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\title{Spectral Estimates Around a Critical Level}

\author{R. Brummelhuis\thanks{Deptartment of Mathematics,
Leiden University, P. O. Box 9512, 2300 RA  Leiden, The Netherlands.
Research made possible by a Fellowship of the Royal Netherlands Academy of
Arts and Sciences}\ ,
\ T. Paul\thanks{CEREMADE, Universit\'e Paris-Dauphine,
Place de Lattre de Tassigny, 75775 Paris Cedex 16, France}
\ \ and A. Uribe\thanks {Mathematics Department,
University of Michigan, Ann Arbor, Michigan 48109.
Research supported by NSF grant DMS-9107600} }

\begin{document}
\setcounter{page}{0}
\date{September 27, 1993}
\maketitle
\tableofcontents
\vfill\break

\section{Introduction and Main Statements}

The semi-classical trace formula of \cite{GU}, \cite{BU}, \cite{PU1},
\cite{PU2} is a rigorous version of the Gutzwiller trace formula which
provides information about the spectral function of Schr\"odinger
operators
in the semi-classical regime.  To be more specific, consider
an operator of Schr\"odinger type on a compact manifold $M$,
that is an
operator of the form
\be\label{I.1}
A_\hbar = \sum_{j=0} ^N \hbar^j A_j
\ee
where, for each $j$, $A_j $ is a differential operator
on $M$ of order $j$. Define the Hamiltonian
$H$ on $T^*M $ by
\be\label{I.2}
H(x,p)\ =\ \sum_{j=0} ^r \sigma _{\mbox{\tiny prin}}(A_j)(x,p),
\quad H \in C^\infty(T^*M),
\ee
the sum of the principal symbols of the $A_j$.	We will assume that
$A_\hbar$ is elliptic, in the sense that there exists a $c > 0$ such
that $H(x,p) \geq c > 0$ on $T^*M$.
The main example is a Schr\"odinger operator,
\[
A_\hbar = - \hbar^2\Delta + V
\]
with $\Delta$ the Laplace-Beltrami operator associated with a Riemannian
metric on $M$ and $V \in C^\infty (M)$ is strictly positive.
$A_\hbar $ then has discrete spectrum,
consisting of eigenvalues  $\{E_j(\h)\}$ with finite multiplicities, and
the trace formula alluded to above states that, under certain assumptions,
the sums
\be\label{I.3}
\Upsilon _{\hbar , E}(\varphi) \ = \
\sum_j\ \varphi\left( {E_j(\h) - E\over\h }\right)
\ee
where $\varphi$ is a test function with compactly supported Fourier
transform, have an asymptotic expansion
in integer powers of $\h$
as $\h\to 0$, and computes
the leading coefficients of these expansions.
One of the assumptions of this Theorem
is that the parameter $E$ appearing in (\ref{I.3}) be
a regular value of the classical Hamiltonian $H(x, p)$, that is, that
there are no equilibria with energy $E$.  In that case the
leading order term of (\ref{I.3}) for $\varphi $'s whose Fourier transform
$\hat{\varphi } $ is supported in a sufficiently small interval around $0$
is:
\be\label{I.4}
{\hat{\vp}(0)\over (2\pi)^n}\; \mbox{LVol}\, (\Sigma_E)\;\hbar^{-n+1}\ ,
\ee
where $\mbox{LVol}\, (\Sigma_E)$ is the Liouville measure of the
energy surface $\Sigma_E\ =\ \{ H=E\}$.  The trace formula also
computes the leading order term of (\ref{I.3}) in case zero is not in
the support of $\hat{\vp}(0)$, under suitable assumptions on the set of
periodic trajectories of the Hamilton flow of $H$,
but we won't recall that formula here.
The goal of the present work is to study the behavior of (\ref{I.3}) in
case there are equilibria with energy $E$.   As a consequence, we will
obtain, in generic cases at least, the limit of the counting function
\be\label{I.5}
N(\h)\ =\ \sharp\,\{\, j\ ;\ |E_j(\h) - E| \leq c\h\,\}
\ee
as $\h\to 0$.

