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\title{The Semi-classical Trace Formula and Propagation of
Wave Packets}
\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.
uribe@math.lsa.umich.edu.
Research supported in part by NSF grants DMS-9107600 and DMS-9303778.}}
\begin{document}
\setcounter{page}{0}
\date{}
\maketitle
\centerline{May 27, 1994.}
\tableofcontents
\vfill\break
\centerline{\large\bf Abstract}
\bigskip
We study spectral and propagation properties of
operators of the form $S_\h = \sum_{j=0}^N \h^j P_j$
where $\forall j$ $P_j$ is a differential operator of order $j$
on a manifold $M$, asymptotically as $\h\to 0$.
The estimates are in terms of the flow $\{\phi_t\}$ of the
classical Hamiltonian $H(x,p) = \sum_{j=0}^N \sigma_{P_j}(x,p)$
on $T^*M$, where $\sigma_{P_j}$ is the principal symbol of $P_j$.
We present two sets of results. (I) The ``semiclassical trace
formula", on the asymptotic behavior of eigenvalues and
eigenfunctions of $S_\h$ in terms of periodic trajectories
of $H$. (II) Associated to certain isotropic submanifolds
$\Lambda\subset T^*M$ we define families of functions
$\{\psi_\h\}$ and prove that $\forall t$
$\{\exp(-it\h S_h)(\psi_\h )\}$ is a family of the same kind
associated to $\phi_t(\Lambda)$.
\vskip .6in
\noindent
{\Large{\bf Introduction and description of results.}}
\bigskip
\noindent
In this paper we present some results concerning spectral
and propagation properties of a class of differential operators
``with small parameter", $\h$ (Planck's constant).
We have in mind operators of the form $a(x,
\hbar D_x)$ on a compact manifold (or on $\bbR^n$ under additional
assumptions at infinity) where $a(x,p)$ is a polynomial in $p$ bounded
below. We are interested in trace formulae, localization of
eigenfunctions and propagation of wave packets in the semi-classical
limit, that is as $\hbar$ tends to zero. We present two sets of
results: (A) The Theorems we announced in \cite{PU}, including a
rigorous proof of a version of the Gutzwiller trace formula, and (B) a
general definition of ``wave packets" on arbitrary manifolds and a
theorem on their propagation under the time-dependent Schr\"odinger
equation.
A secondary goal of this paper is to show how semi-classical
problems can be treated with {\em standard} Fourier integral operator
methods. We first learned of this possibility while reading an
article of Yves Colin de Verdi\`ere, \cite{CdV1}, (proof of Theorem 4.4).
A first version of the Gutzwiller trace formula was obtained by this
method in joint work of one of us with Victor Guillemin, see \cite{GU},
in the course of studying elliptic operators with symmetries.
This approach, which we believe is very natural (e.g. its underlying
geometry is that of the pre-quantum circle bundle)
is developed here systematically.
\medskip
The apparent reason that H\"ormander's theory of Lagrangian distributions
seems unsuited to the treatment of semi-classical problems is that
it is a {\em homogeneous} theory. That is, in H\"ormander's theory
all geometrical objects (e.g. Lagrangian submanifolds, symbols) are
homogeneous under the $\bbR^+$ action on the punctured cotangent
bundle of the manifold where they live. This is clearly false of
the corresponding semi-classical objects. Homogeneity is however
restored if one assigns to $\hbar$ the order $(-1)$. (Because
each $x$ derivative has a factor of $\h$.) The Schr\"odinger
operator becomes then formally of order zero, and therefore the
fundamental solution of Schr\"odinger's equation
$i\hbar \partial u/\partial t = a(x,\h D_x)u$
is the exponential of an operator of formal order one, namely
$-i\hbar^{-1} a(x, \hbar D_x)$. In our method we {\em in fact}
realize $\hbar$ as a pseudodifferential operator of order $(-1)$, microlocally
in a region of bounded energy. This in particular leads to the
proof of the trace formulae of \cite{PU}.
\medskip
To be more specific,
consider a Schr\"odinger operator on, say, a compact manifold M,
-$\hbar^2 {\Delta}$ + V(x), where ${\Delta}$ denotes the Laplacian on
M. The time dependent Schr\"odinger equation is
\begin{equation}
\label{0.1}
\hbar\, i\, {\partial \psi_\hbar\over\partial t}\,= \,
(-\hbar^2 {\Delta} + V(x))\psi_\hbar\ .
\end{equation}
One is interested in the initial value problem
with initial condition $\psi_\hbar^0$ in $L^2 (M)$.
Since we are interested in asymptotics as $\hbar\to 0$, it is natural
to prescribe the small $\hbar$ behavior of the initial data,
$\psi_\hbar^0$. We shall do this by considering the $\hbar$
{\em transform} of the initial data, which by definition is the Fourier
transform in $\hbar^{-1}$, that is, the following distribution
on $M\times S^1$:
\begin{equation}
\label{0.2}
\Psi^0(x,{\theta}) =\ \sum_{m=1}^\infty\ {\psi}_{1/m}(x)\,
e^{im\theta}\ .
\end{equation}
In making this definition, we have tacitly restricted
$\hbar$ to take the values $ 1/m$, for $m=1,2,...$.
We'll see later on that in many cases this restriction on $\h$
can be lifted.
\begin{definition}
We will say that the family $\{ \psi^0_\hbar\}$ is of
{\em semi-classical type} iff the wave-front set of (\ref{0.2})
is contained in the set $\{\ \kappa\neq 0\ \}$, where $\kappa$ is
the variable dual to $\theta$.
\end{definition}
Let us give two examples of such families:
\noindent
A. Wave packets (\S 5): let us consider the function
\[
{\phi}^{\hbar} _{(q,p,A)}(x)= (\det A_i)^{1/4}
e^{-iq.p/2{\hbar}}e^{-(x-q)^tA(x-q)/{\hbar}}e^{ip.x/{\hbar}}\ .
\]
Then the wave front set of
\[
\Psi(\theta, x)=\sum _{m=1} ^\infty
\psi^{m^{-1}}_{(q,p,A)}(x)e^{im\theta}
\]
is the set $\{ (q,\theta=-qp/2;\kappa p,\kappa),\kappa >0\}$.
\noindent
B. WKB states. Consider now a family of vectors of the form
$\psi_\hbar(x)=e^{iS(x)}$,with S(x) real on an open set,
then the wave front set of $\Psi(\theta,x)$ defined as
before is:
\[
\{\ (\theta,x;\kappa,\xi) \ ;\ \xi=\kappa S'(x)\, ,\, \kappa >0\, ,\,
\theta=S(x)\ \}\ .
\]
\medskip
Back to the time-dependent Schr\"odinger equation, (\ref{0.1}),
if we denote by $\Psi$ the $\hbar$-transform of the time-dependent
wave functions $\psi_\hbar$, one easily sees that the equation is
equivalent, formally, to
\begin{equation}
\label{0.3}
- D_t {\Psi}= D_{\theta}[-(D^{-1}_{\theta})^2{\Delta}
+V(x)]{\Psi}
=[-D^{-1}_{\theta}{\Delta} +D_{\theta}V(x)]{\Psi}
\end{equation}
where $D^{-1}_{\theta}$ is a parametrix of $D_{\theta}$,
which will be properly defined later.
One of the main ideas of this paper consists in noting
that the operator
\begin{equation}
\label{0.4}
P=-D^{-1}_{\theta}{\Delta} + D_{\theta}V(x)
\end{equation}
can be considered {\em when acting on semi-classical families}
as a classical pseudo-differential operator on
$M\times S^1$ with symbol
\begin{equation}
\label{0.5}
p(x,\theta;\xi,\kappa)=\kappa^{-1}\xi^2 +\kappa V(x)\ .
\end{equation}
In fact, notice that p is homogeneous of degree one in the
cotangent variable $(\xi,\kappa)$. Of course one has to deal with
the fact that $\kappa^{-1}$ is singular, and that besides that
$\kappa^{-1}\xi^2$ looks not like a classical symbol, but like a
product-type symbol. However, we will see that,
micro locally inside cones of the form
\[
\| \xi\| \kappa^{-1} 0\,\}
\ee
which is the phase space of the homogenized
semi-classical Hamiltonian $\calH (q,\xi; \theta,\kappa)$. We will now
arrive at this manifold more invariantly: $\Sigma$ is the
symplectification of the pre-quantum $\bbR$-bundle of $T^*M$. This point
of view will enable us to systematically homogenize symplectic objects
in $T^*M$.
\medskip
We begin by recalling some basic facts
about pre-quantum bundles. Let $\eta$ ($=\sum\, pdq$) be the canonical
one-form of $T^*M$, so that
\begin{equation}\label{g2.1}
\omega\ =\ -d\eta\ .
\end{equation}
Then the one-form
\begin{equation}\label{g2.5}
\alpha\ =\ \eta + d\theta
\end{equation}
is a connection form with curvature $-\omega$ on the trivial
$\bbR$-bundle
\[
P\ :=\ T^*M\times\bbR\ .
\]
{\em The $\bbR$ bundle with connection $(P,\alpha)$ is the pre-quantum
bundle of $T^*M$.} It possesses the following important property.
Define a
map from $C^\infty (T^*M)$ to vector fields on $P$, as follows:
\begin{eqnarray}
C^\infty (T^*M) & \to & \mbox{Vect}(T^*M\times\bbR)\nonumber\\
H & \mapsto & \Xi^\#_H\ +\ H\partial_\theta \label{g2.6}
\end{eqnarray}
where $\Xi^\#_H$ is the horizontal lift (with respect to the connection
(\ref{g2.5})) of the Hamilton vector field of $H$ and $\partial_\theta$
is the basic vector field along $\bbR$.
\begin{proposition}\label{g22.11} (Kostant, Souriau)
The map (\ref{g2.6}) is an injective morphism of Lie algebras, that is
\[
\Theta_{\{H,G\}}\ =\ [\Theta_H\;,\;\Theta_G]
\]
where
\begin{equation}\label{g2.7}
\forall H\in C^\infty (T^*M)\ ,\quad
\Theta_H\ =\ \Xi^\#_H\ +\ H\partial_\theta \ .
\end{equation}
\end{proposition}
Consider, for example, $M=\bbR^n$. It is well-known that the space
of affine Hamiltonians $H$, that is, of the form
\begin{equation}\label{g2.11}
H(q,p)\ =\ aq + bp + c\ ,
\end{equation}
is a $(2n+1)$-dimensional Poisson subalgebra of $C^\infty(\bbR^{2n})$:
it is in fact the Heisenberg Lie algebra.
>From Proposition \ref{g22.11}, we know that
$\{ \Theta_H\ ;\ H\ \mbox{affine}\ \}$ is then a Heisenberg Lie algebra
of vector fields on $P = \bbR^{2n}\times\bbR$. This of course
is not surprising: $P$ itself is the Heisenberg group and the algebra
above is its Lie algebra.
The space of homogeneous polynomials on $\bbR^{2n}$ of degree two is a
Lie subalgebra of $C^\infty(\bbR^{2n})$ isomorphic to the algebra of the
group of symplectic linear transformations of $\bbR^{2n}$. Under
$\Theta$, one obtains an $sp(n)$ algebra of vector fields on
$P=\bbR^{n+1}$, generating the action of the symplectic group on the
Heisenberg group by automorphisms.
\bigskip
\bigskip
The connection form $\alpha$ makes $P$ a contact manifold. It is
well-known that such manifolds have a homogeneous symplectification,
which in the present case will turn out to be the manifold $\Sigma$
of the previous section. The symplectification is the following subset
of the cotangent bundle of $P$, $\Sigma^0\subset T^*P\setminus \{ 0\}$:
\begin{equation}\label{g3.1}
\Sigma^0\ =\ \{\ (x, \kappa\alpha_x)\ ;\ x\in P\ \mbox{and}\
\kappa > 0\ \}\ .
\end{equation}
We summarize some basic properties of $\Sigma^0$ in the following elementary
Lemma:
\begin{lemma}\label{g33.11}
Under the natural diffeomorphism,
\begin{eqnarray*}
P\times\bbR^{+} & \to & \Sigma^0 \\
(x , \kappa) & \mapsto & (x, \kappa\alpha_x)
\end{eqnarray*}
the canonical one-form of $T^*P$ pulls-back to the form $\kappa\alpha$
on $P\times\bbR^{+}$. Hence:
\hfill
\noindent
(1) $\Sigma^0$ is a symplectic submanifold of $T^*P$, naturally
symplectomorphic to $\Sigma = T^*(M\times\bbR)^+$.
\par\noindent
(2) The Hamiltonian
\begin{eqnarray*}
\kappa : \Sigma^0 & \to & \bbR \\
(x, \kappa\alpha_x) & \mapsto & \kappa
\end{eqnarray*}
generates the $\bbR$-action induced on $\Sigma^0$
by the $\bbR$ action on $P$.
\par\noindent
(3) For every $c>0$, the symplectic quotient $\kappa^{-1}(c)/\bbR$ is
naturally symplectomorphic to $(T^*M, c\omega)$.
\end{lemma}
\smallskip
We will henceforth work with the model $\Sigma = T^*(M\times\bbR)^+$
and will continue to think of $P$, identified with
\[
P\,=\,\{\, (q,\theta ; \xi, \kappa=1)\in T^*(M\times\bbR)^+\,\}\,,
\]
as the prequantum line bundle of $T^*M$, with the connection form
$ \alpha\,=\,\eta + d\theta\,.$
\smallskip
As we saw in the previous section, a classical Hamiltonian $H(q,p)$
lifts to a Hamiltonian $\calH$ on $\Sigma$ homogeneous of degree one.
A general description of this procedure is as follows:
Pull-back $H$ to $P$ by the natural projection $\pi : P\to T^*M$, then
extend uniquely $\pi^* H$ to a homogeneous Hamiltonian of degree-one,
$\calH$, on $\Sigma$. It is clear that this description of $\calH$
coincides with that of the previous section.
\bigskip
We next consider the problem of homogenizing Lagrangian (or, more
generally isotropic) submanifolds of $T^*M$. Recall that a submanifold
$\Lambda_0\subset T^*M$ is isotropic iff it is flat with respect to
the connection on $\pi : P\to T^*M$, that is, if the pull-back to
$\Lambda_0$ of the symplectic form vanishes. This forces
$ \mbox{dim}\,\Lambda_0\leq \mbox{dim}\,M$,
and if equality occurs then $\Lambda_0$ is Lagrangian. In the general
isotropic case, it makes sense to ask if there is a {\em global}
smooth horizontal section of $P$ over $\Lambda_0$. In case $\Lambda_0$
is Lagrangian, the image of such a section would be a Legendrian
submanifold of $P$. Global horizontal lifts do not always exist.
\begin{definition}
We will say that a closed Lagrangian (resp. isotropic) submanifold,
$\Lambda_0\subset T^*M$, is strongly admissible iff there exists
a closed conic Lagrangian (resp. isotropic) submanifold
$\Lambda\subset\Sigma$ such that
\be\label{g4}
\Lambda_0\ =\ \left(\Lambda\cap P\right)/\bbR\ .
\ee
We call $\Lambda$ a homogenization of $\Lambda_0$.
\end{definition}
We now investigate when $\Lambda_0$ is strongly admissible. Let
$\iota : \Lambda_0\hookrightarrow T^*M$ denote the inclusion of a closed
Lagrangian submanifold. Since $0 = \iota^*\omega = -d\iota^*\eta$,
the one-form $\iota^*\eta$ is closed. We leave the following as
an exercise:
\begin{lemma}
$\Lambda_0$ is strongly admissible iff $\iota^*\eta$ is exact. If
$\iota^*\eta\;=\; df$ for some $f : \Lambda_0\to\bbR$, then
\be\label{g4.a}
\Lambda\ =\ \{\; (x\; ,\; \theta\,,\kappa\,)\ ;\ \kappa > 0\ ,\
x\in\Lambda_0\ \mbox{and}\ \theta\;=\;-f(x)\;\}
\ee
is a homogenization of $\Lambda_0$.
\end{lemma}
\smallskip
An example relevant to \S\S 3,4 is the following:
\begin{lemma}
Let $H\in C^\infty(T^*M)$ and let $\{ \phi_t \}$ denote its Hamilton
flow. For a given $t$, let
\[
\Gamma_t^0\, =\, \{\, (x,[\phi_t (x)]')\,;\,x\in T^*M\,\}\,\subset
T^*M\times T^*M \,=\, T^*(M\times M)
\]
be the twisted graph of the map $\phi_t$ (prime denotes reflection across the
zero section). Then $\Gamma_t^0$ is
strongly admissible. The twisted graph of the homogenization
of $H$ is a homogenization of $\Gamma _t^0$.
Equivalently, identifying $\Gamma_0^t$ with $T^*M$ via projection
onto the first factor, a homogenization of $\Gamma_t^0$ of the form
(\ref{g4.a}) is defined by the function
\be\label{g4.b}
f(x)\,=\,\int_0^t\,\calL (\phi_\tau(x))\,d\tau\,,
\ee
where $\calL : T^*M\to \bbR$ is the function
\be\label{g4.c}
\calL \,=\, \partial_rH - H\,,
\ee
where $\partial_r = \sum p{\partial\over\partial p}$ is the radial
vector field in $T^*M$.
\end{lemma}
\begin{proof}
The equivalence of both descriptions of a homogenization of $\Gamma_t^0$
follows from the equation for $\dot{\theta}$ in Hamilton's equations
for $\calH$, namely
\be\label{g5}
\dot{\theta}\,=\,{\partial\over\partial\kappa}\,\kappa\,H(x,\xi/\kappa)\,=\,
H(x,\xi/\kappa) - \kappa^{-1}\xi\cdot\nabla_pH (x,\xi/\kappa)\,.
\ee
(Observe that if $\kappa = 1$ then the right-hand side is $-\calL(x,\xi)$.)
Therefore we only need to prove that
\be\label{g6}
df\,=\, \eta - \phi_t^*(\eta)\,.
\ee
To prove this for all $t$ we observe that it is true for $t=0$ and
compute the derivative with respect to $t$ of each side. On the right
we get (see \cite{KN1}, Proposition 3.6)
\[
\phi_t^*(\calL_\Xi \eta)\,=\,\phi_t^*\Bigl(
d(\eta(\Xi))- \omega(\Xi,\cdot)\Bigr)\, =\,
\phi_t^*(d(\eta(\Xi)))-dH\,.
