\mid(\psi_{(x,\xi)}^a,\psi_j^\hbar)\mid ^2$) share
some of those properties. Corollary 1.4 shows that there is a nonzero
density of eigenfunctions whose Husimi functions are actually pointwise
sensitive to the presence of periodic trajectories. The contributions
to the formulas of Theorems 1.2 and 1.4 are in accordance with the
properties of scars mentioned before:
- The fact that $(x,\xi)$ belongs or not to a periodic trajectory
doesn't affect the order of the expansion, but rather changes the numerical
leading coefficient.
- This coefficient becomes greater as $\mu$ (the highest Liapunov exponent)
tends to $0$.
\bigskip
\noindent
{\bf 2.}
We finish this section by discussing the dependence of the leading coefficient $d_0$
with respect to the symbol $a$.
The extra contribution of a hyperbolic $\gamma$ to the leading term of
$\sum_j\varphi(\frac{E_j(\hbar)-E}{\hbar})\mid(\psi_{(x,\xi)}^a,\psi_j^\hbar)\mid ^2$ is
\be\label{bof}
b(a) :=\sum _{l\neq 0} \hat\varphi(lT_\gamma) e^{i(\frac{lS_\gamma}{\h}+\sigma_\gamma)}
\int e^{-it^2\dot{x}\dot{\xi}}e^{it\eta\dot{x}}\overline{a(\eta)}(U^l a)
(\eta -t\dot{\xi})d\eta dt\, .
\ee
Let us take $(x,\xi) \in \gamma$ and let us suppose
$n=2$ (the case $n>2$ can be treated analogously). Let
$a_0= (4\pi)^{-1/2}e^{-\eta^2/2}$.
We wish to estimate $(a_0,U^lZ(t\dot x,t\dot\xi)a_0)$. First remark that
\be
\mid (a,a')\mid ^2\ =\ \int \calW_a(u,v)\calW_{a'}(u,v)dudv
\ee
where $\calW_a$ is the Wigner function of $a$. Moreover it is well known that
\[
\calW_{M(S)Z(e,f)a}(u,v)\ =\ \calW_a(S^{-1}(u-e,v-f)\,.
\]
$\calW_{a_0}$ has an effective support of size $1$ near the origin, since
\[
\calW_{a_0}\ =\ e^{-(u^2+v^2)}\,.
\]
Consider now $(e,f)=b_s+b_u$ with $b_s$ and $b_u$ tangent vectors
belonging to the stable and
unstable directions at $(x,\xi)$. Then $\calW_{Z(e,f)}a_0$ will have an
effective
support near $b_s+b_u$. It is easy to see that the effective support of
$\calW_{U^lZ(e,f)a_0}$ won't intersect the one of $\calW_{Z(e,f)a_0}$ as soon
as:
\[
\mid b_s\mid >\frac{1+e^\mu}{1-e^\mu}
\]
or
\[
\mid b_u \mid > \frac{1+e^{-\mu}}{1-e^{-\mu}}
\]
If one remarks finally that
\[
\psi_{(x,\xi)}^{Z(e,f)a}(y)\,=\,
\psi_{(x,\xi)}^{a}(y-\sqrt\h e)e^{-i\frac{\sqrt\h f\xi}{\h}}
\]
i.e $\psi_{(x,\xi)}^{Z(e,f)a}$ is microlocalized around the point
$(x +\sqrt\h e;\xi +\sqrt\h f)$, one may conclude as follows:
\par\noindent
{\it If the "effective support" of $a$ is roughly of size $1$ and contains
the origin, then $\sum_j\varphi(\frac{E_j(\hbar)-E}{\hbar})\mid(\psi_{(x,\xi)}^a,\psi_j^\hbar)\mid ^2$ is
sensitive to the presence of $\gamma$ in a tubular neighborhood of $\gamma$ of
size $\sqrt\h \frac{e^\mu +1}{e^\mu -1}$ in the stable direction and
$\sqrt\h \frac{e^{-\mu} +1}{1-e^{-\mu}}$ in the unstable one.}
\section{References}
\begin{thebibliography}{99}
\bibitem{BG} L.\ Boutet de Monvel and V. Guillemin, {\it The spectral
theory of Toeplitz operators.} Annals of Mathematics Studies No.\ 99,
Princeton University Press, Princeton, New Jersey (1981).
\bibitem{BPU} R. Brummelhuis, T. Paul and A. Uribe. Spectral estimates
near a critical level, preprint.
\bibitem{BU} R.\ Brummelhuis and A.\ Uribe. A trace formula for
Schr\"odinger operators. Comm. Math. Phys. {\bf 136} (1991), 567-584.
\bibitem{CdV} Y. Colin de Verdi\`ere. Quasi-modes sur les varietes
Riemanniennes. Invent. Math. {\bf 43} (1977), 15-42.
\bibitem{CF} A. Cordoba and C. Fefferman. Wave packets and Fourier
integral operators. Comm. in P.D.E., {\bf 3} (1978), 979-1005.
\bibitem{D} D. Delande, Comm. Atom. Mol. Phys. {\bf 25} (1991).
\bibitem{Gr} A. Grossmann. Parity operator and quantization of
$\delta$ functions. Comm. Math. Phs. {\bf 48} (1976), 191-194.
\bibitem{G2} V. Guillemin. Symplectic spinors and partial differential
equations. In Coll. Inst. CNRS n. 237, G\'eom\'etrie Symplectique et
Physique Math\'ematique, 217-252.
\bibitem{Gu} M. Gutzviller. {\it Chaos in classical and quantum Mechanics},
Springer Verlag, 1991.
\bibitem{H} G. Hagedorn. Semiclassical Quantum Mechanics, Com. Math. Phys,
{\bf 71} (1980), 77-93.
\bibitem{HMR} B. Helffer, A. Martinez et D. Robert.
Ergodicit\'e et limite semi-classique. Comm. Math. Phys. {\bf 109} (1987),
313-326.
\bibitem{He} E. Heller, Phys. Rev. Letters vol 53 (1984), 1515.
\bibitem{LJ} R. Litteljohn. The semiclassical evolution of wave packets,
Physics reports 138 (1986), 193-291 .
\bibitem{PU1} T.\ Paul and A.\ Uribe. The semi-classical trace formula
and propagation of wave packets. To appear in the J. of Funct. Analysis.
\bibitem{Ra} J.V. Ralston. On the construction of quasimodes associated with
stable periodic orbits. Comm. Math. Phys. {\bf 51} (1976), 219-242.
\bibitem{TA} M. Taylor. {\it Pseudodifferential operators}, Princeton University
Press, 1981.
\bibitem{TR} F. Treves. {\it Introduction to pseudodifferential and Fourier
integral operators}, Plenum Press, New York and London (1980).
\bibitem{V} A. Voros, The WKB-Maslov method for nonseparable systems
Coll. Inst. CNRS n. 237, G\'eom\'etrie Symplectique et
Physique Math\'ematique, 217-252.
\end{thebibliography}
\end{document}