\medskip
When one drops the assumption that $E$ be a regular value,
the behavior of (\ref{I.3}) will depend on the nature of
the singularities of $H$ on $\Sigma_E$, which of course can be extremely
complicated.   Here we will only consider the simplest case: that of a
Hamiltonian $H$ which has a non-degenerate critical manifold $\Theta$ in the
sense of Morse theory.
More specifically, we will work under the following assumption:

\newcommand{\x}{\overline{x}}
\medskip
\par\noindent
{\bf Main Hypothesis}
\begin{quote}
\em
The set of critical points of the Hamiltonian $H(x,p)$
is a smooth submanifold, $\Theta$, and $H$ has a non-degenerate normal Hessian
on $\Theta $, that is,
\be \label{MH}
Q(H)_z := d_z^2H : (T_z(T^*M) / T_z(\Theta ))^2 \to \bbR
\ee
is non-degenerate for all $z \in \Theta$.
Moreover, we will assume that
the multiplicities of the eigenvalues of the normal Hessian
are independent of $z \in \Theta $.
\end{quote}

\medskip \noindent
Since, in
particular, the differential of $H$ is zero in directions tangent to $\Theta$,
the connected components of $\Theta$ are contained in level sets of $H$.
Since we are interested in estimates that are local in the energy we may
assume without loss of generality that
\be\label{I.6}
\Theta\subset H^{-1}(E_c)\ ,
\ee
for some particular critical value of the energy, $E_c$. We will also assume,
in order to simplify the statements of our main theorems, that $\Theta $ is
connected. There is no essential loss of generality in doing this, since the
contributions
of the different components of $\Theta $ to the asymptotics of
$\Upsilon _{\h}$ can just be added: this is so because
we will analyze $\Upsilon _{\h}$ by microlocal techniques.


\begin{theorem}\label{ONE}
Under the Main Hypothesis, let $n=\mbox{dim}\, M$,
$N=\mbox{codim}\,\Theta$, and let $\nu$ be equal
to the number of negative eigenvalues of the Hessian of $H$ on $\Theta$.
Let $\varphi$ be a smooth function on $\bbR$ with Fourier
transform supported in a sufficiently small neighborhood of the origin.
Then, as $\h\to 0$:

\smallskip\noindent
{\bf A.}
If $\nu \geq 1$, $N-\nu \geq 1$ and both are odd, there is an asymptotic expansion
\be\label{I.7}
\sum_j\ \varphi\left( {E_j(\h) - E_c\over\h }\right)\ \sim
\ \h^{-(n-1)}\;\sum^\infty_{j=0}\; \sum_{l=0,1} \ c_{j,l}\;
\h^{j}\; [\log (1/\h)]^l\
\ee
involving logarithms of $\hbar $.
Moreover, the first-non-zero coefficient involving such a logarithm,
$c_{j_0,1}$, occurs for
\[
j_0\ =\ {N\over 2}-1\;.
\]
(Notice that $N$ must be even in this case.)

\smallskip\noindent
{\bf B.}
If $\nu \geq 1$, $N-\nu \geq 1$ and one of them is even, or if the Hessian is
definite on $\Theta$, there is an asymptotic expansion
\be\label{I.7.c}
\sum_j\ \varphi\left( {E_j(\h) - E_c\over\h }\right)\ \sim
\ \h^{-(n-1)}\;\sum^\infty_{j=0}\;  c_j\, \h^{j/2}\;,
\ee
involving half-integer powers of $\h$.

\end{theorem}

\noindent
{\bf Remark.}  The condition that the support of $\hat{\varphi}$ be
in a small neighborhood of zero can be relaxed to: 
$\hat{\varphi}$ of compact support, provided the linearized flow
at every $z\in\Theta$ has no non-zero periodic vectors.  In that case, write
$\varphi = \varphi_1+\varphi_2$ with  $\hat{\varphi_1}$ small, and
treat the $\varphi_2$ term as in Theorem 5.6 of \cite{GU}.


\medskip
We can compute some of the coefficients in the expansions of theorem \ref{ONE},
namely the coefficients of the leading term and of the leading log term
(in part {\bf A}). To state these formulae we need to recall a basic fact
about clean fixed point manifolds of symplectic mappings. Let
$\{ f_t \}  $ denote the Hamilton flow of $H$ on $T^*M $. Then, if $t$ is
sufficiently close to $0$ but not equal to $0$, $\Theta $ is a clean fixed
point manifold of $f_t $ (equivalently, $t$ is not a period of one of
the linearized flows $\{ d(f_s)_z \}_s $ on $T_z(T^*M) / T_z(\Theta )$ for
$z$ in $\Theta $). Recall that by \cite{DG} we can in this situation
associate to $\Theta $ and $f_t $ a canonical
measure $\mu _t$ on $\Theta $; the construction of this
important measure is recalled in the appendix to this paper.
There we will also show that the family of measures
$|t|^{N/2}\mu_t$ is smooth at $t=0$.  In that case, a key r\^ole is played
by the measure
\be\label{I.7.d}
\mu_0\;= \;\lim _{t \to 0^{+}} \ {d^{N/2-1} \over d t^{N/2 - 1}}\;t^{N/2} \,
\mu _t\,.
\ee
One should think of $\mu_0$ as a regularization of Liouville measure on $\Theta$.