\]
On the left, we get
\[
{d\over dt}\,\Bigl[ d(f)_x \Bigr]\,=\, d\Bigl[ {d\over dt} f\Bigr]_x\,=\,
\phi_t^*\Bigl( d(\partial_rH)\Bigr)
-dH\,,
\]
and since $\partial_rH = \eta(\Xi)$ the result follows.
\end{proof}
\medskip
It turns out that many submanifolds of interest (e.g. invariant tori) are
{\em not} strongly admissible; the condition $\iota^*\eta\;=\; df$
is too restrictive. However, there are many cases of interest
where
\[
\iota^*\eta\ =\ d\log \phi
\]
where $\phi : \Lambda_0\to S^1$. That is, one has $\iota^*\eta\;=\; df$
where $f$ is only defined modulo $2\pi\bbZ$. Then the previous
constructions can be carried out if we work on a periodic version of $P$,
that is, if we take $P = T^*M\times S^1$ so that $\theta$ is now
$2\pi$-periodic. (In the language of cohomology, strong admissibility
amounts to $[\iota^*\eta]=0$ in $H^1(\Lambda_0, \bbR)$ while the weaker
version requires that the cohomology class of $\iota^*\eta$ in
$H^1(\Lambda_0, S^1)$ vanishes.) Clearly there's nothing special about
the number $2\pi$; we formulate the results in general:
\begin{definition}
Given $\hbar_0 > 0$, let $S^1_{\hbar_0}\ =\ \bbR/(2\pi/\hbar_0)\,\bbZ$
and
\[
P_{\hbar_0}\ =\ T^*M\times S^1_{\hbar_0}\ ,
\]
\[
\Sigma_{\hbar_0}\subset T^*P_{\hbar_0}
\]
be the obvious periodic variants of $P$, $\Sigma$. We will say that
$\Gamma_0\subset T^*M$ is $\hbar_0$-admissible iff there exists a
closed conic Lagrangian submanifold, $\Lambda\subset \Sigma_{\hbar_0}$
such that (\ref{g4}) holds.
\end{definition}
This is a geometrical version of the Bohr-Sommerfeld condition.
\begin{lemma}
$\Lambda_0$ is $\hbar_0$-admissible iff there exists
$\phi : \Lambda_0\to S^1_{\hbar_0}$ such that
$\iota^*\eta\;=\; d\log\phi$. In that case,
\[
\Lambda\ =\ \{\; (x\; ,\; e^{i\theta}\,,\kappa\,)\ ;\ \kappa > 0\ ,\
x\in\Lambda_0\ \mbox{and}\ e^{i\theta}\;=\;\phi(x)\;\}
\]
is a closed conic Lagrangian submanifold of $\Sigma$ satisfying
(\ref{g4}).
\end{lemma}
\medskip\noi
{\bf Remarks:}
\smallskip\noi
1.- Obviously a strongly admissible $\Lambda_0$ is $\hbar_0$ admissible
for all $\hbar_0$.
\smallskip\noi
2.- In semi-classical asymptotics associated with $\hbar_0$-admissible
objects, it is known that
one has a nice theory if the values of $\hbar $ are restricted
to go to zero along the sequence of values
\[
\hbar\ =\ {\hbar_0\over k}\ ,\ k=1,2,\cdots\ .
\]
We see here a geometric interpretation of this fact: the values
above are the reciprocals of the eigenvalues of $D_\theta$.
\smallskip\noi
3.- For ease of notation, we will say that $\Lambda_0$ is {\em admissible}
if it is $1$-admissible, leaving to the reader the task of rescaling
to obtain the general $\hbar_0$-admissible case.
% section two, on compound asymptotics
\newcommand{\psim}{\{\psi_m\}}
\newcommand{\fs}{\mbox{FS}}
\newcommand{\mts}{M\times S^1}
\section{Semi-Classical States}
In this section we study the relationship between the notion of
compound asymptotics, as developed for example in Chapter 7 of
\cite{GS1}, and the $\h$-transform, which we define as the Fourier
transform in $\kappa$. We will restrict our attention in this section
to compact admissible Lagrangians and the `periodic' version of $P$,
with structure group $S^1$. Therefore the semi-classical homogenization
of $T^*M$ is
\[
\Sigma\,=\, T^*(M\times S^1)^{+}
\]
quantized by $L^2(M\times S^1)$.
\subsection{Definitions and some elementary results.}
We begin by defining a very general concept of {\em semi-classical
family} of wave functions on a manifold $M$. We keep the notation
of the previous section: we denote by $\Sigma$ the open set
in $T^*(\mts)\setminus\{ 0\}$ defined by:
\[
\Sigma\ :\ \kappa > 0
\]
where $\kappa$ is the variable dual to the circular variable $\theta$.
\begin{definition}
Let $M$ be a smooth manifold, and $\psim$ a sequence of smooth
half-densities on $M$. We will say that $\psim$ is a {\em
semi-classical family} (scf) iff the series
\be\label{asy0}
\Psi\ =\ \sum_{m=1}^\infty\ \psi_m(x)\ e^{im\theta}
\ee
converges weakly to a distribution $\Psi$ on $M\times S^1$, and
\be\label{asy0.1}
\wf\, (\Psi)\ \subset\ \Sigma\ .
\ee
We will say that $\Psi$ is the $\h$-transform of the
family.
\end{definition}
\noindent
{\sc Remark:} If one has a distributional half-density, $\Psi$,
on $\mts$ satisfying (\ref{asy0.1}), then its Fourier coefficients
are automatically smooth. This is because the $m$-th Fourier
coefficient of $\Psi$ is the push-forward
\[
\psi_m\ =\ {1\over 2\pi}\ \pi_*\, \left(\, e^{-im\theta}\, \Psi\, \right)
\]
where $\pi : \mts\to M$ is the natural projection, and condition
(\ref{asy0.1}) ensures that no covectors in the wave-front set of
$\Psi$ are conormal to the fibers of $\pi$. Hence there is no loss of
generality in the definition above in assuming that the $\psi_m$ are
smooth.
\begin{lemma}
A sequence $\psim$ of smooth half-densities on $M$ is a scf if
for every compact $K\subset M$ contained in a coordinate
chart $U\, ,\ (x_1,\ldots x_n)$ and every multi-index $\alpha$
there exists a constant $C_{K,\alpha}$ such that
\be\label{asy1}
\sup_{x\in K}\ {1\over m^{|\alpha|}}\, \left|\, D^\alpha_x\,\psi_m(x)\,
\right|\ \leq\ C_{K,\alpha}\ .
\ee
\end{lemma}
Before presenting the proof, a couple of remarks.
Implicitly in (\ref{asy1}) is the identification of half-densities
and functions on $U$ provided by the coordinate functions.
Notice that as a corollary we get that any expression
of the sort arising in WKB theory, that is
\be\label{asy2}
\psi_m(x)\ =\ a(x,m)\, e^{imS(x)}
\ee
where $a$ has symbolic behavior on $m$, defines a semi-classical family.
\noindent
{\sc Proof of the Lemma: }
Condition (\ref{asy1}) with $\alpha=0$ says that $\psi_m$ is bounded
on compact sets uniformly on $m$. This implies that the series
(\ref{asy0}) converges weakly: if $u(x,\theta)$ is a test function
on $\mts$, then
\[
\left|\ \int_0^{2\pi}\, e^{im\theta}\, u(x,\theta)\ d\theta\ \right|\
\leq\ {2\over m^2}\ \sup\,\left(D_\theta^2\, u\right)\ ,
\]
which shows that the series paired against $u$ converges and is bounded
by the $C^2$ norm of $u$.
\smallskip
Working locally in some coordinates $(x_1,\ldots ,x_n,\theta)$
take now a point $(x^0,\theta^0, \xi^0 ,\kappa^0)$. To see if it is not
in $\wf\, (\Psi)$, we consider the large positive $\lambda $ behavior of
the pairing
\[
I(\lambda)\ =\ \left(\, \rho_1(\theta)\,\rho_2(x)\,\Psi\ ,\
e^{i\lambda\, (x\xi +\theta\kappa)}
\,\right)
\]
where $\rho_1(\theta)\,\rho_2(x)$ are smooth cut-off functions non
vanishing at $(x^0,\theta^0)$. Clearly,
\be\label{asy3}
I(\lambda)\ =\ \sum_{m=1}^\infty\ \hat{\rho}_1(\lambda\kappa - m)\
\widehat{\rho_2\psi_m}(\lambda\xi)\ .
\ee
We must show that the estimates (\ref{asy1}) imply that this is
$O(\lambda^{-N})$ for each $N>1$, uniformly in $(\xi,\kappa)$ in
a neighborhood of $(\xi^0,\kappa^0)$. In fact, it suffices to take
$\xi$ of Euclidean norm one:
\be\label{asy3.1}
\|\,\xi\,\|\ =\ 1\ ,
\ee
a fact that will simplify the writing below.
Since $\rho_2\in C_0^\infty$, for every $K>0$ there exists $A_K>0$
such that
\[
\left|\,\hat{\rho_1}(\mu)\,\right|\ \leq\ {A_K\over (1+|\mu |)^K}\ .
\]
It follows that
\[
\forall\,\lambda > 0\ ,\ \kappa < 0 \quad
\left|\,\hat{\rho_1}(\lambda\kappa - m)\,\right|\ \leq\
{A_K\over (1+m)^K}\ .
\]
Hence
\be\label{asy4}
\left|\, I(\lambda)\,\right|\ \leq\ A_K\ \sum_{m=1}^\infty\
{1\over (1+m)^K}\, \left|\,\widehat{\rho_2\psi_m}(\lambda\xi)\,\right|\ .
\ee
On the other hand, by assumption (\ref{asy3.1}), we have that
\[
\lambda^{2N}\, \hat{f} (\lambda\xi)\ =\
\widehat{\left(\Delta^N f\right)}\, (\lambda\xi)
\]
where $\Delta$ is the Euclidean Laplacian, this
for any $f\in C_0^\infty$. It follows that
\be\label{asy5}
\lambda^{2N}\, \left|\hat{f} (\lambda\xi)\right|\ \leq\
\int_{\bbR^n}\ \left|\Delta^N f\right|(x)\, dx\ \leq\
\mbox{Vol}\;(\mbox{supp}\,f)\ \sup|f(x)|\ .
\ee
Apply this inequality with $f = \rho_2\psi_m$, using the fact that
\[
\sup\left|\Delta^N\rho_2\psi_m\right|\ \leq \ C_N\, m^{2N}\ .
\]
Hence we get that
\[
\lambda^{2N}\,\left|\,\widehat{\rho_2\psi_m}(\lambda\xi)\,\right|\ \leq\
B_N\ m^{2N}\ ,
\]
for some constant $B_N$. Multiplying (\ref{asy4}) by $\lambda^{2N}$
and taking $K=2N+2$ we see that $I(\lambda)$ is $O(\lambda^{-2N})$.
\smallskip
\hfill{$\Box$}
\medskip
It would be desirable to have a converse to this lemma, or otherwise
establish local criteria on $\psim$ on $M$ to be a scf. A moment's
thought however shows that this is subtle: estimates of the type
(\ref{asy1}) are unchanged if we multiply $\psi_m$ by a phase
factor $e^{i\omega_m}$ with an $m$-dependent phase. Such phase
factors however can be expected to change significantly the
$\h$-transform of the family. We will see in the next subsection
how to choose the phase of a family $\psim$ in a ``coherent" fashion.
\bigskip
\bigskip
Attached to any semi-classical family is its {\em frequency set},
$\fs(\psim)$, which we define next. For simplicity we state the
definition in the case $M=\bbR^n$, referring the reader to
\cite{GS1},Chapter 7 Proposition 2.2 for the proof that the local
definition below extends to manifolds, the frequency set of a
semi-classical family being an invariantly-defined closed set of the
cotangent bundle $T^*M$ (possibly intersecting the zero section).
\begin{definition} (see also \cite{R}, Definition IV-7.)
The frequency set of a semi-classical family, $\psim$, on $\bbR^n$
is defined by the following condition: $(q^0,p^0)\in\bbR^{2n}$ is
{\em not} in $\fs(\psim)$ iff there exist a cut-off function,
$\rho\in C_0^\infty(\bbR^n)$ and a neighborhood $V$ of $p$ in
$\bbR^n$ such that $\rho(q^0)\neq 0$ and $\forall N>0$ $\exists
C_N>0$ such that
\[
\forall p\in V\quad
\left|\, \widehat{\rho\psi_m}\, (m p)\,\right|\ \leq\ {C_N\over m^N}\ .
\]
\end{definition}
\medskip
As we will prove below, there is a close relationship between the
frequency set of a scf and the wave-front set of its $\h$-transform.
The result is as follows. Identify $P$ with the submanifold
of $\Sigma$ where $\kappa =1$:
\[
P\ =\ \{\ (x,\xi ; \theta,\kappa=1)\ \}
\]
\begin{proposition}
Let $\psim$ be a semi-classical family on a manifold $M$,
and let $\Psi$ be its $\h$-transform. Then the projection of the
intersection of the wave-front set of $\Psi$ with $P$ contains
the frequency set of $\psim$:
\[
\fs(\psim)\subset \pi\, \left(\, \wf\, (\Psi)\cap P\,\right)\ .
\]
\end{proposition}
To prove this proposition we will need to use the $\h$-transform of the
asymptotic Fourier transforms. We now digress to define it.
\begin{definition}
If $u\in C_0^\infty(\bbR^n\times S^1) $, define
\[
\calF (u) (p,s)\ =\ \int_{\bbR^n}\ u(x\, ,\, s-p\cdot x)\ dx\ .
\]
\end{definition}
If we decompose $u$ in its Fourier coefficients,
\[
u(x,\theta)\ =\ \sum_m\ u_m(x)\, e^{im\theta}\ ,
\]
then
\[
\calF (u) (p,s)\ =\ \sum_m\ \hat{u}_m\, (mp)\, e^{ims}\ .
\]
The following is an easy exercise:
\begin{lemma}
The operator $\calF$ is an elliptic Fourier integral operator associated
to the canonical relation
\be\label{asy6}
\{\ (q, \sigma p\, ;\, \theta , \sigma\, ;\, p,\sigma q\, ;\, s, \sigma)
\ ;\ s=\theta + qp\,,\; \sigma\neq 0\ \}\ .
\ee
\end{lemma}
\noindent
{\sc Proof of the Proposition.} Since the statement is local, we can
assume $M$ is an open set in $\bbR^n$. By ellipticity of $\calF$, the
wave-front set of $\calF\,(\Psi)$ is the image of $\wf\,(\Psi)$ under
the canonical relation (\ref{asy6}).
Let $(q^0,p^0)$ be a point in the complement of
$\pi\, \left(\, \wf\, (\Psi)\cap P\,\right)$. We will prove that it
is {\em not} in the frequency set of $\psim$. The assumption implies
that there exist: a cut-off function $\rho (q)$ such that
$\rho(q^0)\neq 0$ and a neighborhood $V$ containing $p^0$ such that
\be\label{asy7}
\forall\ \theta\ ,\ p\in V\ ,\ q\in\bbR^n\quad
(q,p\, ;\ \theta,1)\not\in\wf\, (\rho\Psi)\ .
\ee
By (\ref{asy6}), it follows that $\wf\, ( \calF\, (\rho\Psi))$ must
be empty: on the one hand, by (\ref{asy7}) it must be included
in $\{\ \sigma = 0\}$ but since by assumption
$\wf\,(\rho\Psi)\subset\Sigma$, again by (\ref{asy6})
$\wf\, ( \calF\, (\rho\Psi))$ is disjoint from this set. Hence
$\calF\, (\rho\Psi)$ is {\em smooth} On the other hand, it is easy to
see that
\[
\calF\,(\Psi)\, (p,s)\ =\ \sum_{m=1}^\infty\
\widehat{\rho\psi}_m\, (mp)\, e^{ims}\ .
\]
Hence the coefficients $\hat{\psi}_m\, (mp)$ are rapidly decreasing
in $m$, uniformly in $p$ in a perhaps smaller neighborhood $V$.
\smallskip
\hfill{$\Box$}
\smallskip
The proof of the previous proposition actually includes a
characterization of the projection of the wave front set of the
$\h$-transform of a scf:
\begin{corollary}
With the previous notations, $(q^0,p^0)$ is not in the projection
of $\wf\, (\Psi)\cap P$ iff there exist a cut-off function $\rho$
with $\rho(q^0)\neq 0$ and a neighborhood $V$ of $p^0$ such that
\[
\forall\ N>0\;,\;\alpha\quad\exists C>0 \quad\forall\ p\in V\ ,\ \ \quad
\left|\, D^\alpha\, \widehat{\rho\psi}_m\, (mp)\,\right|\leq
{C\over m^N}\ .
\]
\end{corollary}
\begin{definition}
The semi-classical wave-front set of $\psim$ is defined to be
the set of the previous corollary, that is, the projection of
$\wf\, (\Psi)\cap P$ to $T^*M$. It will be denoted $S\wf\, \psim$.
\end{definition}
Thus the $S\wf$ of a scf is a finer invariant than the frequency set of
the family: $\wf\, (\Psi)$ contains information on the phases of the
$\psi_m$, while the frequency set only pertains to their size.
\subsection{Lagrangian semi-classical states.}
\begin{definition}
Let $\Lambda_0\subset T^*M$ be an admissible Lagrangian submanifold.
A {\em Lagrangian semi-classical state} (scs) associated with
$\Lambda_0$ is a sequence $\psim$ of (square integrable) smooth
half-densities on $M$ such that its $\h$-transform,
\[
\Psi\ =\ \sum_{m=1}^\infty\ \psi_m\, e^{im\theta}
\]
is a classical Lagrangian distribution associated to a horizontal lift of
$\Lambda_0$,
\[
\Psi\in I^l\, (M\times S^1\; ,\; \Lambda)\ ,\quad \Lambda
\subset\Sigma\ .