\begin{theorem} \label{TWO} Keeping the hypotheses of Theorem \ref{ONE}, the leading term of
the asymptotics of Theorem \ref{ONE} is of the order $h^{-(n-1)} \log h^{-1}$
if $N=2$ and the normal Hessian of $H$ is indefinite. In all other
cases the leading term is of order $h^{-(n-1)}$. Furthermore, we have

\smallskip \noindent
(i) In case {\bf A},
if $\hat{\varphi }$ is flat at $0$ (i.e. $\hat{\varphi }^{(k)}(0) = 0$ for all $k \geq 1$),
the coefficient of the leading logarithmic term is
\be \label{1.8a}
c_{N/2 - 1, 1} \, = \, C_{N,n,\nu}\cdot \hat{\varphi}(0)\
\mu_0(\Theta)\;,
\ee
with
\[
C_{N,n,\nu}\,=\,
(2\pi )^{-n} (2i)^{N/2 - 1} \cdot
{{\Gamma \left( {{\nu } \over 2} \right) \,
\Gamma \left( {{N - \nu } \over 2} \right)} \over
{\Gamma \left( {N \over 2} \right)}} \cdot
\Sigma _{\nu -1} \cdot
\Sigma _{N - \nu - 1}
\]
where if $k>1$, $\Sigma _{k-1}$ is the surface measure of the unit sphere
in $\bbR ^k$:
$\Sigma _{k-1} = 2\pi ^{(k-1)/2} / \Gamma \left( (k-1)/2 \right)$,
and where $\Sigma _0 := 2$.

\smallskip \noindent (ii) If $N = 2$ but the normal Hessian is definite,
say positive definite, then the coefficient of the leading term is
\be \label{1.8b}
c_0 = (2\pi)^{-n}\, \left( \mbox{LVol}\, (\Sigma _{E_c}) \, + \, c(\Theta ) \right),
\ee
where
\be \label{1.8c}
c(\Theta ) \,  = \,
\pi \hat{\varphi }(0) \, \mu_0 (\Theta) \, + \,
i \int _{\bbR } \,
|t| \, \hat{\varphi }(t) \, \mu _t(\Theta ) \, { {dt} \over t} \ .
\ee

\smallskip\noindent (iii)
Finally, if $N > 2$, the
leading coefficient $c_{0,0}$ is in both cases of Theorem \ref{ONE} the same
as in the regular case:
\be\label{I.8e}
c_{0,0}\ = \ c_0 \ = { \hat{\varphi}(0)\over (2\pi)^n}\;
\mbox{LVol}\, (\Sigma_{E_c})\ .
\ee
\end{theorem}

\medskip \noindent
{\bf Remarks.}	(a)  The integral (\ref{1.8c}) should be interpreted as the
principal value distribution $PV (1/t)$ paired with the smooth function
$|t| \hat{\varphi }(t) \mu _t(\Theta )$.
\par\noindent
(b)  At first sight the coefficient (\ref{1.8c}) looks complex even if
$\varphi$ is real-valued, which would of course contradict the definition
of $\Upsilon _{\hbar }$. However, using the fact that $\mu _t = \mu_{-t}$ and
that $\overline{\hat{\varphi }(t)} = \hat{\varphi }(-t) $ for real-valued
$\varphi $, one easily shows that (\ref{1.8c}) is in fact real for such
$\varphi $.
\par\noindent
(c)  Note that if the normal
Hessian is definite, then $\Theta $ is itself already a component of
$\Sigma _{E_c}$. In this case $\Theta $ contributes to the expansion
of $\Upsilon _\hbar$ starting from the order $\hbar ^{-n + N/2} $.
One can compute the part of the coefficient of this term coming from
$\Theta $ for arbitrary $N$, cf. section 3.4 below, but we have omitted this
result, which did not seem very useful if $N > 2$.

\bigskip\noindent
{\bf Summary:}
The picture that emerges from theorems \ref{ONE} and \ref{TWO} is as
follows.  First, the Weyl law (\ref{I.4}) for the top order asymptotics
of $\Upsilon _{\hbar , E_c}$ changes if and only if $N = 2$. This is in
agreement with the easily verified fact that the singularity of the
Liouville measure on $\Sigma _{E_c} $ is integrable if $N > 2$, but not
if $N = 2$.  Secondly, in case {\bf A} of Theorem \ref{ONE}, the leading
contribution of the critical manifold $\Theta$ to the asymptotics changes  by a
logarithmic term, with coefficient proportional to $\mu_0(\Theta)$.

\medskip\noindent
{\bf Example.}	Consider a Schr\"odinger operator
$- (\hbar ^2/2) \Delta + V(x) $ on an $n$-dimensional compact Riemannian
manifold $M$, with strictly positive potential $V$ having a local maximum
$E_c $
at the point $x_0 \in M$ and suppose that in fact $\Theta $ consists of the
single point $z_0 = (x_0, 0) \in T^*M $. Then $N = 2n $ and $\nu = n $ and if $\{ f_t \} $ is the Hamilton flow
of $H = p^2/2 \, + \, V(x) $ then
\be\label{I.7.2}
\mu _t (\{z_0 \}) =
|\det(I-d(f_t)_{z_0})| ^{-1/2}
\ee
(cf. the appendix).
If the dimension of $M$ is odd, then logarithms of $\h $ will appear in the
expansion of $\Upsilon _{\hbar, E_c} $, the first one with index
$j_0 = n - 1$ and coefficient $c_{n-1, 1} $ which can be read off from
(\ref{1.8a}). A particularly interesting case here is that of a
one-dimensional Schr\"odinger operator, since if $n = 1$ then $N = 2$ and
the Weyl law (\ref{I.4}) changes; in fact, this is the only case where this
happens for Schr\"odinger operators. An easy computation shows that in this
case, if $\hat{\varphi }(0) = 1$,
\be
c_{0, 1}\; =\; { 2 \over {\sqrt{|\det d^2_{z_0} H (z)|}}}\;=\;
{2 \over \sqrt{ |V'' (x_0)| } }.
\ee

\bigskip

Using Theorems \ref{ONE} and \ref{TWO} and a Tauberian argument, we will show:

\begin{theorem}\label{THREE}
Under the Main Hypothesis,
let $f(\h)$ denote the leading order term in the expansion of $\Upsilon _\hbar $
in theorem $\ref{ONE}$
with $\varphi$ such that $\hat\varphi (0) = 1$ and $N$ and $\nu$ of any parity.
Assume that the linearized flows
$d(f_t)_z$ on $T_z(T^*M) / T_z(\Theta )$, $ z \in \Theta $,
do not have any non-zero periods. Furthermore, assume that if $N > 2$ then the set
of non-trivial periodic trajectories of $\{ f_t \}$ on
$\Sigma _{E_c} \setminus \Theta $ has Liouville measure $0$. Under these
assumptions,

\[
N(\h)\ =\ 2c\,f(\h)  + o(f(\h) )\ .
\]
\end{theorem}

\medskip \noindent
{\bf Example.} If $N = 2$ and the normal Hessian $Q(H)$ is indefinite then
the linearized flows in theorem \ref{THREE} are hyperbolic, and the
hypothesis of the theorem are automatically satisfied. Thus in this case
\[
N(\hbar ) \ \simeq \ { {4c} \over  {(2\pi )^{n-1}}} \, \mu_0(\Theta ) \,
\hbar ^{-(n-1)} \log \hbar ^{-1}
\]
and inspection of the proof below shows that one can actually improve the
$o(\hbar ^{-(n-1)} \log \hbar ^{-1})$-estimate for the error to
$\calO (\hbar ^{-(n-1)})$.
If $Q(H)$ is definite then of course the linearized flows are completely
periodic and this theorem does not apply.

\medskip
The hypothesis on the linearized flow in Theorem \ref{THREE} can surely
be weakened.  An open problem in this direction is to find a full trace
formula, i.e. an asymptotic expansion of $\Upsilon_{\h}(\varphi )$
without the restriction on the support of the Fourier transform
in Theorems \ref{ONE} and \ref{TWO}.
One expects additional contributions, e.g. logarithmic terms,
arising from the periods of the linearized flow along the critical
manifold.