\]
\end{definition}
Notice that, in particular, a scs is a semi-classical family; of course
one of a very special kind. We will see in fact that a scs is in
particular an oscillatory integral of the type considered by
Duistermaat, \cite{Du}, or a {\em compound asymptotic} in the
terminology of Guillemin and Sternberg, \cite{GS1}. For convenience,
we recall the local definition: locally a sequence $\psim$ is a compound
asymptotic or a classical oscillatory function associated with a
Lagrangian $\Lambda_0\subset T^*M$ iff it is of the form
\be\label{asy2.5}
\psi_m\, (x)\ =\ \int\ e^{im\phi_0\,(x,w)}\; a(x,w,m)\ dw
\ee
where $\phi_0$ is a non-degenerate phase function parametrizing
$\Lambda_0$ and $a(x; w,\mu)$ is a classical symbol:
\[
a(x; w,\mu)\ \sim\ \sum_{j=0}^\infty\ a_j(x,w)\; \mu^{l-j}
\]
in $C^\infty$ as $\mu\to\infty$. (The symbolic estimates on
$a$ are uniform on $(x,w)$ in compact sets.)
The point we wish to make in this section is that {\em by the
$\h$-transform, the theory of compound asymptotics reduces to the standard
theory of Lagrangian distributions}. We will derive a local
expression for a scs, and observe that it is identical to that of a
compound asymptotic. Before starting in that direction, we first notice
that scs's are invariant under multiplication/division by $m$, which
corresponds, on the $\h$-transform side, to differentiation/integration
with respect to the variable $\theta$:
\begin{lemma}\label{asy2200}
If $\psim$ is a scs, so are $\{\,m\psi_m\,\}$ and
$\{\, m^{-1}\psi_m\,\}$. The latter are associated to the same
Lagrangian $\Lambda\subset\Sigma$ as $\psim$.
\end{lemma}
\begin{proof}
If $\Psi$ is a Lagrangian distribution associated to a {\em closed}
Lagrangian $\Lambda\subset\Sigma$, then so is $D_\theta\Psi$, which
gives that $\{\,m\psi_m\,\}$ is a scs. To see that
\[
\sum_{m=1}^\infty\ {1\over m}\;\psi_m(x)\; e^{im\theta}
\]
is Lagrangian, we must use the fact that the obvious relative
parametrix of $D_\theta$ is a pseudodifferential operator associated
to a pair of intersecting Lagrangians: the diagonal in
$[T^*(M\times S^1)\setminus 0] \times [T^*(M\times S^1)\setminus 0]$
and the flow-out of $\{\;\kappa =0\;\}$, see \cite{GUh}.
Since $\Lambda$ by assumption does {\em not} intersect
$\{\;\kappa =0\;\}$, the action of this operator on $\Psi$ is identical
to that of a $\Psi$DO of order (-1).
\end{proof}
\medskip
\noindent
{\sc Remark:} The fact that the relative parametrix of $D_\theta$ acts
on semi-classical families as a pseudodifferential operator of order
$(-1)$ is fundamental for our approach to semi-classical problems,
and we will make extensive use of this fact.
\bigskip
We will now prove the equivalence of scs and compound asymptotics.
\begin{lemma}\label{asy2211}
Let $\phi_0(x;w)\in C^\infty(U\times W)$, $U\subset\bbR^n$ and
$W\subset\bbR^N$ open, be a non-degenerate phase function parametrizing
an admissible $\Lambda_0$. Then the phase
\be\label{asy2.1}
\phi(x,\theta\,;\, w ,\kappa)\ =\ \kappa\left(\, \theta + \phi_0(x,w)
\right)
\ee
is a non-degenerate phase function parametrizing a horizontal lift
of $\Lambda$. Here $(w ,\kappa)\in W\times\bbR^{+}$.
\end{lemma}
\begin{proof}
Consider the set $C_\phi$ defined by the equations
$\partial_\kappa\phi = 0$, $\partial_w\phi = 0$. Clearly
\be\label{asy2.2}
C_\phi\ =\ \{\; (x,\theta\, ;\, \kappa , w)\ ;\ \theta\,=\,-\phi_0(x,w)
\ \mbox{and}\ \partial_w\phi_0\,=\,0 \;\}\ .
\ee
It is obvious that the $\phi$ is non-degenerate if $\phi_0$ is, and
that the Lagrangian submanifold parametrized by $\phi$ is
\be\label{asy2.3}
\Lambda\ =\ \{\; (x,-\phi_0(x,w)\; ;\;\kappa\partial_w\phi_0(x,w) ,
\kappa) \ ;\ \partial_w(x,w)=0\; \kappa > 0\;\}\ .
\ee
If one here sets $\kappa = 1$ and divides by the action of $S^1$,
one obtains the Lagrangian parametrized by $\phi_0$. This is
sufficient to prove the lemma. We would like however to point out
explicitly that the rule
\[
C_{\phi_0}\ni\theta\ \mapsto\ \phi_0(x,w)
\]
(where $C_{\phi_0} = \{ (x,w)\;;\;\partial_w\phi_0 (x,w)=0\}$)
defines a function $f$ on $C_{\phi_0}$ such that
\be\label{asy2.4}
df\ =\ \iota^*(xdp)\ ,
\ee
where $\iota (x,w)\ =\ (x, (d_x\phi_0)_{(x,w)})$ is the map parametrizing
$\Lambda_0$ and $xdp$ is the canonical one-form of $T^*M$. Equation
(\ref{asy2.4}), which we leave as an easy exercise, implies that
on a horizontal lift of $\Lambda_0$, $\theta = \theta_0 -\phi_0(x,w)$,
which is satisfied by points on $\Lambda$ by (\ref{asy2.3}).
\end{proof}
\begin{proposition}\label{asy2222}
$\psim$ is a Lagrangian scs associated with $\Lambda_0$ iff it is a compound
asymptotics, that is iff it can be written locally in the form
(\ref{asy2.5}).
\end{proposition}
\begin{proof}
Assume $\psim$ is a Lagrangian scs.
By Lemma \ref{asy2211}, the $\h$-transform of $\psim$ is given locally
by an oscillatory integral of the form
\be\label{asy2.6}
\Psi\, (x,\theta)\ =\ \int\ e^{i\kappa\,(\theta + \phi_0(x,w)\,)}\
b(x,\theta\, ;\, w,\kappa)\ dw\;d\kappa\ ,
\ee
where $b$ is a classical symbol compactly supported in
$(x,\theta,w)$. Taking the $m$-th Fourier coefficient
of this, we find that $\psi_m$ can be written locally as
\[
\int\ e^{i\kappa\phi_0(x,w)}\ \tilde{b}\,(x,m-\kappa\,;\, w,\kappa)\
dw\; d\kappa\, ,
\]
where $\tilde{b}$ is the Fourier transform of $b$ in the $\theta$
variable:
\[
\tilde{b}\,(x,\mu\,;\, w,\kappa)\ =\ {1\over 2\pi}\ \int_0^{2\pi}\
e^{-i\mu\,\theta}\ b(x,\theta\,;\, w,\kappa)\ d\theta\ .
\]
(Without loss of generality we can take $b$ with small support in
$(x,\theta)$, and hence in the previous integral we might as well
integrate over the real line.)
Substituting $\kappa$ by $m-\kappa$, we obtain (\ref{asy2.5}) with
\be\label{asy2.7}
a(x,w,m)\ =\ \int\ e^{-i\kappa\phi_0(x,w)}\
\tilde{b}\,(x,\kappa\,;\, w, m-\kappa)\ d\kappa\ .
\ee
Our task is to show that this is a symbol, as indicated.
Since $b$ is a classical symbol, by assumption, we have an asymptotic
expansion
\be\label{asy2.8}
b(x,\theta\;,\;w,\kappa)\,\sim \,\sum_{j=0}^\infty\ b_j(x,\theta;w)\;
\kappa^{d-j}\ ,
\ee
as $\kappa\to\infty$. It is clear that a completely analogous
expansion holds for the Fourier transform of $b$ in the $\theta$
variable. Hence
\[
a(x,w,m)\ \sim\ \sum_{j=0}^\infty\ \int\ e^{-i\kappa\phi_0(x,w)}\
\tilde{b}_j\,(x,\kappa\,;\, w) (m-\kappa)^{d-j}\ d\kappa\ ,
\]
which by the inversion of the Fourier transform gives
\be\label{asy2.9}
a(x,w,m)\ \sim\ \sum_{j=0}^\infty\ \left(\; (m-D_\theta)^{d-j}\;
b_j\; \right) \; (x\,,\, -\phi_0(x,w)\,,\,w) \ .
\ee
Conversely, take an oscillatory integral of the form (\ref{asy2.5}).
Assume first that the amplitude $a$ is independent of $m$. Then
(by summing a geometric series) the $\h$-transform of $\psim$ is
\be\label{asy2.10}
\Psi\,(x,\theta)\ =\ \int\ {a(x,w)\over 1-e^{i[\theta + \phi_0(x,w)]}}
\ dw\ .
\ee
The function
\[
a(x,w)\over 1-e^{[i\theta + \phi_0(x,w)]}
\]
represents a distribution in $(x,\theta, w)$-space conormal to the
hypersurface $\theta = i\phi_0(x,w)$, and (\ref{asy2.10}) shows that
$\Psi$ is the push-forward of it to $(x,\theta)$ space. By the
assumption that $\phi_0$ is non-degenerate, this shows that
$\Psi$ is a Lagrangian distribution associated to
\[
\Lambda\ =\ \{\ (x,\theta\,;\,\kappa\,p , \kappa\ ;\ \exists w\
\theta = -\phi_0(x,w)\;,\; \partial_w\phi_0(x,w)\;,\; p=\partial_x
\phi_0(x,w)\ \}\ ,
\]
which is precisely a horizontal lift of $\Lambda_0$.
The general case (i.e. when $a$ depends symbolically on $m$) is
an asymptotic sum of derivatives and integrals (with respect to
$\theta$) of the previous case. Since these operations preserve
the class of scs, see Lemma \ref{asy2200}, the proof is finished.
\end{proof}
Finally, we note that Lagrangian scs's obviously have a symbol:
\begin{definition}
The symbol of a scs $\psim$ is defined as the symbol of its
$\h$-transform, $\Psi$.
\end{definition}
\subsection{Hermitian semi-classical states}
In this section we will construct semi-classical families associated
to admissible isotropic submanifolds, $\Gamma_0 \subset T^*Q$,
generalizing the construction of Lagrangian semi-classical states of
\S 2.2. Our
construction is similar in spirit to that of Ralston, \cite{Ra}, and Colin
de Verdi\`ere, \cite{CdV}, with the
difference that a symbol calculus (albeit a complicated one) exits in the
present case.
\medskip
Let $M$ be a smooth manifold, and $\Gamma\subset T^*M\setminus\{ 0\}$
a closed conic isotropic submanifold. In \cite{Gu}, \cite{BG}
Boutet de Monvel and Guillemin define classes of distributions on $M$,
$I^p(M,\Gamma)$, with wave-front set contained in $\Gamma$ and
generalizing the concept of Lagrangian distributions. The elements of
$I^p(M, \Gamma)$ are called Hermite distributions associated to $\Gamma$.
\begin{definition} A family $\psim$ in $L^2(Q)$ is a Hermite s.c state
associated to an admissible $\Gamma_0 \subset T^*Q$ iff
\[
\Psi\ =\ \sum_{m=1}^\infty\; \psi_m(x)\ e^{im\theta}
\]
converges weakly to a Hermite distribution on $Q\times S^1$
associated to a horizontal lift of $\Gamma_0$.
\end{definition}
Our treatment of Hermite families will be very brief. We will simply
indicate the local nature of the Hermitian states and give some examples.
It turns out that coherent states at a point, or wave-packets, form a
Hermite family. In the next section we will use this fact, together with
the invariance of the class of Hermite distributions under Fourier integral
operators, to derive a theorem of propagation of wave packets similar to
those of Cordoba-Fefferman, \cite{FC}, and Hagedorn, \cite{Ha}.
\medskip
We begin by discussing the local expressions of Hermite distributions.
For all that follows we refer the reader to the source, \cite{BG}.
Let $\Gamma\subset T^*U\setminus\{ 0\}$, $U\subset\bbR^m$ open,
be a closed isotropic submanifold of dimension $n-l$. Let
$\phi(x,w)\in C^\infty(U\times W)$ with $W\subset\bbR^N$ open
be a smooth phase function. Hence the critical set
\be\label{2c.1}
C_\phi\ =\ \{\; (x,w)\;;\; d_w\phi\ =\ 0\; \}
\ee
is a smooth manifold and the map
\be\label{2c.2}
\begin{array}{ccc}
C_\phi & \to & T^*U\setminus\{ 0\}\\
(x, w) & \mapsto & (x, d_x\phi)
\end{array}
\ee
is a Lagrangian immersion, which, since we are working locally, will be
assumed to be an embedding. To parametrize $\Gamma$, split the $w$
variables in two sets,
\be\label{2c.3}
w\ =\ (\tau\,,\, \eta)\ \in\ \bbR^{N-l}\times\bbR^l\ .
\ee
\begin{definition}
We say that $\phi$ is a non-degenerate function parametrizing $\Gamma$
iff:
\noindent
(a) $C_\phi$ intersects transversely the set
\[
\Omega \ :=\ \{ \ \eta\;=\;0\ \}\ ,\ \mbox{and}
\]
\noindent
(b) The embedding (\ref{2c.2}) maps $C_\phi\cap\Omega$ onto
$\Gamma$.
\end{definition}
\smallskip
Let us now assume that $\Gamma$ and $W$ are conic, and $\phi$ is positive
homogeneous of degree one in $w$.
Keeping the notations of the definition, Hermite distributions on $U$
associated with $\Gamma$ are given by oscillatory integrals of the form
\be\label{2c.4}
\int\; e^{i\phi (x,\omega)}\;
a\left(x,\tau , {\eta\over\sqrt{|\tau|}}\right)\; d\tau\, d\eta
\ee
where $a(x,\tau,\eta)$ has symbolic behavior on $\tau$ and is rapidly
decreasing in $\eta$. A bit more precisely, $a$ satisfies estimates of
the form
\be\label{2c.5}
|\;D_x^\alpha\, D_\tau^\beta\, D_\eta^\gamma\;
a(x,\tau,\eta) | \leq C\; |\tau|^{m-|\gamma|}\,
\left( 1+ |\eta |\right)^{-r}
\ee
on every compact $K\subset\subset U$ and for all $r>0$ and multiindices
$\alpha\,,\,\beta\,,\,\gamma$.
It is not hard to see that the wave-front set of (\ref{2c.4}) is contained in
$\Gamma$: by rapid decrease in $\eta$, the amplitude
\[
a\left(x,\tau , {\eta\over\sqrt{|\tau|}}\right)
\]
is rapidly decreasing uniformly in cones in $(\tau, \eta)$-space
with closure contained in the complement of $\Omega$. We will denote
the space of such symbols by $\frakH ^m (n-l, l)$.
\bigskip
We now turn to Hermitian semi-classical families. Let $\Gamma_0
\subset T^*Q$ be an admissible isotropic submanifold.
\begin{proposition}
A family $\psim$ is a Hermitian semi-classical state associated with
$\Gamma_0$ iff it can locally be written in the form
\be\label{2c.6}
\psim (x)\ =\ \int\; e^{im\phi_0(x,\eta)}\
a\left(x, m , {\eta\over \sqrt{m}}\right)\, d\eta
\ee
where $\phi_0$
is a non-degenerate phase function parametrizing an open subset of
$\Gamma_0$, and
\[
a(x,m,\eta)\ \sim\ \sum_{j=0}^\infty\; a_j(x,\eta)\; m^{l-j/2}
\]
where $\forall j\ \;a_j\in\frakH^j (n-l, l)$.
\end{proposition}
\begin{proof}
By the theorem of equivalence of phase functions of \cite{BG}, all
we need to notice is that the phase function
\[
\phi(x,\theta ; \eta, \kappa)\ =\ \kappa\left( \theta + \phi_0(x,\eta)\right)
\]
is a non-degenerate phase function parametrizing a horizontal lift of
$\Gamma_0$. The rest of the proof is identical to that of Proposition
\ref{asy2222}.
\end{proof}
\bigskip
We recall that the symbol of an Hermite distribution is a symplectic
spinor on the associated isotropic. By \cite{BG} the symbol
of an Hermitian scs can be identified with a symplectic spinor on
the isotropic $\Gamma_0\subset T^*M$. We make this explicit in the
following case:
\bigskip
\noindent
{\bf Example: Wave packets.} The simplest co-isotropic is of
course a single point, $\Gamma_0 = \{ (x_0,\xi_0)\}$.
\begin{definition}\label{def22cc1}
A generalized wave packet centered at $(x_0,\xi_0)\in T^*M$ is a Hermitian
semi-classical state associated with $\Gamma_0 = \{\,(x^0,\xi^0)\,\}$.
\end{definition}
As we will see (Lemma \ref{WP} below), the standard wave packets
(or coherent states) in
$\bbR^n$ are a particular case of Hermitian scs.
In \cite{PU1} we estimate the matrix
elements between eigenfunctions of a Hamiltonian and wave packets.
\medskip
Specializing (\ref{2c.6}), we see that a wave packet can be written
in local coordinates in the form
\be\label{w1}
\psi_m (x)\,=\,e^{im[(x-x_0)\cdot \xi_0 + \theta_0]}\,
\int e^{im(x-x_0)\cdot \eta}\,
a(x,m,{\eta\over \sqrt{m}})\, d\eta\,.
\ee
if we lift the point $(x_0,\xi_0)$ to $(x_0,\xi_0,\theta_0 )$.
\bigskip
Turning to the symbol of a wave packet, a symplectic spinor on
$\{ (x_0,\xi_0)\}$ is simply a smooth vector in the metaplectic
representation of the symplectic group of $W:=T_{(x_0,\xi_0)}(T^*M)$.
If we introduce a Riemannian metric on the manifold $M$, we get a
direct sum decomposition $W = V\oplus H$, where $V\cong T^*_{x_0}M$ is
the kernel of the cotangent fibration and $H$ the horizontal subspace.
Both $V$ and $H$ are Lagrangian subspaces; therefore (see \cite{LV} \S 1.2.5)
there is a natural realization of the metaplectic representation of
Mp$(W)$ in $L^2(T^*_{x_0}M)$. {\em Therefore the symbol of a wave packet
centered at $(x_0,\xi_0)$ is a Schwartz function on $T^*_{x_0}M$.}
The following definition is therefore natural. We mention that,
independently and simultaneously, Karasev and Vorobjev have
arrived at a similar concept, see \cite{Kar} \S 2v.
\begin{definition}
A Gaussian wave packet is a Hermitian scs whose symbol is a Gaussian.