\medskip
The result of Theorem 1.3 is precisely the one predicted by
the uncertainty principle, a form of which says that one can put one
semi-classical state per unit volume of phase space, measured in units of
$(2\pi\h)^n$.  Hence the function $N$ should satisfy
\be\label{I.8}
{\h^n\,N(\h)}\
\sim (2\pi)^{-n}\,\mbox{Liouville Volume of}\; H^{-1}[E_c-\h c , E_c+\h c]\ .
\ee
This is justified by Theorem \ref{THREE}. If we take for example a one-dimensional
Schr\"odinger operator whose potential $V$ has a local maximum $E_c$ at the
point $x_0$ (and at no other points) then by the above Theorem:
$
N(\h ) \simeq  \, \mbox{const} \cdot \log(1/ \h),
$
and the $\log(1/ \h)$ is predicted by (\ref{I.8}).

\bigskip
Our next results are on the behavior of the eigenfunctions corresponding
to the eigenvalues counted by $N(\h)$. In case $E$ non-critical, and
under the assumption that the Hamiltonian flow is ergodic on the energy
surface $H^{-1}(E)$, it is known that almost all such eigenfunctions
become uniformly distributed in a suitable sense, see \cite{Sch},
\cite{Z4} \cite{CdV2} and specially \cite{HMR}.  In the present
setting, and if $N>2$, the notion of ergodicity of the classical flow
is still valid since the Liouville density, although singular,
remains integrable, and it
turns out that almost all the eigenfunctions are still uniformly
distributed.  If $N=2$ and the Hessian is indefinite then, as Theorem 1.5 below shows, the singularity of the
Liouville measure ``traps" the wave functions on $\Theta$
(without any considerations of ergodicity). 
This second result, in dimension one,  has been found independently by
Colin de Verdi\`ere and Parisse, \cite{CdVP}, who use it to show that
there can exist eigenfunctions of the Laplacian on a compact manifold
that concentrate along an unstable periodic geodesic.  We also mention
earlier work of Duclos and Hogreve, on the eigenfunctions with
eigenvalue precisely $E_c$, \cite{Du}.

To state our Theorems, we need to quote a construction from the appendix of
\cite{PU2}.  We denote by $\calS (T^*M)$ the space of smooth functions
that satisfy Schwartz estimates in the fiber directions, uniformly in
the $M$ variables.  In \cite{PU2}, we defined a quantization
procedure of functions in $\calS (T^*M)$, $ a\mapsto \Op (a,\h)$,
where $\Op (a,\h)$ is an $\h$-dependent operator with semi-classical
symbol $a$.  The operators $\Op (a,\h)$ are a manifold version of the
$\h$-admissible operators of Robert and Helffer, \cite{R}.  Roughly, the
Schwartz kernel of $\Op (a,\h)$ has the local expression
$ \int\, e^{i \xi\cdot (x-y)}\,a(x,\h\xi)\,d\xi $.
We first need a slight generalization of Theorem \ref{THREE}.

\begin{theorem}\label{THREE'}
Let $\{ \psi^{\h}_j\}$ be an orthonormal basis of $L^2(M)$ such that
$A_\h \psi^{\h}_j\ = E_j(\h) \psi^{\h}_j$, and let $a\in\calS (TM)$.
Under the main hypothesis, the conclusions of Theorems \ref{ONE} and
\ref{TWO} remain valid for the asymptotics as $\h\to 0$ of
\be\label{I.8.a}
\sum_j\, <\psi^{\h}_j, \Op (a,\h)\psi^{\h}_j>\,
\varphi\left( {E_j(\h) - E_c\over\h }\right)\,,
\ee
provided the formulae for the coefficients in the expansion
are modified as follows:

\smallskip\noindent
(i)  If $\nu\geq 1$, $N-\nu\geq 1$ and both are odd,
\be\label{I.8.b}
c_{N/2-1,1}\,=\, C_{N,n,\nu}\cdot \hat{\varphi}(0)\,\int_\Theta a\,
d\mu_0\,.
\ee