\end{definition}
\medskip
Observe that in the representation (\ref{w1}), the parametrizing map
\be\label{w2}
\begin{array}{rcc}
C_{\phi_0}= \{ (x_0,\eta)\}& \to & T^*M\\
(x_0,\eta) & \mapsto & (x_0, \xi_0+\eta)
\end{array}
\ee
identifies the space of the $\eta$ variables with $T_{x_0}^*M$.
\begin{lemma}
Under (\ref{w2}), the symbol of the Hermitian scs (\ref{w1}) is
the Schwartz function $\eta\mapsto a_0(x_0,\eta)$.
\end{lemma}
The proof follows from the definition of the symbol of an Hermite
distribution given by an oscillatory integral representation, \cite{BG}.
\bigskip
Let $\{\psi_m\}$ be a Gaussian wave packet centered at $(x_0,\xi_0)$, and
introduce local coordinates near $x_0$.
Then $\psi_m$ is given by an expression of the form (\ref{w1}) where,
by hypotheses, $a_0(x_0,\eta)$ is a Gaussian in $\eta$, that is
\[
a_0(x_0,\eta)\,=\, e^{-\eta\cdot A\eta/2}\,,
\]
where $A$ is a positive definite symmetric matrix.
Modeling on the case of $\bbR^n$, let us pick
\[
\theta_0\,=\, {1\over 2}\, x_0\cdot\xi_0\,,
\]
a choice dictated by Heisenberg group considerations that we
won't go into.
\begin{lemma}\label{WP}
There exists $\rho\in C_0^\infty$ such that $\rho(x_0)=1$ and
\[
\psi_m\,=\,(2\pi)^{n/2}\,|A|^{-1/2}\,
\rho(x)\, e^{im[x\cdot\xi_0- x_0\cdot\xi_0/2]}\,
e^{-m(x-x_0)\cdot A^{-1}(x-x_0)/2} +
O(m^{l-1/2}).
\]
\end{lemma}
\begin{proof}
Write
\[
a_0(x,\eta)\,=\,a_0(x_0,\eta) + \sum_{j=1}^n\,(x-x_0)_j\,b_j(x,\eta)\,,
\]
where $(x-x_0)_j$ is the $j$th coordinate of $x-x_0$. This can be
done in a neighborhood of $x_0$. Let $\rho$ have support in that
neighborhood and satisfy $\rho(x_0)=1$. We must estimate the integrals
\[
I_j(x,m)\:=
\int\, e^{im(x-x_0)\cdot\eta}\,\rho(x)\,(x-x_0)_j\,b_j(x,\eta/\sqrt{m})
\,d\eta\,.
\]
We integrate by parts: Since $D_{\eta_j}e^{im(x-x_0)\cdot\eta} =
m(x-x_0)_j e^{im(x-x_0)\cdot\eta}$,
\[
I_j(x,m)\,=\,{-1\over m^{3/2}}\,\int\,
e^{im(x-x_0)\cdot\eta}\,\rho(x)\,(D_{\eta_j}b_j)(x,\eta/\sqrt{m})
\,d\eta\,,
\]
and the integral is easily seen to be $O(1)$, uniformly on $x$ near $x_0$.
\end{proof}
%As we will now see
%on any Riemannian manifold there is a fairly natural way of constructing
%``Gaussian wave packets" at every $\{(x_0,\xi_0)\}\in T^*M\setminus\{ 0\}$,
%wich are special cases of Hermitian sc states. (See also \cite{Kar} for
%an analogous construction.) Let $\rho\in C^\infty (M)$ with
%$\rho(x_0)\not= 0$ supported in a domain of normal coordinates centered
%at $x_0$. A Gaussian wave packet is
%a Hermitian scs associated with $\Gamma_0$ with an amplitude $a(x,\tau,
%\eta/\sqrt{\tau})$ of the type (in normal coordinates)
%\[
%a(x,\tau,\eta)\,\sim\,\sum_{j=0}^\infty\, a_j(x,\eta)\,\tau^j\,,
%\]
%where $a_0(x,\eta)$ is of the form $\rho(x)$ times a Gaussian in $\eta$.
%By the previous Proposition, such a state can be written in the form
%\be\label{2c.7}
%\psi^{(x_0,\xi_0)}_m\,=\,\rho(x)\,e^{im\langle x, \xi_0\rangle }\,
%e^{-m Q(x-x_0)}\,+\,O(1/\sqrt{m})
%\ee
%where $Q$ is a quadratic form on $T_{x_0}M$ with positive definite real part.
%In case $M=\bbR^n$, the leading term is precisely the usual definition
%of coherent states.
\section{Semi-Classical Fourier Integral Operators}
\newcommand{\hFIO}{scFIO}
We now discuss the operators whose Schwartz kernels are semi-classical
states associated to non-compact Lagrangians. We begin with a
preliminary remark:
\begin{lemma}\label{FIO1}
Let $M_1$ and $M_2$ be manifolds, and
\[
\calF\,: C^\infty(M_1\times S^1)\to C^\infty(M_2\times S^1)
\]
an operator. Then $\calF$ intertwines the $S^1$ actions iff its
Schwartz kernel is the pull-back by the map
\[
\begin{array}{rcl}
\pi\,: M_2\times S^1\times M_1\times S^1 &\to & M_2\times M_1\times S^1\\
(x_2,e^{i\theta_2},x_1,e^{i\theta_1}) & \mapsto &
(x_2,x_1,e^{i(\theta_1-\theta_2)})
\end{array}
\]
of a distribution $\calF_0$ on $M_2\times M_1\times S^1$. If this is
the case, $\calF$ is a Fourier integral operator associated to the
canonical relation $\overline{\Gamma}\subset
T^*(M_1\times S^1\times M_2\times S^1)$
iff $\calF_0$ is a Lagrangian distribution associated to the Lagrangian
submanifold
\be\label{fio1}
\Lambda\,=\,\{\,(x_2,x_1,e^{i\theta};\xi_2,\xi_1,\kappa)\,;\,
(x_2,e^{i\theta},x_1,1,\xi_2,\kappa,-\xi_1,\kappa)\in\overline{\Gamma}\,\}\,.
\ee
\end{lemma}
\begin{proof}
Part one of the lemma follows from the fact that the operator $\calF$
intertwines the $S^1$ actions iff its Schwartz kernel is annihilated by
the vector field
${\partial\over\partial\theta_1}+{\partial\over\partial\theta_2}$.
>From this it also follows that its wave-front set is contained in
$\{\kappa_1=-\kappa_2\}$. We leave the rest of the proof as an
exercise (introduce local coordinates $\theta_1-\theta_2$ and
$\theta_1+\theta_2$).
\end{proof}
We will usually identify the Schwartz kernel of an operator $\calF$
that intertwines the $S^1$ actions with the distribution $\calF_0$ of
the Lemma.
\bigskip
Let $\Gamma\subset T^*M_1\times T^*M_2$ be a canonical relation.
Recall the notation: $\Gamma'$ is the set of all $(x_1,p_1;x_2,p_2)$ such
that $(x_1,p_1;x_2,-p_2)\in\Gamma$. By definition $\Gamma$ is a canonical
relation iff $\Gamma'$ is a Lagrangian submanifold of $T^*M_1\times T^*M_2$.
\begin{definition}
(a) A canonical relation $\Gamma\subset T^*M_1\times T^*M_2$ will be
called admissible iff $\Gamma'$ is admissible in the sense of \S 1.2.
(b) If $\Gamma\subset T^*M_1\times T^*M_2$ is an admissible canonical
relation, a sequence of operators $\{F_k\}$,
\[
F_k\,: C^\infty(M_1)\to C^\infty(M_2)\quad k=1,2,\ldots
\]
will be called a semi-classical Fourier integral operator (\hFIO) from
$M_1$ to $M_2$ iff the $\h$-transform ($\h=1/k$)
of the Schwartz kernels of the
$F_k$ is a Lagrangian scs associated with a horizontal lifting of
$\Gamma'$.
\end{definition}
Notice that, by Lemma \ref{FIO1}, if $\{F_k\}$ is a scFIO then the
operator $\calF$ from $M_1\times S^1$ to $M_2\times S^1$ which is defined
by $\calF |_{D_{\theta_1}=k=D_{\theta_2}}= F_k$
is an ordinary Fourier Integral Operator.
Our main example will be the approximation to the propagator for a
Schr\"odinger-type operator that will be constructed in \S 4.
\medskip
Directly from their definition, and by the general theory of Fourier
Integral Operators, we see that scFIOs have a symbol calculus.
We won't write down the the details of this, since it would be
largely a repetition of the corresponding results for standard FIOs.
As an indication of how the theory of scFIOs reduce to that of standard
FIOs, we'll simply state the following result on propagation of
wave-packets and of Lagrangian scs's:
\begin{theorem}
Let $F_k : C^\infty(M_1)\to C^\infty(M_2)$ be a scFIO associated to the
admissible canonical relation $\Gamma$, and let $\{\psi_k\}$ be a Lagrangian
scs associate to an admissible $\Lambda\subset T^*M_1$.
Assume that $\Gamma$ intersects $\Lambda\times T^*M_2$ cleanly. Then
\[
\Gamma (\Lambda)\,=\,\{\,\overline{\xi}\in T^*M_2\,;\,\exists\,
\overline{\eta}\ \mbox{such that}\ (\overline{\eta},\overline{\xi})\in
\Lambda\,\}\,,
\]
is an immersed Lagrangian submanifold in $T^*M_2$, and
the sequence $\{F_k(\psi_k)\}$ is a Lagrangian scs associated to it.
\end{theorem}
\bigskip
By a straightforward generalization of
the proof of Proposition (\ref{asy2222})
we obtain
an oscillatory integral representation of the Schwartz kernels of
a scFIO, $F=\{F_k\}$:
\begin{lemma}\label{intrep}
If $\phi(x,y,w)$ is a phase function locally parametrizing the
Lagrangian relation associated with $F$, then locally the Shwartz
kernel of $F_k$ is of the form
\[
F_x(x,y)\,=\, \int e^{ik\phi(x,y,w)}\,a(x,y,w,k)\,dw\ + O(k^{-\infty})\,,
\]
where $a$ is a symbol as before (compactly supported in $(x,y,w)$).
The error is in the $C^\infty$ topology.
\end{lemma}
Observe that the estimates are microlocally on compact
subsets of the Lagrangian relation associated with $F$.
We will discuss the case of sc$\Psi$DOs (the scFIOs with $M_1=M_2=M$
and $\Gamma$ the diagonal) in the Appendix.
\section{The Propagator.}
\newcommand{\sh}{S_\hbar}
%\newcommand{\psim}{\{\psi_m\}}
In this section we will construct semi-classical approximations to
the propagator of a Schr\"odinger-type operator on a manifold $M$.
The $\h$-transform of the approximating operators will be an
ordinary Fourier integral operator on the circle bundle $M\times S^1$.
In fact the sequence of the Schwartz kernels of their Fourier
coefficients will be a Lagrangian scs. The approximation will be valid
in a compact range of energies, which accounts for the
finite-propagation speed of singularities proper of FIOs.
\bigskip
The setting is as follows. Let $M$ be a compact manifold, and consider
an $\hbar$-dependent self-adjoint operator of the form
\be\label{3.1}
\sh\ =\ \sum_{l=0}^N\; \hbar^l\; P_l\ ,
\ee
where for each $l$ $P_l$ is a differential operator of order $l$. The
symbol of $\sh$ is defined to be the smooth function on $T^*M$
\be\label{3.2}
H(x,p)\ =\ \sum_{l=0}^N\; \sigma_{P_l} (x,p)
\ee
where $\sigma_l$ is the principal symbol of $P_l$.
We will make henceforth the following
\par\smallskip\noindent
{ \large \bf Assumptions:}
\be\label{3.3.1}
\forall\; (x,p)\in T^*M\quad H(x,p)\; >\; 0\,,\quad\mbox{and}
\ee
\be\label{3.3.2}
P_N\quad\mbox{is elliptic.}
\ee
\noindent
{\sc Remarks:} (a) Condition (\ref{3.3.1}) can be weakened to:
$H$ bounded below.
Indeed it suffices to add a large enough constant to $\sh$ to ensure
(\ref{3.3.1}); the reader can check that the statements of the trace
formula and the propagation of semi-classical families theorem below
are insensitive to the subtraction of a constant from the
Hamiltonian. We assume (\ref{3.3.1}) for convenience of exposition.
\par\noindent
(b) Condition (\ref{3.3.1}) does not imply that $P_N$ must be
elliptic: consider $S_{\h} = \h^2D_1^2 + 1$.
\par\noindent
(c) Condition (\ref{3.3.1}) implies that
the symbol of the zeroth order operator $P_0$ must be a positive
function: $H(x,0)=\sigma_{P_0}(x)>0$.
\medskip
The ellipticity condition implies that, for every $\h>0$, the spectrum
of $S_\h$ is discrete (see below).
We will denote by $\{ \psi_{j,\h}\}$ an orthonormal
basis of $L^2(M)$ of eigenfunctions of $S_\h$, and by $E_{j,\h}$ the
associated eigenfunctions:
\be\label{3.3.3}
S_\h (\psi_{j,\h})\,=\,E_{j,\h}\,\psi_{j,\h}\,.
\ee
\medskip
The $\h$-transform of $\sh$ is defined to be the following operator
on $M\times S^1$:
\be\label{3.4}
P\ :=\ \sum_{l=0}^N\; D_\theta^{-l}\; P_l\ ,
\ee
where we denote by $D_\theta^{-l}$ the obvious relative parametrix of
$D_\theta$: the operator equal to $k^{-1}$ on the $k$-th eigenspace of
$D_\theta$ for all non-zero $k$ and equal to zero on the kernel of
$D_\theta$. Explicitly,
\be\label{3.5}
\left(\, D_\theta^{-1}\, u\,\right)\, (x, \theta)\ =\
{i\over 2\pi}\;\int_0^{2\pi}\; dt\; \int_0^t\; \left(
u(x, \theta - s)-\overline{u}(x)\right)\; ds
\ee
where
\be\label{3.5.1}
\overline{u}(x)\,=\,{1\over 2\pi}\;\int_0^{2\pi}\;u(x,\theta) d\theta
\ee
is the projection of $u$ onto the kernel of $D_\theta$. The operator
$D^{-1}_\theta$ is {\em not} an ordinary pseudodifferential operator
since $D_\theta$ is not elliptic. $D_\theta^{-1}$ is however a
$\Psi$DO with singular symbol, in the sense of Melrose-Uhlmann and
Guillemin- Uhlmann, \cite{MUh}, \cite{GUh}. It will become an ordinary
$\Psi$DO after suitable microlocalization, which amounts to the
localization in energy for $\sh$ alluded to above.
\medskip
We call $P$ the $\h$-transform of $\{\sh\}$ because $\forall k\in\bbZ^{+}$
\be\label{c.1}
P|_{\calH_k}\cong S_{1/k}\,,
\ee
where $\calH_k$ is the $k$-th eigenspace of $D_\theta$; indeed in
$P|_{\calH_k}$ one can replace $D_\theta$ by $k$, thereby obtaining
$S_{1/k}$. To be more specific, let us now introduce the joint
eigenvalues and eigenfunctions of the commuting pair $(P, D_\theta)$.
By the previous assumptions, the operator
\be\label{3.6}
R\,=\,\sum_{l=0}^N\; D_\theta^{N-l}\; P_l \,=\, D_\theta^NP
\ee
is an
$N$-th order {\em elliptic differential operator with positive symbol}
equal to
\be\label{3.7} \sigma_R\, (x,\xi; \theta,\kappa)\,=\,\;
\sum_{l=0}^N\; \kappa^{N-l}\sigma_{P_l} (x, \xi)\,=\, \kappa^N\,
H(x,\xi/\kappa)\,. \ee
Therefore the spectrum of $R$ is bounded below
and consists of eigenvalues with finite multiplicities and converging
to infinity. Let $\{ \Psi_{k,j}(x,\theta)\}$
(where $k\in\bbZ$ and $j=1,2,\ldots$) be an orthonormal basis of
$L^2(M\times S^1)$ consisting of joint eigenvectors for the pair
$(R,D_\theta)$ with joint eigenvalues $\{(\lambda_{k,j},k)\}$:
\be\label{c.2}
\left\{
\begin{array}{rcl}
R\Psi_{k,j} & = & \lambda_{k,j}\,\Psi_{k,j} \\
D_\theta\Psi_{k,j} & = & k\,\Psi_{k,j}\;.
\end{array}
\right.
\ee
>From the second equation follows that $\Psi_{k,j}$ is of the form
\be\label{c.3}
\Psi_{k,j}(x,\theta)\,=\, e^{ik\theta}\,\psi_{k,j}(x)
\ee
for some function $\psi_{k,j}$ on $M$. Plugging back in the first
equation and recalling the definition of $D_\theta^{-1}$, get
\be\label{c.4}
R\Psi_{k,j} \,=\, e^{ik\theta}\,k^N\sum_{l=0}^N\,k^{-l}\,P_l(\psi_{k,j})
\ee
and so the first eigenvalue equation in (\ref{c.2}) becomes (upon division
by $e^{ik\theta}k^N$)
\be\label{c.5}
S_{1/k}\,\psi_{k,j}\,=\, E_j(k)\,\psi_{k,j}\ ,\quad\mbox{where}\quad
E_j(k)\,=\,k^{-N}\,\lambda_{k,j}\,.
\ee
This procedure can be reversed, and we see that the eigenvalue problems
(\ref{c.2}) and (\ref{c.5}) are equivalent via (\ref{c.3}).
\subsection{Energy localization}
We now proceed to microlocalize $P$ in an open cone away from $\{\kappa =0\}$.
The result is an {\em ordinary} $\Psi$DO, $P_f$, that approximates $P$ within a
bounded range of energies. Begin by defining
\be\label{c.6}
Q\, :=\, R^{1/N}\,.
\ee
This is a first-order, elliptic, classical self-adjoint pseudodifferential
operator with eigenfunctions $\Psi_{k,j}$ and eigenvalues
\be\label{c.7}
\mu_{k,j}\,=\,(\lambda_{k,j})^{1/N}\,=\, |k|\,(E_j(k))^{1/N}\,.