\smallskip\noindent
(ii) If $N>2$,
\be\label{I.8.d}
c_{0,0}\;=\; = \; c_0 \; = \; {\hat{\varphi}(0)\over (2\pi)^n}\,\int_{\Sigma_{E_c}}\, a\,
d\lambda
\ee
where $\lambda$ denotes Liouville measure on $\Sigma_{E_c}$.
\end{theorem}

Using this result, the Egorov-type theorem of \cite{PU2}, a refined Tauberian
theorem proved in \S 6 and a well-known argument of Colin de Verdi\`ere
and Zelditch, we prove:

\begin{theorem}\label{FOUR}
Let $\{ \psi_j^{\h}\} $ denote a basis of $L^2(M)$ of eigenfunctions of
$A_{\h}$ with associated eigenvalues $E_j(\h)$, and make the same assumptions
as in Theorem \ref{THREE}.  Then:
\par\noindent
(1) If $N>2$ and the flow on $\Sigma_{E_c}$ is ergodic, there exists
a density-one subset
\be\label{I.9}
L(\h)\subset \{\, j\;;\; |E_j(\h)-E_c| < c\h\,\}
\ee
such that for all $a\in \calS (T^*M)$ and all integer-valued
functions $\h \to j(\h)$ for which $j(\h ) \in L(\h )$ for all $\h $ we have
\be\label{I.10}
\lim_{\h\to 0} \langle \psi_{j(\h )}^{\h}\,,\,
\Op (a,\h)\psi_{j(\h )}^{\h}\rangle\;=\;
{1\over\mbox{LVol}\,\Sigma_{E_c}}\,\int_{\Sigma_{E_c}}\, a\, d\lambda \ ,
\ee
$\lambda$ denoting Liouville
measure.
\par\noindent
(2) If $N=2$ and the Hessian $d^2H$ is indefinite along $\Theta $, there exists a density-one subset $L(\h)$ such that
for all $a\in \calS (T^*M)$ which restricted to $\Theta$
are constant, equal to $a_\Theta$ say, and for all
$\h \to j(\h ) \in L(\h )$,
\be\label{I.11}
\lim_{\h\to 0} \langle \psi_{j(\h )}^{\h}\,,\,
\Op (a,\h)\psi_{j(\h )}^{\h}\rangle\;=\;
a_\Theta
\ee
uniformly with $j\in L(\h)$.
\end{theorem}
\noindent
{\bf Remarks.}	(a) That $L(\h)$ have density one means that
\[
\lim_{\h\to 0}\, {\# L(\h) \over N(\h)}\;=\; 1\,.
\]
The statements in (\ref{I.10}) and (\ref{I.11}) are equivalent with the following:
$\forall \epsilon >0\ \exists \delta >0$ such that $\h\in(0,\delta)$ and
$j\in L(\h)$ imply that the distance between
$\langle \psi_j^{\h}\,,\, \Op (a,\h)\psi_j^{\h}\rangle$
and the corresponding right-hand side is less than $\epsilon$.
These follow easily from
\be\label{I.10.a}
\lim_{\h\to 0}\, {1\over N(\h)}\, \sum_{j\,;\, |E_j(\h)-E_c|<c\h }
\left|\, \langle\psi^{\h}_j\,,\,\Op (a,\h)\psi^{\h}_j\rangle - m(a) \,\right|\,=
\, 0\,,
\ee
where $m(a)$ denotes the right-hand side of (\ref{I.10}), (\ref{I.11}),
respectively.
\par\noindent
(b) In case $N=2$ and $n=1$ (so that $\Theta =	\{ z_0\}$) the assumption
of $a|_\Theta$ being constant is automatic, and the limit in
(\ref{I.11}) is simply $a(z_0)$.
\par\noindent
(c) In case $N=2$ this result implies localization of the
eigenfunctions on $\Theta$, since the right-hand side of (\ref{I.11})
is zero if $a$ vanishes on $\Theta$.

\bigskip
Finally, we wish to note that all the above results immediately extend to
Schr\"odinger operators on $\bbR ^n$, by the technique of
\cite{BU}, section 5, provided we restrict attention to the discrete
spectrum below $\liminf _{|x| \to \infty } V(x)$ and modify
the definition of $\Upsilon _{\hbar, E}$, by summing only
over $\{j : |E_j(\hbar ) - E| \leq \hbar ^{1 - \varepsilon } \} $ in
(\ref{I.3}), for some fixed but arbitrary $\varepsilon \in (0, 1) $.

\medskip
\bigskip
The paper is organized as follows.  In \S 2, we reduce the study of
(\ref{I.3}) to a stationary phase problem with a degenerate phase.
Precisely, the phase in question is $\psi = t\,Q(w)$ in $(t,w)$ space,
where $Q$ is a non-degenerate quadratic form related to the normal
Hessian, $t$ represents time and $w$ is a variable normal to $\Theta$.
In $\S 3$ we do a careful study of such oscillatory integrals with
main result a stationary phase lemma for them. We refer to the
introduction to \S 3 for a discussion of the relationship of our
results with those already in the literature. Together, \S\S 2 and 3
prove Theorem \ref{ONE}, but do not compute any of the coefficients in
terms of the Hamilton flow of $H$. This is because in \S 2 we use the
Morse lemma with parameters, which introduces a Jacobian in the
calculation over which we have no a priori control.

This is circumvented by the results of \S 4, where we show that a
suitable Fourier transform of (\ref{I.3}) is a singular Lagrangian
distribution $\Upsilon(s,E)$ in two variables, associated to a pair of
intersecting Lagrangians.   We then compute the symbol of $\Upsilon$,
in the sense of the symbol calculus for singular Lagrangian
distributions due to Guillemin, Melrose and Uhlmann.  We then show that
that symbol is intimately related to those coefficients of the
asymptotic expansions of Theorem \ref{ONE} that we compute in Theorem
\ref{TWO}.  As a by-product of this approach, we show in \S 5 that our
estimates can be made uniform in a neighborhood of size $\h$ of $E_c$.
In \S 6 we prove a Tauberian lemma that will demonstrate Theorem
\ref{THREE}, and \S 7 is devoted to the proof of Theorems
\ref{THREE'} and \ref{FOUR}.

\medskip\noindent
{\sc Acknowledgments.}	We are would like to thank Gunther Uhlmann for
explaining to us the calculus of singular Lagrangian distributions, and to
Yves Colin de Verdi\`ere and Richard Melrose for useful discussions.
Some of the present results were announced in \cite{PUTr}.


\section{Oscillatory Integral Representations of Spectral Functions}


In this section we recall some results from \cite{PU2} on oscillatory
integral representations of various spectral objects.  Let $M$ be a
compact manifold, and $A_\hbar$ an $\hbar$-dependent differential
operator on $M$ of the form (\ref{I.1}).
We will in fact assume	$H(x,p) \geq c > 0, \forall (x ,p) \in T^*M$.
By ellipticity and compactness, for each positive value of $\hbar$ the
operator $A_\hbar$ has discrete spectrum.  We will denote by $E_1(\hbar)
\leq E_2(\hbar) \leq\cdots$ the eigenvalues (repeated according to their
multiplicity) and $\psi_j^\hbar$, $j = 1,2,\ldots$, corresponding
eigenfunctions of $A_\hbar$.  We wish to obtain an oscillatory integral
description of
\be\label{II.1}
\sum_j \varphi\bigg({{E_j(\hbar)-E}\over\hbar}\bigg)
\ee
where $\check\varphi
\in C_0^\infty(\bbR)$, $E$  is a parameter, and the
large parameter in the integral we are after is of course
$\kappa = \hbar^{-1}$. Here $\check\varphi (t) =
(2\pi )^{-1} \hat{\varphi }(-t) =
= (2\pi )^{-1} \, \int _{\bbR } \, \varphi (s) e^{ist} \, ds$ is the inverse
Fourier transform of $\varphi $.
In \cite{PU2} it is shown that (\ref{II.1}) is given,
modulo $O(\hbar^\infty)$, by an integral of the form
\be\label{II.2}
{ 1 \over {2\pi } }
\int_0^{2\pi} \int e^{-iks} \check{\varphi} (t)\;
\tr\,\left[\, \chi(\tilde P)\,
e^{i\,(-tD_\theta\tilde{P} + (s+tE)D_\theta)}\, \right]\; ds\, dt
\ee
where:

\smallskip
a)  $\hbar^{-1} = k \in \bbZ^+$.

\smallskip
b)  $\tilde P$ is a certain classical pseudodifferential operator on
$M\times S^1$, of order zero, commuting with $D_\theta$ ($\theta$ the angular
variable on $S^1 = \bbR/2\pi\bbZ)$.