\ee
\begin{lemma}\label{tres.1}
Let $f\in C_0^\infty(\bbR^{+})$ be
identically one on an open interval $I_f$ and $0\leq f\leq 1$ everywhere.
Then the operator
\[
Q_f\ =\ f\left( D_\theta Q^{-1}\right)
\]
(defined by the spectral theorem)
is a classical zeroth order pseudodifferential operator whose symbol
is zero in a neighborhood of $\{\kappa = 0\}$ and
\[
\sigma_{Q_f} (x,\xi; \theta, \kappa)\ =\ f\left(\ H(x, \xi/\kappa)^{-1/N}
\right)
\]
elsewhere. Moreover:
\par\noindent
(i) $Q_f$ is microlocally supported in the set
\be\label{3.8}
\calU_f\ =\ \{\; (x, \xi ; \theta ,\kappa)\ ;\ H(x , \xi/\kappa)^{-1/N}\in
\supp (f)\;\}\ ,\quad\mbox{and}
\ee
\par\noindent
(ii) $Q_f$ is microlocally equal to the identity in the set
\be\label{3.9}
\calV_f\ =\ \{\; (x, \xi ; \theta ,\kappa)\ ;\ H(x , \xi/\kappa)^{-1/N}\in
I\;\}\,.
\ee
\end{lemma}
\noindent
{\sc Remarks:} (a) The statement about the microlocal support
of $Q_f$ means that
\be\label{3.10}
\forall u\in\calD' (M\times S^1)\quad\quad
\wf (u)\cap \calU_f\; =\;\emptyset \Rightarrow \wf \left[ Q_f(u)\right]\; =\;
\emptyset\ ,
\ee
while the statement about $Q_f$ being microlocally the identity means that
\be\label{3.11}
\forall u\in\calD' (M\times S^1)\quad\quad
\wf (u)\subset \calV_f \Rightarrow \wf \left[ u - Q_f(u)\right]\; =\;
\emptyset\ .
\ee
\par\noindent
(b) The satements (i) and (ii) in the lemma also hold for any power of
$Q_f$.
\par\noindent
(c) $Q_f$ clearly commutes with both $P$ and $D_\theta$.
\begin{proof} The operator inside $f$ is a classical zeroth order
self-adjoint $\Psi$DO, and the first conclusion about $Q_f$ is a direct
application of Theorem I of the appendix of \cite{GS9}. The statements
(\ref{3.10}) and (\ref{3.11}) follow from Theorem 0.6 of \cite{CdV1},
in the following way. The couple of operators $(Q, D_\theta)$ form
a commuting elliptic pair of first-order self-adjoint pseudodifferential
operators, with joint spectrum $\{(\mu_{k,j},k)\}$. Let
\be\label{c.8}
J\,: T^*(M\times S^1)\setminus\{0\}\to\bbR^2\setminus\{0\}
\ee
be the map with components $J=(q,\kappa)$ where $q$ is the symbol of
$Q$. The Theorem cited says the following:
{\em Let $u=\sum_{k,j}u_{k,j}\Psi_{k,j}$ and $\Gamma\subset\bbR^2\setminus
\{ 0\}$ be an open cone. Then}
\be\label{c.8.1}
\bigl(\,\forall\ (k,j)\ \mbox{such that}\
(\mu_{k,j},k)\in \Gamma\ \ u_{k,j}=0\,\bigr)\ \Rightarrow\
\wf (u)\cap J^{-1}(\Gamma) = \emptyset\,.
\ee
Let $C_f$, $C_I\subset\bbR^2\setminus\{0\}$ be the cones
\be\label{c.8.2}
C_f\,=\, \{\,(z_1,z_2)\,;\, z_2/z_1\in\mbox{supp}\,(f)\,\} \quad
\mbox{and}\quad
C_I\,=\, \{\,(z_1,z_2)\,;\, z_2/z_1\in I\,\}\,.
\ee
Then
\be\label{c.9}
\calU_f\,=\,J^{-1}(C_f)\quad \mbox{and}\quad
\calV_f\,=\,J^{-1}(C_I)\,,
\ee
and $\forall\; (k,j)$
\be\label{c.10}
(\mu_{k,j},k)\not\in C_f\Rightarrow Q(\Psi_{k,j})\,=\,0\,,\quad
\mbox{while}\quad
(\mu_{k,j},k)\in C_I\Rightarrow Q(\Psi_{k,j})\,=\,\Psi_{k,j}\,.
\ee
Consider now $u\in\calD '(M\times S^1)$,
write $u =\sum_{k,j} u_{k,j}\Psi_{k,j}$ and
$Q(u) = \sum_{k,j} a_{k,j}\Psi_{k,j}$. By (\ref{c.10}),
$\forall (k,j)$
\be\label{c.11}
(\mu_{k,j},k)\not\in C_f\Rightarrow a_{k,j}\,=\,0\,,\quad
\mbox{and}\quad
(\mu_{k,j},k)\in C_I\Rightarrow a_{k,j}\,=\,u_{k,j}\,.
\ee
(\ref{3.10}) and (\ref{3.11}) follow easily from statements
(\ref{c.8.1}) through (\ref{c.11}).
\end{proof}
\medskip
We will use the operator $Q_f$ to microlocalize the operator $P$. As
mentioned earlier, the operator $D_\theta^{-1}$ is a pseudodifferential
operator with singular symbol. Precisely, its Schwartz kernel is in a
class $I^{p,l}((M\times S^1)^2 , \Lambda_0, \Lambda_1)$
of singular Lagrangian distributions of Guillemin and Uhlmann,
where $\Lambda_0$ is the conormal
to the diagonal in $(M\times S^1)^2$ and $\Lambda_1$ if the flow-out
from $\{ \kappa = 0\}$. We refer the reader to \cite{GUh} for the theory
of such distributions. The properties of these
operators that we presently need are: (1) they are stable under
the action of FIOs, and in particular $\Psi$DOs, and (2) microlocally
away form the intersection $\Lambda_0\cap\Lambda_1$ they are ordinary
Fourier integral operators. These facts together with Lemma \ref{tres.1}
imply:
\begin{lemma}
The operator
\be\label{3.12}
B\ :=\ Q_f\, D_\theta^{-1}
\ee
is a classical pseudodifferential operator of order
$(-1)$. Moreover $B$ is microlocally supported in $\calU_f$ and it is
microlocally equal to $D_\theta^{-1}$ in $\calV_f$.
\end{lemma}
We now define the microlocalized version of $P$:
\be\label{3.13}
P_f\ :=\ B^N\,R\ =\ B^N\; \sum_{l=0}^N\; D_\theta^{N-l}\, P_l\ ,
\ee
which is easily seen to equal
\be\label{3.14}
P_f\ =\ Q_f^N \, P\, .
\ee
The Fourier coefficients of $P_f$ are (with $\h=1/m$)
\be\label{3.13a}
\Bigl( P_f \Bigr)_m\,=\, [\, f(S_\h)^{-1/N}\,]^N\, S_\h\,.
\ee
We summarize the properties of $P_f$ that we need in the following:
\begin{proposition}
The operator $P_f$ is a classical zeroth order pseudodifferential operator
on $M\times S^1$ commuting with $D_\theta$. It is microlocally supported
in $\calU_f$ and it is microlocally equal to $P$ in $\calV_f$. In particular,
in $\calV_f$ its symbol is
\be\label{3.15}
\sigma_{P_f} |_{\calV_f} (x, \xi; \theta, \kappa)\ =\ H(x, \xi/\kappa)\ .
\ee
\end{proposition}
In the next paragraph we use $P_f$ to approximate $\exp itD_\theta P$.
\subsection{Propagation of sc states}
As a first application of the previous construction of $P_f$, we will
now prove the following theorem:
\begin{theorem}\label{propaga}
Let $\Lambda_0\subset T^*M$ be an admissible Lagrangian (resp. isotropic)
submanifold, and let $\psim$ be an associated Lagrangian (resp. Hermitian)
semi-classical state. Then, for every $t$, the family
\be\label{3.16}
\{\;\exp (-itm S_{1/m})(\psi_m)\; \}
\ee
is a Lagrangian (resp. isotropic) family associated to the manifold
$\phi_t(\Lambda_0)$, where $\{\phi_t\}$ denotes the Hamilton flow of
$H(x,p)$.
\end{theorem}
\medskip
\noindent
{\sc Remarks:}
(1) The exponential appearing in (\ref{3.16}) is the fundamental
solution of Schr\"odinger's equation
\[
i\hbar {\partial u\over\partial t}\ =\ S_\hbar\, (u)
\]
with $\hbar = 1/m$. Again, the restriction of $\h$ to the reciprocals
of integers is inessential.
\smallskip\noindent
(2) Although we won't do so here, we could compute the symbol of the image
family. For the wave packets of \S 2 this yields formulae analogous
to those of Cordoba-Fefferman and Hagedorn, \cite{FC}, \cite{Ha}.
See \cite{PU1} for estimates of matrix elements between a wave
packet and the eigenfunctions of the Hamiltonian.
\bigskip
The proof of this Theorem is based on the following lemma:
\begin{lemma}
Let $f\in C_0^\infty(\bbR^{+})$ and
let $\psi\in\calD ' (M\times S^1)$ be such that $\wf (\psi)\subset
\calV_f$. Then
\[
\left[\; e^{itD_\theta P} - e^{itD_\theta P_f} \;\right]\, (\psi)
\]
is smooth.
\end{lemma}
\begin{proof}
Write $e^{itD_\theta P_f} - e^{itD_\theta P}=
e^{itD_\theta P}\left[ e^{itT} - I\right]$, where $
T\, = \,D_\theta( P_f - P)\, =\, D_\theta P (Q_f^{2N} - I)$.
Then $\Psi := \left[e^{itT} - I \right] (\psi)$ can be written as
\[
\Psi\,=\, ie^{itT}\,\left(\,\int_0^t\, e^{-isT}\,T(\psi)\,ds
\,\right)
\]
(Duhamel's principle).
Since $\wf (\psi)\subset\calV_f$, $Q_f^{2N}\psi -\psi$ is smooth; hence
$T(\psi)$ is smooth. Since the operator $\exp(isT)$ maps smooth functions
into smooth functions, $\Psi$ is smooth and hence
\[
\left[\, e^{itD_\theta P} - e^{itD_\theta P_f} \,\right]\, (\psi)
\,=\, -e^{itD_\theta P}(\Psi)
\]
is smooth.
\end{proof}
Although we won't need it right away, we note the following corollary
that we will use in the proof of the trace formula:
\begin{corollary}\label{crucial}
With the notations of Lemma \ref{tres.1}, let
$f,g\in C_0^\infty(\bbR^{+})$ be such that $\mbox{supp} (f)\subset I_g$.
Then
\be\label{3.16.1}
\left[\; e^{itD_\theta P} - e^{itD_\theta P_g} \;\right]\, Q_f
\ee
is a smoothing operator.
\end{corollary}
\medskip\noindent
{\sc Proof of Theorem} \ref{propaga} Choose $f\in
C_0^\infty(\bbR)$ as in \S 3.1, making sure that the set $\calV_f$
contains $\Lambda$, the horizontal lift of $\Lambda_0$ with respect to
which the $\h$-transform of $\psim$ is Lagrangian (resp. Hermitian). This
is indeed possible since by admissibility $\Lambda$ is a closed conic
set in $\Sigma$. From the lemma follows that the $\hbar$ transform of
$\{\;\exp (itm S_{1/m})(\psi_m)\; \}$ and
\be\label{3.17}
\exp (itD_\theta P_f)\, (\psi)\ ,
\ee
where $\psi$ is the $\hbar$ transform of $\psim$, differ by a smooth
function on $M\times S^1$, and hence one is a Lagrangian (resp.
Hermitian) sc state iff the other one is. We have seen that $Q_f$
is a classical self-adjoint zeroth order pseudodifferential operator, so
$W = D_\theta P_f$ is a first order operator and $\exp (itW)$ is
a Fourier integral operator whose canonical relation is the graph
of $G_t$, the time $t$ map of the Hamilton flow of the symbol of $W$.
By the invariance of Lagrangian (resp. Hermitian) distributions under
FIO's associated to canonical graphs, we conclude that the $\hbar$
transform of (\ref{3.16}) is a Lagrangian (resp. Hermitian) distribution
associated to $G_t(\Lambda)$. On $\calV_f$ however the symbol of $W$ is
\[
\kappa\, H(x, \xi/\kappa)\ ,
\]
from which it is clear that the reduction of $G_t(\Lambda)$ is
$\phi_t(\Lambda_0)$, and the proof of the Theorem is complete.
\par\nobreak\hfill $\Box$
\medskip
The sequence of Fourier coefficients of the Schwartz kernel of the
operator $\exp (i t D_\theta P_f)$ is therefore a Lagrangian
semi-classical family associated to the graph of the flow at time $t$
of the reduction of the symbol of $P_f$. The latter agrees with the
classical Hamiltonian an the open set of the form $H^{-1}(E_1, E_2)$,
and is zero outside a larger set of the same form.
\bigskip
We finish with a local oscillatory integral representation of the
Shwartz kernel of the operator $\calU_f:=\exp (i t D_\theta P_f)$. We
are only interested in an expression valid microlocally in the open
cone $\calV_f$, which is invariant under the Hamilton flow of the
symbol of $D_\theta P_f$. By the previous paragraph and
Lemma (\ref{intrep}), microlocally in the open set
\[
X_I:= \{\,(x,p)\in T^*M\,;\, H(x,p)^{-1/N}\in \mbox{int} I\,\}
\]
(which is invariant under the Hamilton flow of $H$)
the Schwartz kernel of the operators $\calU_{f,m} = \exp(itm (P_f)_m)$
can be written as oscillatory integrals of the form
\[
\int\, e^{im\psi(t,x,y,w)}\,a(t,x,y,w;m)\,dw
\]
where $\psi$ is a non-degenerate phase function parametrizing the
graph of the Hamilton flow of $H$ in $X_I$. For $|t|$ small,
it is natural to take $\psi$ of the form
\[
\psi(t,x,y,w)\,=\, S(t,x,w)-y\cdot w
\]
where $S$ is the solution of the Hamilton-Jacobi equation
\[
\left\{
\begin{array}{lcc}
{\partial S\over\partial t}+ H(x,\nabla_xS) & = & 0\\
S(t=0,x,w) & = & x\cdot w
\end{array}
\right.
\]
The resulting expression (valid for small time)
is well known on $\bbR^n$, in the theory
of Helffer and Robert, \cite{HR1}. (See also recent work of Meinrenken,
\cite{Mein}, for an extension of that theory to all $t$).
Our formulation however shows that $\calU_f$ is globally a
Lagrangian distribution, and therefore can be treated with
standard FIO techniques.
\section{The Semiclassical Trace Formula}
\subsection{The statement}
The setting and notations in this section are the same
as is \S 3: we consider a Schr\"odinger-type operator
$S_\h$ on a compact manifold $M$ with associated classical
Hamiltonian $H(x,p)$ a smooth function on $T^*M$. Both
assumptions (\ref{3.3.1}) and (\ref{3.3.2}) will be in
effect.
\smallskip
The semiclassical trace formula (STF) is a statement about the asymptotic
distribution of the spectral data of $S_\h$, as $\h\to 0$,
more precisely, about the asymptotic behavior of sums of the form
\be\label{4a.1}
\sum_{j=1}^\infty\,\,
\varphi \left( {E_{j,\h}-E\over \h}\right)
\quad\quad \mbox{as}\ \h\to 0\,.
\ee
This formula generalizes results of Chazarin, \cite{C2}, Colin de
Verdi\`ere, \cite{CdV0}, and Duistermaat-Guillemin, \cite{DG}, in the
context of large-eigenvalue asymptotics of an elliptic operator.
Recent work on the STF on $\bbR^n$, using different techniques, has
been done by Meinrenken, \cite{Mein} and Dozias, \cite{DO}.
\medskip
In (\ref{4a.1})
$A_\h$ is an $\h$-admissible zeroth order pseudodifferential
operator (c.f. appendix), $\varphi$ is a test function and
$E$ is a parameter. Each periodic trajectory with action
$\oint_\gamma pdq$ in the support of $\hat{\varphi}$ contributes to
the small $\h$ behavior of (\ref{4a.1}). In the proof, we analyze the
contribution of the trajectories with a given action, and then sum the
estimates. Accordingly, we will assume that the Fourier transform of
$\varphi$, $\hat{\varphi}$, is in $C_0^\infty(\bbR)$. This restriction
on $\varphi$ is the main difference between our result and Gutzwiller's
trace formula, \cite{Gut}.
The restriction is necessary, in the sense that the sum
of the contributions of infinitely-many trajectories in general is
divergent.
\newcommand{\se}{\Sigma_E}
\medskip
The remaining assumptions in the STF are on the Hamilton flow of
$H$ on the energy shell
\be\label{4a.2}
\se :=\, H^{-1}(E)\,.
\ee
To begin with, we assume:
\be\label{4a.3}
E\ \mbox{ is a regular value of}\ H\,,
\ee
so that $\se$ is a smooth $(2n-1)$-dimensional manifold. It is
compact by the ellipticity assumption and the compactness of $M$.
Physically, this assumption says that there are no equilibria with
energy $E$. Without this assumption, the asymptotics of (\ref{4a.1})
can include logarithmic terms in $\h$, and half-integral powers of
$1/\h$; with R. Brummelhuis we have analyzed a first case of
this situation in \cite{BPU}. Finally, we'll assume
\be\label{4a.4}
\mbox{the Hamilton flow of}\ H\ \mbox{is clean on}\ \se\,,
\ee
a notion that we now recall.
\begin{definition}
The Hamilton flow of $H$, $\{\phi_t\}$, is said to be clean on
$\se$ iff:
\begin{enumerate}
\item The set
\be\label{4a.5}
\calP\, :=\, \{\, (z,T)\in \se\times\bbR\, ;\, \phi_T(x)=x\,\}
\ee
is a submanifold of $\se\times\bbR$ (the dimension is allowed
vary from one connected component to another), and
\item For every $(z,t)\in\calP$ the tangent space $T_{(z,T)}\calP$
is the set of all $(\zeta, \tau)\in (T_z\se)\times\bbR$ such that
\be\label{4a.6}
\tau\Xi_z + d(\phi_T)_z(\zeta)\, =\, \zeta\,.
\ee
\end{enumerate}
\end{definition}
{\bf Remarks:} (a) The dimensions of the various connected components
of $\calP$ can range from one, dimension of a component of the form
$\gamma\times\{ T\}$ where $\gamma$ is an isolated trajectory,
to $(n-1)$, corresponding to the component $\se\times\{ 0\}$.
(b) Condition 2 in the previous definition, applied to a connected component
of $\calP$ of the form $\gamma\times\{ T\}$, is equivalent to the
non-degeneracy of $\gamma$.
(c) Equation (\ref{4a.6}) is the differentiated version of
equation $\phi_T(z)=z$.
\medskip
In order to state the STF, we need to review two facts implied by
the ``cleanness" assumption.
\begin{lemma}\label{acuatro1}
\hfill
\par\noindent
{\bf 1.} With the notation $\phi_t(z) = (q(t),p(t))\in T^*M$, the map
\be\label{4a.7}
\begin{array}{rcl}
\calP & \to & \bbR\\
(z,T) & \mapsto & \int_0^T p\dot{q}\,dt
\end{array}
\ee
is locally constant, (i.e. it's constant on each connected component
of $\calP$).
\par\noindent
{\bf 2.} The connected components of $\calP$ inherit smooth densities.
\end{lemma}
Part 1 is classical: all members of a smooth family of periodic
trajectories have the same action. For part 2 in general we refer to
\cite{GU}, Lemma 2.7; here we will simply describe the densities in the
following three cases:
\begin{enumerate}
\item For the connected component $\se\times\{ 0\}$, the density is
the Liouville density on $\se$.
\item For the connected component $\gamma\times\{ T\}$ corresponding to
a non-degenerate trajectory, the density is
\be\label{4a.8}
dt\over \sqrt{|\det (I-P_\gamma)|}
\ee
where $t$ is the time variable along the trajectory (with an arbitrary
initial condition) and $P_\gamma$ is the linearized Poincar\'e map of
$\gamma$.
\item In case the Hamilton flow of $H$ on $\se$ is periodic,
let $T^{\#}:\se\to\bbR$ be the ``primitive period" function. Then the
connected components of $\calP$ are of the graphs of the maps
$mT_0$, $m\in\bbZ$, and as such are naturally identified with $\se$
itself. Under that identification, the density is just the
Liouville density on $\se$ (see \cite{UZ}, section 3.4.1).
\end{enumerate}
\newcommand{\Hsub}{H_{\mbox{\tiny sub}}}
\smallskip
Finally, the formula involves the sub-principal symbol of $\sh$, which is
\be\label{3.3}
\Hsub (x,p)\ =\ \sum_{l=0}^N\; \sigma_{s,l}\, (x,p)
\ee
where $\sigma_{s,l}$ is the subprincipal symbol of $P_l$. From $\Hsub$
we define a function, $\beta$, on $\calP$, by averaging it:
\be\label{3.3bis}
\begin{array}{rcl}
\beta: \calP & \to &\bbR\\
(z,T) & \mapsto & \int_0^T\,\Hsub \pi_t(z)\,dt\,.
\end{array}
\ee
We are now ready to state the STF:
\begin{theorem}\label{lapapa}
Under the assumptions (\ref{4a.3}) and (\ref{4a.4}), for each
test function $\varphi$ such that $\hat{\varphi}\in C_0^\infty(\bbR)$ and
each $\h$-admissible zeroth-order pseudodifferential operator $A_\h$,
the sum (\ref{4a.1}) has an asymptotic expansion as $\h\to 0$ of
the form
\be\label{4a.9}
\sum_{j=1}^\infty\,\,
\varphi \left( {E_{j,\h}-E\over \h}\right)
\sim \sum_\nu\, e^{i \alpha_\nu \h^{-1}}\,\h^{-d_\nu}\,e^{\pi i m_\nu/4}
\, \sum_{j=0}^\infty\, c_{\nu,j}\,\h^j\,,
\ee
where:
\begin{enumerate}
\item The sum $\sum_\nu$ is over the connected components, $\calP_\nu$, of
$\calP$ containing at least a point $(z,T)$ with $T$ in the support of $\hat{\varphi}$.
This sum is finite.
\item $d_\nu = (\mbox{dim}\,\calP_\nu - 1)/2$.
\item $m_\nu$ is an integer, the common Maslov index of the trajectories in
$\calP_\nu$, and finally
\item The leading coefficient of the $\nu$-th term is
\be\label{4a.10}
c_{\nu,0}\,=\, (2\pi)^{-(d\nu+1)/2}\,
\int_{\calP_\nu}\,e^{i\beta}\,a\,\hat{\varphi}(T)\,d\mu_\nu\,.
\ee
where $d\mu_\nu$ is the natural density on $\calP_\nu$ and $a$ the principal
symbol of $A_\h$.
\end{enumerate}
\end{theorem}
\noindent
{\bf Remarks.} (a) In the integral (\ref{4a.10}), the function
$\hat{\varphi}$ is considered as a function of the second
variable, $T$, on $\calP_\nu\subset\se\times\bbR$, while $a$ and $\beta$
are functions of the first. In each of the
cases above, the result of the integral is as follows:
\begin{enumerate}
\item For the component $\se\times\{ 0\}$, the integral is
$\hat{\varphi}(0)\int_{\se}a d\lambda$, where $d\lambda$ is
the Liouville measure of $\se$.
\item For the component $\gamma\times\{ T\}$ where $\gamma$
is non-degenerate, the integral equals
\be\label{4a.11}
\hat{\varphi}(T)\, {1\over\sqrt{|\det (I-P_\gamma)|}}\,\int_0^{T^{\#}}\,
e^{i\beta}\,a(q(t),p(t))dt\,,
\ee
where $T^{\#}$ is the primitive period of $\gamma$.
\item In case the flow on $\se$ is periodic with primitive
period function $T^{\#}$, the integral over the $m$-th component
is
\be\label{4a.12}
\int_{\se}\,e^{i\beta}\,a(z)\,\hat{\varphi}\left(mT^{\#}(z)\right)\,
d\lambda_z\,.
\ee
\end{enumerate}
\noindent
(b) The degree $d_\nu$ is zero for a non-degenerate periodic trajectory,
and $(n-1)$ for $\se\times\{ 0\}$.
\medskip
Applying the technique of \cite{BU} of passing to a compact
manifold, one immediately gets:
\begin{corollary}
For a Schr\"odinger operator $-\h^2\Delta +V$ on $\bbR^n$ with
$V\in C^\infty$ bounded below, if $E<\limsup_{|x|\to\infty} V(x)$,
the above statements remain true of the sums
\[
\sum_{j;|E_{j,\h}-E|\leq \h^{1-\epsilon}}\,
\varphi \left( {E_{j,\h}-E\over \h}\right)
\]
for any $0<\epsilon<1$.
\end{corollary}
%\newpage
\subsection{The $\h$ transform of the spectral function}
In the setting of the last section, let us introduce the notation
\be\label{4b.0}
E_j(k)\,:=\,E_{j,1/k}\quad\mbox{and}\quad
A_j(k)\,:=\,\,.
\ee
In accordance with the spirit of this paper and \cite{GU}, we
will study the asymptotic behavior of
\be\label{4b.1}
\sum_{j=1}^\infty\,A_j(k)\,\varphi (k(E_j(k)-E))\quad\quad
\mbox{as}\ k\to\infty
\ee
by studying the singularities of the Fourier series with
coefficients (\ref{4b.1}), namely
\be\label{4b.1.1}
\Upsilon_E(s)\, =\, \sum_{k=0}^\infty
\sum_{j=1}^\infty\,A_j(k)\,\varphi (k(E_j(k)-E))\, e^{iks}\,.
\ee
We call this distribution
{\em the $\h$ transform of the spectral function}.
Our immediate goal is to show how, up to a smooth function,
$\Upsilon_E$ can be constructed from the Schwartz kernel
of the approximation to the propagator of \S 3.3.
\medskip
We first must say a few words about the matrix coefficients $A_j(k)$.
By the definition of $\h$-admissible $\Psi$DO of the Appendix, there
exists a zeroth order, classical $\Psi$DO on $\calM$, $A$, which
commutes with $D_\theta$ and such that for every $k\in\bbZ^{+}$ the
operator $A_{1/k}$ is the restriction of $A$ to $L^2(M)\otimes
(\bbC\,e^{ik\theta})$.
\begin{proposition}\label{cuatro0}
Let $\calM = M\times S^1$, and define the operator
\[
\calF_E\,: C^\infty(S^1\times\bbR\times\calM\times\calM)
\to C^\infty(S^1)
\]
by the formula
\be\label{4b.16}
\calF_E(u)(s)\,=\,\int\,u(s-tE,t,z,z)\,\hat{\varphi}(t)\,dt\,dz
\ee
where $z=(x,\theta)$ denotes the variable in $\calM$.
Choose $f,g \in C_0^\infty(\bbR^{+})$ as in Corollary \ref{crucial}
and such that $E^{-1/N}\in I_f$,
and let $\calK_{f,g}(s,t,z,z')$ denote the
Schwartz kernel of $A\,Q_f\, e^{iD_\theta(tP_g+sI)}$.
Then $\calF_E$ extends by continuity to a class of distributions
containing $\calK_{f,g}$ and
\be\label{4b.18}
\Upsilon_E \; =\; \calF_E(\calK_{f,g})\quad\quad\mbox{modulo}\ \
C^\infty(S^1)\,.
\ee
\end{proposition}
\medskip
We will break the proof into a series of lemmas.
If $\chi\in C_0^\infty(\bbR^{+})$, define
\be\label{4b.2}
\Upsilon_E^\chi (s)\,=\, \sum_{k=1}^\infty\sum_j\, A_j(k)\,\chi(E_j(k))\,
\varphi (k(E_j(k)-E))\,e^{iks}\,.
\ee
\begin{lemma}\label{cuatro1}
If $\chi\in C_0^\infty (\bbR)$ is identically $1$ on an interval
$I$ containing $E$ in its interior and $0\leq \chi\leq 1$ elsewhere,
then
\be\label{4b.2bis}
\sum_j\,A_j(k)\, \chi(E_j(k))\, \varphi (k(E_j(k)-E)) -
\sum_j\, A_j(k)\,\varphi (k(E_j(k)-E))\,=\, O(k^{-\infty})\,,
\ee
where the estimate is locally uniform in $E$.
Hence $\Upsilon_E- \Upsilon_E^\chi$ is a smooth function.
\end{lemma}
\begin{proof}
To lighten-up the notation a bit, we'll take $A_j(k)=1$; all we
need for the proof is that the matrix coefficients are bounded by a
constant independent of $j$ and $k$, because $A$ is zeroth order.
We must show the rapid decrease in $k$ of
\be\label{4b.3}
\sum_j (1-\chi)(E_j(k))\,\varphi (k(E_j(k)-E))\,=\,
\sum_{j ; E_j\not\in I}\,(1-\chi)(E_j(k))\,\varphi (k(E_j(k)-E))\,.
\ee
We will first establish the rough estimate
\be\label{4b.4}
\exists\ C,\,l,\,m\quad \forall\ \lambda,\,k\quad\quad
\#\,\{\,j\,;\, \lambda \leq E_j(k) \leq \lambda +1\,\}\leq
C\,\lambda^l\,k^m\,,
\ee
which is a consequence of standard Weyl-type estimates on the
eigenvalues of the elliptic differential operator $R$
(defined in (\ref{3.6})). More specifically,
recall that the eigenvalues of $R$ are the numbers
$k^N E_j(1/k)$, indexed by pairs of indices $(k,j)$.
There exists then a constant $C$ such that $\forall \lambda'$
\be\label{4b.5}
\#\,\{\,(j,k)\,;\, \lambda' \leq k^NE_j(k) \leq \lambda' +1\,\}\leq
C\,(\lambda'+1)^{(n/N)}
\ee
(see for example \cite{Schu} Theorem 21.2).
Fix now $k$ and set $\lambda' = k^N\lambda$. Since the
pairs $(j,k)$ in the set (\ref{4b.5}) with a given $k$
are not more than with $k$ variable, get the inequality
\be\label{4b.6}
\#\,\{\,j\,;\, \lambda \leq E_j(k) \leq \lambda +1/k^N \,\}\leq
C\,(k^N \lambda+ 1)^{(n/N)}\,.
\ee
Summing $k^N$ of such estimates, each for the number of
$E_j(k)$ ranging in an interval of length $k^{-N}$ between
$\lambda$ and $\lambda +1$, gives
\be\label{4b.7}
\#\,\{\,j\,;\, \lambda \leq E_j(k) \leq \lambda +1/k^N \,\}\leq
C\,k^N\left(\, k^N(\lambda+1) +1\right)^{(n/N)}
\ee
with the same constant and for all $\lambda, k$.
This proves (\ref{4b.4}). Now recall that $\varphi$ is
rapidly decreasing.
Let $\epsilon>0$ be such that $[E-\epsilon, E+\epsilon]\subset I$. Since
\be\label{4b.8}
\forall\,p>0\ \ \exists\,C_p>0 \ \ \forall\,t\not\in [-\epsilon,\epsilon]\quad\quad
|\varphi (t)|\leq C_p\,|t|^{-p}\,,
\ee
then $\forall p>0$ and $\forall E_j\not\in I$
\be\label{4b.9}
|\varphi (k(E_j(k)-E))|\leq C_p\,k^{-p}\,|E_j(k)-E|^{-p}\,.
\ee
It follows that the absolute value of (\ref{4b.3}) is less than or
equal to
\be\label{4b.10}
\sum_{j;E_j\not\in [E-\epsilon,E+\epsilon]} \left|\,\varphi\left( k(E_j(k)-E)\right)\,
\right| \leq C_p\,k^{-p}\left( I + II\right)\,,
\ee
where
\be\label{4b.11}
I\,=\,\sum_{0E+\epsilon}\, | E_j(k)-E|^{-p}\,.
\ee
We now use the rough bound (\ref{4b.4}) to estimate these sums.
First $\exists C_1$ such that
\be\label{4b.12}
I\leq \#\{\,j\,;\, E_j(k)\leq E-\epsilon\,\}\,(E-\epsilon)^{-p} \leq
C_1k^m\,(E-\epsilon)^{-p}.
\ee
Secondly,
\be\label{4b.13}
\begin{array}{rcl}
II& \leq &\sum_{r=0}^\infty\,\sum_{\epsilon+r+1>E_j-E\geq \epsilon+r}\, (E_j(k)-E)^{-p}\\
& \leq & C k^m\, \sum_{r=0}^\infty\, (\epsilon+r)^{l-p}\,.
\end{array}
\ee
Hence if we define
\[
\tilde{C_p}\,=\, C_p\left( C_1\,(E-\epsilon)^{-p} + C\sum_{r=0}^\infty\,
(\epsilon+r)^{l-p}\right)
\]
(which is finite if $p\geq l+2$) we have
\[
\sum_{j;E_j\not\in [E-\epsilon,E+\epsilon]} \left|\,\varphi\left( k(E_j(k)-E)\right)\,
\right| \leq \tilde{C_p}\,k^{m-p}\,.
\]
Since we can choose $p$ arbitrarily large, we have established
the rapid decrease of (\ref{4b.3}); local uniformity on $E$ is
obvious from the proof.
\end{proof}
We now pick $\chi^f(x)=f(1/x^N)$ where $f\in C_0^\infty(\bbR^{+})$.
We record the following consequence of the definitions
(recall that $Q_f$ commutes with $P$ and $D_\theta$):
\begin{lemma}\label{cuatro2}
With the previous choice of $\chi$
\be\label{4b.15}
\Upsilon^{\chi^f}_E(s)\,=\,\sum_{k=0}^\infty\sum_j
\,\varphi(k(E_j(k)-E))\,e^{iks}\,.
\ee
\end{lemma}
Turning to the operator $\calF_E$:
\begin{lemma}\label{cuatro3}
$\calF_E$ is a Fourier integral operator associated with the canonical
relation $\calC_E\subset T^*S^1\times
T^*(S^1\times\bbR\times\calM\times\calM)$
set of all points of the form
\[
\left(\,(s = \theta + tE, \kappa)\ ,\
(\theta, \kappa\,;\, t, \kappa E\,;\, \x \, ;\, \x)\,\right)
\]
with $\x\in T^*\calM$ and $t\in\mbox{supp}\, \hphi$.
In particular, $\calF_E$ extends by continuity to
distributions $u\in\calD' (\calM)$ satisfying
\be\label{4b.19}
\wf (u) \cap \{\,\kappa = 0\,\}\,=\,\emptyset.
\ee
\end{lemma}
\begin{proof}
The Schwartz kernel of $\cal F$ is locally given by
oscillatory integrals of the form
\be\label{4b.17}
\hphi (t) \int e^{i\,[\kappa(s - tE)\, +\, (x - y) \cdot \xi]}
\; d\xi d\kappa.
\ee
\end{proof}
Next we construct the right-hand side of (\ref{4b.15})
from the Schwartz kernel of the operator $A\,Q_f\, e^{iD_\theta(tP+sI)}$.
\begin{lemma}\label{cuatro4}
$\Upsilon^{\chi^f}_E$ is the result of applying $\calF_E$
to the Schwartz kernel of the operator $A\,Q_f\,e^{iD_\theta(tP+sI)}$.
\end{lemma}
\begin{proof} Notice that since the microlocal support of $Q_f$
does not intersect the set $\{\kappa =0\}$, one can indeed apply
the operator $\calF_E$ to the Schwartz kernel of
$Q_f\,e^{iD_\theta(tP+sI)}$. The latter is
\[
\sum_k\sum_j\, e^{ik(tE_j(k)+s)}
(A\circ Q_f)(\Psi_{k,j})(z)\,\overline{\Psi_{k,j}(z')}\,,
\]
and an easy calculation shows that applying $\calF_E$ to this series
term by term yields $\Upsilon^{\chi^f}_E$.
\end{proof}
\smallskip\noindent
{\sc Proof of Proposition }\ref{cuatro0}.
By Corollary \ref{crucial},
\be\label{4b.20}
\calF_E\,\left(\, A\,Q_f\,e^{isD_\theta}\,
\left[\, e^{itD_\theta P} - e^{itD_\theta P_g} \;\right]
\right) \in C^\infty(S^1)\,.
\ee
By lemmas \ref{cuatro1} and \ref{cuatro4}, the proof of Proposition
\ref{cuatro0} is complete.
\subsection{Proof of the trace formula}
Proposition \ref{cuatro0} reduces the problem of studying the
singularities of $\Upsilon_E$ to a problem of composing two Fourier
integral operators: we must show that, under the hypotheses of the
trace formula, the result of applying $\calF_E$ to $\calK_{f,g}$ is a
Lagrangian distribution. We must then locate its singularities, and
compute the associated symbols. Finally, one uses an elementary fact
that the Fourier coefficients of a Lagrangian distribution on the
circle have an asymptotic expansion governed by the singularities of
the distribution. All this analysis is in fact identical to that
contained in \cite{GU}, here we will simply highlight the main features
for completeness. We will continue to use the notation
$\calM = M\times S^1$.
\newcommand{\T}{\dot{T}^*}
\begin{theorem}
Without having to assume (\ref{4a.3}) or (\ref{4a.4}),
the distribution $\calK_{f,g}$ is a Lagrangian distribution on
$S^1\times\bbR\times\calM\times\calM$, with associated Lagrangian
submanifold, $\Gamma$, set of all points
$(s,\sigma\,;\,t,\tau\,;\,\x:\y)\in
\T S^1\times\T\bbR\times\calU_f\times\calU_f$ such that, with
the notation $\x=(x,\xi;\theta,\kappa)$ and $\y=(y,\eta ;\theta,\kappa')$
\be\label{4c.1}
\begin{array}{rcl}
s& = & \theta + tE\\
\y' & = & \phi_t(\x)
\end{array}
\ee
\end{theorem}
\begin{corollary}
Without having to assume (\ref{4a.3}) or (\ref{4a.4}),
the wave-front set of $\Upsilon$ is contained in the set of
all $(s,\sigma)$ such that $\kappa > 0$ and
$\exists z\in\se ,\; T\in\mbox{supp}\,\hat {\varphi}$ such that
\[
\phi_t(z)\,=\,z\quad\mbox{and}\quad s=\int_0^T\,p\,\dot{q}\,dt
\]
where $\phi_t(z) = (q(t),p(t))$.
\end{corollary}
By the composition Theorem of Fourier integral operators,
$\calF_E(\calK_{f,g})$ is a Lagrangian distribution provided
$\calC_E$ meets $\Gamma$ cleanly, see \cite{GU} for the precise
meaning of this. Because of the simple $\theta$ dependence in
all the operators, this condition reduces to one in $T^*M$:
\begin{proposition}
The clean intersection condition \cite{Ho} ensuring that the result of
applying $\calF_E$ to $\calK_{f,g}$ is a Lagrangian distribution is
equivalent to (\ref{4a.3}) and (\ref{4a.4}).
\end{proposition}
Proceeding exactly as in \cite{GU} (the only difference being the
presence of the $\Psi$DO $A$, which at the symbolic level merely
multiplies the symbol of $\exp(iD_\theta(tP_g+s\mbox{Id})$ by $a$) one proves
the Theorem with $\h\to 0$ along the values $\h=1/k$. It remains to
show that the estimates are valid as $\h\to 0$ continuously.
\bigskip
For that effect consider, for $\alpha\in (0,1)$,
\be\label{4c.2}
\Upsilon_\alpha (s) \, =\,
\sum_{k=0}^\infty\sum_j
\,\varphi
\Bigl((k+\alpha)\,(E_j(1/(k+\alpha))-E)\Bigr)\,
e^{i(k+\alpha)s}\,.
\ee
This is the $\Upsilon$ of \cite{GU} associated to the pair of commuting
operators $D_\theta+\alpha$ and $D_\theta P_f$. Observe that the
dependence on $\alpha$ is at the subprincipal symbol level. Therefore
the dependence on $\alpha$ in the oscillatory integral representation
of $\Upsilon_\alpha$ is only on the amplitude (not the phase), and the
amplitude clearly depends smoothly on $\alpha$. Therefore the
estimates on the Fourier coefficients of $\Upsilon_\alpha(s)
e^{-i\alpha s}$ are uniform in $\alpha$, which amounts to the desired
estimates as $\h\to 0$ continuously in the trace formula. This
argument is a direct analogue of Colin de Verdi\`ere's in \cite{CdV1},
Theorem 4.4.
%\newpage
\section{Variations on the STF}
We have gathered below three variations on the STF. The last two are
generalizations of Theorem 2.3 in \cite{BU} and Theorem 6.3 in \cite{TU},
respectively, to the present setting; the main difference is
that the present results contain no square roots of the eigenvalues,
which have no sense physically. The proofs of these
generalizations are identical to those of the original Theorems, one
simply has to work with the approximant $AQ_f\exp[itD_\theta(tP_g+sI)]$
of \S 4. Therefore we simply state the results.
\subsection{The STF and the Bohr-Sommerfeld condition.}
We place ourselves in the setting of the STF. Then $\Upsilon$
is a distribution on the circle with finitely-many
classical conormal singularities at the points $exp(it\oint_\gamma pdq)$,
where $\gamma$ is a periodic trajectory on $\se$ with period in
the support of $\hat{\varphi}$. Hence the singularity of $\Upsilon$
created by $\gamma$ is at $1\in S^1$ iff $\gamma$ satisfies the
Einstein-Bohr-Sommerfeld quantization condition that
\be\label{5.1}
\oint_\gamma pdq \in\ 2\pi\bbZ\,.
\ee
In this section we exploit this fact to prove that the
{\em averages} of the Fourier coefficients of $\Upsilon$ have a
leading asymptotic behavior governed by the periodic trajectories
satisfying (\ref{5.1}). The idea is to apply F\'ejer summation
to the Fourier series of $\Upsilon$:
\begin{lemma}\label{cinco0}
Suppose the numerical sequence $(a_k)$ has the asymptotic behavior
\be\label{5.2}
a_k\,\sim\,c_0\,\omega^k\,k^d + O(k^{d-1})\quad\quad\mbox{as}\
k\to\infty\,.
\ee
where $\omega\in S^1$ is a complex number of modulus one. Then,
if $\omega\not =1$
\be\label{5.3}
{1\over m}\,\sum_{k=1}^m\,a_k\,\sim\,\left\{
\begin{array}{ll}
c_0\,m^{d-1}\,{\omega^{m+1}\over\omega-1}+O(k^{d-1})&\quad\mbox{if}\ d>0 \\
c_0\,m^{d-1}\,{\omega^{m+1}\over\omega-1}+O(k^{-1}\log k)&
\quad \mbox{if}\ d=0\,,
\end{array}
\right.
\ee
while if $\omega =1$,
\be\label{5.3a}
{1\over m}\,\sum_{k=1}^m\,a_k\,\sim\,{c_0\over d+1}\,m^d + O(m^{d-1})\,.
\ee
\end{lemma}
\begin{proof}
If $b_k= O(k^{d-1})$, then ${1\over m}\,\sum_{k=1}^m\,b_k$ is
$O(m^d-1)$ if $d>0$ and $O(m^{-1}\log m)$ if $d=0$, so WLOG
$a_k = c_0\,\omega^k\,k^d$. But then, if $\omega \not =1$,
\be\label{5.4}
{1\over m}\,\sum_{k=1}^m\,a_k\, =\, c_0\,\left(\omega
{\partial\over \partial\omega}\right)^d\,
{\omega^{m+1}-1\over\omega-1}\,.
\ee
(Here of course we are abusing the notation, regarding $\omega$ as a
variable, what we mean is an identity of functions evaluated
at the comlex number $\omega$.) The estimates (\ref{5.3}) follow easily.
\end{proof}
As corollary we obtain:
\begin{corollary}\label{cras2}
Besides the assumptions of the STF assume $\hat{\varphi}\in C_0^\infty(\bbR)$
vanishes to order $n=\mbox{dim}\,(M)$ at zero. Let
\be\label{5.5}
L(\varphi, m)\,:=\,{1\over m}\,
\sum_{k=1}^m\, \sum_j\,\varphi\left( k(E_j(k)-E)\right) \,.
\ee
Then:
\par\noindent
(a) If there is no periodic trajectory $\gamma\subset\Sigma_E$ whose action
is in $\pi\bbQ$ and satisfying $\hat{\varphi}(T_\gamma)\not= 0$, then
\be\label{5.5a}
L(\varphi, m)\, =\,O(1/m)\,.
\ee
\par\noindent
(b) Suppose $\calP_\nu\subset\calP$ is connected component of action
$S\in 2\pi\bbZ$, and that all other such components have dimension
strictly less than $\mbox{dim }\calP_\nu$. Then
\be\label{5.5b}
L(\varphi, m)\, =\, m^d_\nu \,{c_{\nu,0}\over d_\nu +1}\,+\, R(m)\,,
\ee
where $c_{\nu,0}$ is the coefficient (\ref{4a.10}) (with $a=1$),
$d_\nu = (\mbox{dim}\calP_\nu -1)/2$ and
\be\label{5.5c}
R(m)\,=\,\left\{
\begin{array}{ll}
O(m^{d_\nu-1}) & \mbox{if } d_\nu\geq 1\\
O(\log(m)/m) & \mbox{if } d_\nu = 0
\end{array}
\right.
\ee
\end{corollary}
The proof follows from the STF and the previous lemma.
\subsection{Estimates Uniform in $E$.}
In Theorem \ref{lapapa}, the parameter $E$ was constant: Assumptions
on $H^{-1}(E)$ translate into estimates with fixed $E$. Here we show
that assumptions on a band of energies $H^{-1}[E_0-\epsilon,E_0+\epsilon]$
translate into the local uniformity in $E$ of the asymptotic expansion
(\ref{4a.9}). The idea is to consider $\Upsilon$ as a distribution on
the $(s,E)$-plane:
\be\label{5a.1}
\Upsilon(s,E)\,=\,\sum_{k=0}^\infty\sum_j\, A_j(k)\,
\varphi (k(E_j(k)-E))\, e^{iks}\,.
\ee
(This distribution is also of use in the study of the case when $E$
is a critical value, see \cite{BPU}.)
The analysis of (\ref{5a.1}) is analogous to that of $\Upsilon_E$,
so we will be brief. First, the operator
\[
\calF (u)(s,E)\,=\,\calF_E(u)(s)
\]
(where $\calF_E$ is as in \S 4) is a Fourier integral operator given
by the same oscillatory integrals (\ref{4b.17}). The canonical relation,
$\calC$, of $\calF$ is the set of all points of the form
\be\label{5a.2}
\left(\,(s = \theta + tE, \kappa)\ ,\
(\theta, \kappa\,;\, t, \kappa E\,;\, \x \, ;\, \x)\,\right)\,.
\ee
A version of Lemma \ref{cuatro1} also holds for the derivatives
of (\ref{4b.2bis}) with respect to $E$, and so the localization process
of \S 4.2 can be applied to $\Upsilon$. The conclusion is:
\begin{proposition}
If $g$ is such that $[E-\epsilon, E+\epsilon]\subset I_f$ and
$\mbox{supp}\,f\subset g^{-1}(1)$,
$\Upsilon = \calF(\calK_{f,g})$ modulo $C^\infty$.
\end{proposition}
\newcommand{\ve}{\lambda}
\begin{corollary}\label{cinco1}
The series (\ref{5a.1}) converges weakly to a distribution $\Upsilon (s,E)$
on the cylinder $S^1\times\bbR$ with wave-front set contained in the
set of all $(s,E; \kappa,\ve)$ such that there exists $(x,p)\in T^*M$
satisfying
\be\label{5a.3}
\left\{
\begin{array}{ccc}
\kappa\neq 0 & \mbox{and} & t:= \ve/\kappa \in\mbox{supp}\hphi \\
\phi_t(x,p)\ =\ (x,p) & & \\
H(x,p)\ =\ E & &\\
s\ =\ \oint pdq & & \\
\end{array}
\right.
\ee
where the integral is along the periodic trajectory defined by
$(x,p)$ and $t$.
\end{corollary}
Notice that the inverse of the slope $\ve/\kappa$ of a covector
in $\wf (\Upsilon)$ has the interpretation of the {\em period} of the
classical trajectory with energy $E$ producing the singularity.
Next we turn to the clean intersection condition that will ensure
that $\Upsilon$ is a Lagrangian distribution.
\begin{proposition}
$\Upsilon$ is a Lagrangian distribution
when restricted to the open set $S^1 \times (E_0-\epsilon, E_0+\epsilon)$
provided that:
\medskip\noindent
(1) The set
\[
\calP \ : =\ \{ (z\,;\,t) \in H^{-1}(E_0-\epsilon, E_0+\epsilon) \times J\,;\,
\phi_t(z) = z \}\ ,
\]
where $J$ is some open neighborhood of $\mbox{Supp} \hphi$ and
$\{ \phi_t \}$ the Hamilton flow of $H$ on $T^*M$, is a submanifold, and
\medskip\noindent
(2) $ \forall (z,t) \in \cal P$, the tangent space $T_{(z,t)} \cal P$
equals the space of all $ (\zeta, \tau)\,\in\,T_{(z,t)}(T^*M \times \bbR)$
such that
\be\label{5a.4}
\tau \Xi_z\; +\; d(\phi_t)_z (\zeta)\ =\ \zeta,
\ee
\end{proposition}
\begin{proof}
Recall the following basic theorem of FIO theory (see for example
\cite{Ho} Vol. IV).
Let $\cal F$ be an FIO from a manifold $Y$ to a manifold $Z$,
and let $\Gamma \subset T^*Y \backslash \{ 0 \}$ be a closed conic
Lagrangian submanifold. The condition for $\calF (u)$ to be
Lagrangian, where $\mu \in I^.(Y,\Gamma)$, is that the canonical
relation of $\calF$, $\calC \subset (T^*Y \backslash \{ 0 \}) \times
(T^*Z \backslash \{ 0 \})$, intersects $\Gamma \times (T^*Z
\backslash \{ 0 \})$ cleanly. In the present case,
\[
Y\ =\ S^1 \times \bbR \times M \times M, \qquad Z\ =\ X^1 \times \bbR
\]
and one can easily show (as in \cite{GU}) that this condition is
equivalent to the clean intersection of the immersed manifolds
\be\label{5a.5}
\begin{array}{ccc}
& & T^*M \times J \\
& &\ \downarrow \Phi \\
T^*M \times T^*\bbR & \hookrightarrow & T^*M \times T^*M \times T^*\bbR
\end{array}
\ee
where
\[
\Phi(z,t)\ =\ (z\,;\, \phi_t(z)\,;\, t, H(z))
\]
and the horizontal arrow is the diagonal inclusion.
This condition is precisely (1) and (2) in the Theorem.
\end{proof}
It is straightforward that this condition implies the clean
intersection conditions on the energy shell
$H^{-1}(E)$ for all $E\in [E_0-\epsilon, E_0+\epsilon]$. Moreover,
the clean intersection condition implies that the manifolds of
periodic trajectories on $H^{-1}(E)$ depend smoothly on $E$.
With this in mind, we can state:
\begin{theorem}\label{uniform}
The coefficients of the asymptotic expansion of (\ref{4a.1}) at every
$E\in [E_0-\epsilon, E_0+\epsilon]$ depend smoothly on $E$, and the
expansion is uniform in $E$.
\end{theorem}
The proof is completely analogous to that of Theorem 6.3 in
\cite{TU}.
\subsection{The periodic case.}
We consider here the case when $E$ is a regular value of $H$ and the
Hamilton flow of $H$ on $\se$ is simply periodic with smallest period
normalized to $2\pi$.
\begin{theorem}\label{periodic1}
There exist constants $C_1>0$, $C_2>0$ such that
\be\label{5b.1}
\{\,j\,;\,|E_j(\h)-E-\h\pi\sigma/2T|\leq C_1\h\,\}\subset
\{\,j\,;\,|E_j(\h)-E-\h\pi\sigma/2T|\leq C_2\h^2\,\}\,.
\ee
\end{theorem}
This is a clustering result: eigenvalues that are $\h$-close to $E$
are actually $\h^2$ close. The second result is a geometric
description of the asymptotic distribution of the eigenvalues in these
clusters. Let $B = \se/S^1$, where the action of $S^1$ on $\se$ is by
the Hamilton flow of $H$. Then $B$ is a smooth manifold (the manifold
of trajectories), and it inherits a symplectic structure.
\begin{theorem}
There is a smooth function, $\rho : B\to \bbR$, such that
$\forall f\in C_0^\infty(\bbR)$
\be\label{5b.2}
{1\over \#J}\,\sum_{j\in J}\,f\left( {E_j(\h)-E\over h^2}\right)\,\sim\,
{1\over \mbox{LVol}(B)}\,\int_B\,f\circ\rho\,d\lambda_B
\ee
where $J$ is the set on the right-hand side of (\ref{5b.1}),
$\mbox{LVol}(B)$ is the Liouville measure of $B$ and $d\lambda_B$ the
Liouville density.
\end{theorem}
The proofs are a straightforward generalization of the corresponding
statements in \cite{BU}, working with the operator $P_f$ of \S 4.
\appendix
\newcommand{\classic}{\calS_{\mbox{\tiny cl}}}
\newcommand{\scpsudo}{sc$\Psi$DO }
\newcommand{\hinv}{\hbar^{-1}}
\section{Appendix A: Semi-classical $\Psi$DOs on Manifolds.}
We now discuss scFIOs in the case $M_1=M_2=M$ and $\Gamma$ the diagonal.
The corresponding scFIOs will be called {\em semi-classical
pseudodifferential operators}, or \scpsudo. Our goal is to associate,
to certain functions $a$ on $T^*M$, a \scpsudo with principal symbol $a$.
\medskip
Pseudodifferential operators with a small parameter, $a(x,\h D_x)$,
are usually set up on $\bbR^n$, within the Weyl calculus, under the
name of $\h$-admissible operators (\cite{HR1}, \cite{Iv}).
They differ from the usual $\Psi$DO theory by the lack of
homogeneity of the symbol: The principal symbol of $a(x,\h D_x)$
is the full amplitude of $a(x,D_x)$. On manifolds, $\Psi$DO theory
is well established by the classical result on the invariance of the
symbol classes under coordinate changes, and the invariance of
the principal symbol as a function on the cotangent bundle.
The latter fails for the lower order terms of the symbol and this does not
allow a naive definition of operators of semi-classical type.
In this appendix we make explicit the theory of semi-classical $\Psi$DOs
on manifolds, within the framework used for the scFIOs of \S 3.
After writing a direct definition of such operators we prove an
Egorov type theorem. Finally we apply this machinery to prove the
uniform distribution of eigenfunctions in case the classical flow
is ergodic, that is the manifold version of the Theorem of
Helffer, Martinez and Robert in \cite{HMR}.
\subsection{Definitions and first properties}
Let $M$ be a smooth compact manifold, and let $P$ be a classical,
properly-supported pseudodifferential operator of order zero
on $M\times\bbR$.
(Recall that ``properly supported" means that the projections from the
support of the Schwartz kernel of $P$ to $M\times\bbR$ are proper.)
Therefore $P$, defined originally on $C_0^\infty(M\times\bbR)$,
extends to $C^\infty(M\times\bbR)$.
Denoting the variable on $\bbR$ by $\theta$, we also assume that $P$
commutes with $D_\theta$.
\begin{definition}
For every $\h>0$, define the operator $P_\h$ on $C^\infty (M)$ by:
\be\label{A1}
P_\h (f)\,=\, e^{-i\h^{-1}\theta}\, P[e^{i\h^{-1}\theta} f]\,.
\ee
The family $\{ P_\h\}_\h$ will be called a \scpsudo. If $\sigma_P$
denotes the principal symbol of $P$, the sc symbol of $\{ P_\h\}_\h$
is defined as the function
\be\label{A2}
H_P(x,p)\,=\,\sigma_P(x,\xi=p,\kappa=1)\,.
\ee
\end{definition}
\noi
{\bf Remarks.}
\smallskip\noi
1.- Since $[P,D_\theta]=0$ the right-hand side of
(\ref{A1}), as well as $\sigma_P$, are independent of $\theta$.
\smallskip\noi
2.- The reason to insist that $P$ be of order zero is because,
morally speaking, in $P_\h$ every derivative should be accompanied
by a factor of $\h$, and the latter have virtual degree $(-1)$.
\smallskip\noi
3.-
Observe that according to our definition a Schr\"odinger operator
\[
-\h^2\Delta + V
\]
is {\em NOT} a \scpsudo. However it becomes one after micro-localization
in an energy band (i.e. the Fourier components of the
operator $P_f$ of \S 4 is a \scpsudo). As we have already seen,
such energy-localized operators
are the only ones that have a reasonable semi-classical behavior.
\smallskip\noi
4.- It is immediate that the class of \scpsudo s forms an algebra,
and that the standard symbol calculus formulae apply to the sc symbol.
\medskip
We next obtain the local oscillatory integral representations of a
\scpsudo.
\begin{proposition}\label{Arep}
If $\{ P_\h\}$ is a \scpsudo, then on functions $f$ supported in
a fixed compact in a coordinate chart of $M$
$P_\h$ has the form
\be\label{A3}
P_\h (f)\,=\, \int\, e^{i\h^{-1}p\cdot x}\hat{f}(\h^{-1}p)\,
a(x,h^{-1}p, h^{-1})\, dp
\ee
modulo $O(\h^\infty)$ in the $C^\infty$ topology.
Here $a(x,\xi,\kappa)$ is a classical symbol of order zero.
\end{proposition}
\begin{proof}
Since $P$ commutes with $D_\theta$, its Schwartz kernel
$\calK(x,\theta, y,\theta')$ is
a distributional function of $(x,y,\theta-\theta')$.
More precisely, just as in Lemma \ref{FIO1}, $\calK$ is of the form
\[
\calK(x,\theta, y,\theta')\,=\,\calK_0(x,y,\theta-\theta')
\]
where $\calK_0$ is a Lagrangian distribution on $M\times M\times\bbR$
associated with the Lagrangian submanifold
\be\label{A4}
\{\, (x,y=x,\theta=0\,;\,\xi,-\xi,\kappa)\,\}\,.
\ee
Furthermore, since $P$ is properly supported $\calK_0$ is
compactly supported.
Therefore $\calK$ is a {\em finite} sum of distributions of the form
\be\label{A5}
\calK_\alpha (x,\theta, y,\theta')\,=\, \int\,e^{i[\xi\cdot (x-y) + \kappa
(\theta -\theta')]}\, A(x,y,\theta - \theta';\xi,\kappa)\, d\xi\,d\kappa\,,
\ee
where each integral is supported in a local coordinate patch of the
form $U\times U$ (where $U$ is a coordinate patch on $M\times\bbR$) and
$A$ is a classical symbol of order zero. Applying e.g. Theorem 3.2. in
\cite{Schu} we see that we can write (\ref{A5}) in the form
\be\label{A6}
\calK_\alpha (x,\theta, y,\theta')\,=\, \int\,e^{i[\xi\cdot (x-y) + \kappa
(\theta -\theta')]}\, a(x;\xi,\kappa)\, d\xi\quad \mbox{mod}\, C^\infty\,,
\ee
in the support of $\calK_\alpha$ (here $a$ is another classical symbol of
order zero). Let $f\in C^\infty (M)$; then, by the above,
$P_\h (f)$ is a finite sum of integrals of the form
\be\label{A7}
e^{-i\hinv\theta}\, \int\,e^{i[(x-y)\cdot\xi + (\theta-\theta')\kappa]}\,
e^{i\hinv\theta'}\,f(y)\,a(x;\xi,\kappa)\,d\xi \, d\kappa\,dy
\ee
which, after the change of variables $\xi = \h p$, becomes the desired
expression.
\end{proof}
\medskip\
It is natural to ask which functions $b(x,p,\kappa)$ arise as
amplitudes $b(x,p,\hinv) = a(x,h^{-1}p, h^{-1})$ in
(\ref{A3}). A first observation is that they must satisfy the following
estimates: Given a compact
$K$ in $x$-space and given $\alpha, \beta, \gamma$ there is
a $C>0$ such that $\forall x\in K$ and $\forall \kappa> 1$ one has
\be\label{A5a}
\left| D^\alpha_\kappa D^\beta_p D^\gamma_x\,b (x,p,\kappa)\right|\,
\leq\, C\,\kappa^{-\alpha}\,(1+|p|^2)^{-|\beta|/2}\,.
\ee
Since $a$ is a classical symbol an amplitude $b$ has an asymptotic expansion
of the form
\[
b(x,p, \kappa)\,\sim\,\sum_{j=0}^\infty\, \kappa^{-j}\,b_j(x,p)
\]
in the following sense:
$\forall N>0\,,\,\alpha\,,\,\beta\,,\,\gamma$ $\exists C$ such that
$\forall\,\kappa>1$
\be\label{A5b}
\left| D^\alpha_\kappa D^\beta_p D^\gamma_x
\Bigl(b(x,p,\kappa)-\sum_{j=0}^N\, \kappa^{-j}\,b_j(x,p)\Bigr)\right|
\leq C\,\kappa^{-\alpha-N-1}\, (1+|p|^2)^{-(|\gamma|+N+1)/2}\,.
\ee
$b_0(x,p)$ is the sc principal symbol.
Furthermore, all functions $b$ above have the property that the function
$b(x,\xi/\kappa)$ extends to a smooth function away from $(\xi,\kappa)=(0,0)$.
For our purposes the following converse is sufficient:
\begin{definition}
Given $U\subset\bbR^n$ open, define $\calS^m (U)$ as the
set of functions $b(x,p,\kappa)\in C^\infty( U\times\bbR^n\times \bbR^{+})$
satisfying : $\forall \alpha,\,\beta,\,\gamma,\,N,\, K$ ($K$ compact)
$\exists C$ such that $\forall (x,p,\kappa)\in K\times\bbR^n\times (1,\infty)$
\be\label{apa3}
\left|\,\partial_\kappa^\alpha\,\partial^\beta_p\,\partial^\gamma_x\,
b(x,p,\kappa)\,\right|\,\leq\,C_{N,\alpha,\beta,\gamma}\,(1+ |p|)^{-N}\,
\kappa^{m-\alpha}\,.
\ee
\end{definition}
Elements in $\calS^m(U)$ are symbols in $\kappa$, uniformly Schwartz in $p$.
We are interested in elements in $\calS^0 (U)$ having an expansion
in decreasing powers of $\kappa$.
\begin{definition}
Let $M$ be a smooth manifold. We denote by $\calS(T^*M)$ the
set of functions $H(x,p)\in C^\infty( T^*M)$
satisfying : $\forall \,N$ $\exists C_N>0$ such that
\be\label{apa1}
\left|\, H(x,p)\,\right|\,\leq\,C_N\,(1+|p|)^{-N}\,,
\ee
where $|p|$ refers to an arbitrary Riemannian metric on $M$.
\end{definition}
It is obvious that the class $\calS$ is independent of the metric
involved in (\ref{apa1}). Elements in $\calS$ are smooth functions
uniformly Schwartz in the fiber direction of $T^*M$.
\begin{lemma}
Given $b\in C^\infty(U\times\bbR^n)$,
$k^{-j}\,b(x,p)\in\calS^{-j}(U)$ iff $b\in\calS (T^*U)$.
\end{lemma}
\begin{definition}
We define $\classic (U)$ as the subset of all $b\in\calS^m$ having
an expansion of the form
$ b\sim\sum_{j=0}^{\infty} \kappa^{-j} b_j(x,p)$ where $\forall j$
$b_j\in\calS(T^*U)$, and the expansion as usual means that
\[
\forall N>0\quad b(x,p,\kappa)\,\equiv
\,\sum_{j=0}^Nk^{-j}\,b_j(x,p)\quad\mbox{mod}
\ \calS^{-N-1}\,.
\]
\end{definition}
\begin{lemma}\label{Alemma1}
If $b\in \classic (U)$, then the function $a$ defined by:
\be\label{apa4}
a(x; \xi,\kappa)\,=\,\left\{
\begin{array}{lr}
0 & \mbox{if}\, \kappa = 0\,,\\
b(x,\xi/\kappa,\kappa) & \mbox{if}\, \kappa \not = 0
\end{array}
\right.
\ee
is a classical symbol of order zero (the fiber variables being
$(\xi,\kappa)$). Moreover, $a$ and each homogeneous term $a_j$ in its
asymptotic expansion vanishes to infinite order on $\Sigma := \{ \kappa
=0\}$. Conversely, any classical symbol $a$ of order zero with this
vanishing property on $\Sigma$ is asymptotic to one of the form (\ref{apa4}).
\end{lemma}
\begin{proof}
The derivatives $D^\alpha_x D^\beta_\xi D^\gamma_\kappa\, a$ are
sums of terms of the form
\[
\kappa^{-\gamma'}\xi^{\beta'}\tilde{b}(x,\xi/\kappa,\xi)\,,
\]
where $\tilde{b}$ is some derivative of $b$. Since $b\in\calS$,
all such terms tend to zero as $\kappa\to\infty$ (provided that
$\xi\not=0$), we see that $a$ is smooth on
$U\times(\bbR^n\setminus \{0\})\times\bbR^{+}$, vanishes to
infinite order on $\Sigma$, and furthermore we obtain
the following rough estimates on the derivatives:
\be\label{apa5}
\left| \,D^\alpha_x\,D^\beta_\xi\, D^\gamma_\kappa\, a(x,\xi,\kappa)\,\right|
\leq C(1+|\xi|^2+\kappa^2)^{\mu}
\ee
where $\mu=\mu(\alpha,\beta,\gamma,K)$ with $x\in K$, ($K$ compact).
Next, let $b\sim\sum_{j=0}^{\infty} \kappa^{-j}\,b_j$. Just as before,
since $b_j\in\calS (T^*U)$ the functions
\[
a_j(x; \xi,\kappa)\,=\,\left\{
\begin{array}{lr}
0 & \mbox{if}\, \kappa = 0\,,\\
\kappa^{-j}\,b_j(x,\xi/\kappa) & \mbox{if}\, \kappa \not = 0
\end{array}
\right.
\]
are smooth on $U\times(\bbR^n\setminus \{0\})\times\bbR^{+}$.
They are clearly positive homogeneous in $(\xi,\kappa)$ of degree $(-j)$.
We now will use Proposition 3.6 in \cite{Schu}: Given (\ref{apa5}),
to prove that $a$ is a classical symbol
it suffices to establish rough estimates of the form
\be\label{apa6}
\left|\, a(x,\xi,\kappa) - \sum_{j=0}^{N} a_j(x;\xi,\kappa)\,\right| \leq
C_N\,(1+|\xi|^2+\kappa^2)^{\mu_N}
\ee
where $\mu_N\to -\infty$ as $N\to\infty$. This however is immediate
from the assumption that $b\sim\sum_{j=0}^{\infty} \kappa^{-j}\,b_j$.
\medskip
We now prove the converse. Thus let $a\sim \sum_{j=0}^\infty a_j$ be
a classical symbol such that every $a_j$
vanishes to infinite order on $\Sigma$. Define $\forall j$
\be\label{apa7}
b_j(x,p)\,=\, a_j(x,\xi=p,\kappa =1)\, .
\ee
Observe that
\[
b_j(x,p)\, =\, |p|^{-j}\, a_j(x,{p\over |p|}, {1\over |p|})\,
\]
Since $a_j$ vanishes to infinite order at $\Sigma$, $b_j$ is in
$\calS (T^*U)$. Now let $b\in\classic$ asymptotic to $\sum b_j$.
\end{proof}
\begin{lemma}\label{Alemma2}
The space of $\Psi$DOs with amplitudes as in Lemma \ref{Alemma1}
is a $*$ algebra under composition and transposition, and it's invariant
under changes of coordinates of the form
$(x,\theta)\mapsto (\Phi (x), \theta)$.
\end{lemma}
\begin{proof}
Indeed this is the algebra of amplitudes of $\Psi$DOs on $M\times\bbR$
whose full symbol vanishes to all orders on $\Sigma$.
\end{proof}
\begin{definition}
The algebra of \scpsudo s on $M$ defined by the previous Lemma will be
called the algebra of admissible observables, and denoted $\calO (M)$.
\end{definition}
Obviously the sc symbols of admissible observables are precisely the
elements of $\calS (T^*M)$.
Finally, observe that an operator $P$ on $M\times \bbR$ of
giving rise to a \scpsudo induces a $\Psi$DO on $M\times S^1$
commuting with $D_\theta$, the Fourier components of which are
naturally identified with $P_\h$ with $\h=1/m$.
\subsection{Egorov's Theorem, positive quantization and
uniformity of eigenfunctions.}
Consider now the operator $P_f$ of \S 4. Recall that it is
a microlocalisation in a region $\calV_I$ corresponding to an energy band I
of an operator of the form
$ \sum_{j=1}^N\, h^j P_j$ where $P_j$ is a differential operator
of order $j$ on $M$.
\begin{theorem}(Egorov's theorem)
If $A_h$ is a \scpsudo microlocally supported in $\calV_I$ and
with sc symbol $a(x,p)$, then
\[
e^{-itP_f^\h}\,A_\h\,e^{itP_f^\h}
\]
is a \scpsudo microlocally supported in $\calV_I$ and
with sc symbol $\phi_t^* a(x,p)$, where $\{\phi_t\}$ is the Hamilton flow
of the symbol p of $P$.
\end{theorem}
\begin{proof}
By the same arguments as in section 3,
\be
e^{-itP_f^\h}\,A_\h\,e^{itP_f^\h}
\ =\ e^{-i\h^{-1}\theta}\, e^{-itP_f}\,A_f\,e^{itP_f}
e^{i\h^{-1}\theta}
\ee
By the classical Egorov theorem on $M\times S^1$,
$e^{-itP_f}\,A\,e^{itP_f}$ is a zeroth order $\Psi$DO with principal
symbol $\Phi_t^*\tilde{a}$ where $\tilde{a}$ is the symbol of $A$,
namely $a(x,\tau^{-1}\xi)$ and $\Phi_t$ is the Hamilton flow of the
symbol of $ P_f$ (which we recall is $\tau p(x,\tau^{-1}\xi)$ on
$\calV_I$). The computations of \S 3 give the result.
\end{proof}
\bigskip
\begin{theorem}(On positive quantization.)
Let $A_\h$ is a \scpsudo with s.c.symbol a$\geq$0.
Then it exists an operator $\widehat A_\h$ such that:
-$\widehat A_\h$ is a non-negative operator.
-$A_\h - \widehat A_\h \ =\ O(\h)$ in the topology of bounded
operators in $L^2(M)$.
\end{theorem}
\begin{proof} By definition
$A_\h \,=\, e^{-i\h^{-1}\theta}\, Pe^{i\h^{-1}\theta}$, where P is a zeroth
order pseudodifferential
operator on $M\times\bbR$, commuting with $D_\theta$.
Then by the Friedrich
quantization procedure (see \cite{CdV2}) one can find a non-negative
$\widehat P$ such that
$P - \widehat P$ is of order $(-1)$. Since the
Friedrich construction involves a
pseudodifferential calculus with several frequency variables whose symbol
is obtained from the symbol of P
without altering the space variables, the symbol of
$\widehat P$ doesn't depend of $\theta$ and so $\widehat P$ also commutes with
$D_\theta$. Define $\widehat A_\h$ by
$\widehat A_\h \,=\, e^{-i\h^{-1}\theta}\, \widehat Pe^{i\h^{-1}\theta}$.
Since $D_\theta[P-\widehat P]$ is of order 0, $\h^{-1}[A_\h\ -\ A]$ is bounded.
\end{proof}
\bigskip
With these ingredients at hand, the following result of Helffer, Martinez and
Robert \cite{HMR} extends immediately to manifolds:
\begin{theorem}(Uniformity of eigenfunctions in the ergodic case.)
Let $S_\h$ be as in \S4. Let us suppose that
the flow of $H$ (the symbol of $S_\h$) is ergodic on a regular energy surface
$\Sigma_E$.
Then there exists a density one sequence of eigenvalues $E_{j_k}$ of $S_\h$
tending to E as $\h$ tends to 0, such that for every sc observable
$A_\h$ with sc symbol $a$ one has:
\be
\lim_{\h\to 0}\;
(\phi_{j_k},A_\h\phi_{j_k})\ =\ \int_{\Sigma_E}\, a(x,\xi)d\mu^L
\ee
where $\phi_{j_k}$ are the eigenfunctions corresponding to $E_{j_k}$ and
$d\mu^L$ is the Liouville measure on $\Sigma_E$.
\end{theorem}
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\end{document}