\smallskip
c)  $\chi\in C_0^\infty(\bbR)$ a suitable cut off function.

\medskip
The operator $\tilde P$ is a microlocalized version of the operator
\[
P = \sum_{j=0}^r D_\theta^{-j} A_j
\]
on $M\times S^1$, where  $D_\theta^{-1}$  is the obvious relative parametrix
of  $D_\theta$.  We won't describe here in detail how  $\tilde P$  is
constructed; it will suffice to say that its principal symbol has the
following property:  there exist constants  $C_1 > C_2 > 0$  such that
\be\label{II.3}
\sigma_{\tilde P}(x,\theta\, ;\, \xi,\kappa)\ = \
\left\{
\begin{array}{lr}
H(x,\xi/\kappa) & \mbox{if}\ |H(x,\xi/\kappa)|\, <\, C_2\\
0 & \mbox{if}\ |H(x,\xi/\kappa)|\, \geq\, C_1
\end{array}
\right.
\ee
In fact $\tilde P$ is microlocally supported in the cone $\vert
H(x,\xi/\kappa)\vert \leq C_1$.  The representation (\ref{II.2}) of (\ref{II.1}) is valid
modulo $O(h ^\infty)$  for $0\leq E \leq C_2-\varepsilon$  for every
$\varepsilon > 0$.

\medskip
In the application of (\ref{II.2}) presented in this paper, we will be
in situations where the main term in the stationary phase expansion of
(\ref{II.2}) (for large $k$) arises from critical points at which
$s = 0$ and $E$ is equal to a critical value  $E_c$ of the Hamiltonian
$H$.

\medskip
To proceed further, we need an oscillatory integral formula for the Schwartz
kernel ${\calK}(t,s,x,y,\theta,\theta')$ of the operator
\[
\calU\ =\ e^{i(-tD_\theta\tilde P + sD_\theta)}\ .
\]
In local coordinates with $\vert t \vert$ small, $\calK$ can be written in the
form
\be\label{II.4}
\calK  = \int e^{i\kappa[\theta-\theta'+s+S(t,x,p)-y\cdot p]}\
\kappa^n\ \alpha (x,y,t;\kappa p, \kappa)\ dp\, d\kappa
\ee
where $\alpha$ is a classical symbol supported in the cone
\[
\vert H(x,\xi/\kappa)\vert \leq C_1.
\]
The function $S(t,x,p)$ is the solution of the Hamilton-Jacobi
equation
\be\label{II.5}
\left\{
\begin{array}{rcl}
\partial_t S + H(x,\nabla_x S) & = & 0\\
S\vert_{t=0} & = & x\cdot p
\end{array}
\right.
\ee
All this follows from the standard construction of
$e^{-itD_\theta\tilde P}$ as a Fourier integral operator.  The Cauchy
problem (\ref{II.5}) corresponds to a different choice of phase function
from that of H\"{o}rmander in \cite{Ho1}.  The phase
(\ref{II.5}) has more physical significance in the present context.
Indeed it is well-known that $S$ is the generating function of $\{\phi_t\}$,
the Hamilton flow of $H(x,p)$:
\be\label{II.5.1}
\phi_t\left({\partial S\over \partial p} ,  p\right)\ =\
\left(x, {\partial S\over \partial x}\right)\ .
\ee

\medskip
One can solve the Cauchy problem (\ref{II.5}) by the method of
characteristics.  The amplitude
\[
\alpha = \alpha(x,y,t\, ;\,\xi,\kappa)\ ,\quad \xi = \kappa \, p,
\]
$\alpha $ a classical symbol of order $0$
with respect to $(\xi, \kappa )$,
of the FIO $\mbox{exp}(itD_{\theta } \tilde {P})$
is obtained by solving the transport equations associated with
the identity $D_t e^{itD_\theta\tilde{P}}$ $= -D_\theta\tilde{P}
e^{itD_\theta\tilde P}$, with initial condition
\[
\alpha \big| _{t=0} \, = \, (2 \pi )^{-(n+1)}.
\]
Next, in formula (\ref{II.2})
we need to compose with  $\chi(\tilde P)$.  However, we are interested
in (\ref{II.1}) with  E  in a
small interval	$[E_1,E_2]$ and the cut-off function $\chi$ is chosen so
that $\chi(\tilde P)$ is microlocally the identity in a small cone
\be\label{II.5.3}
E_1 -\varepsilon \leq H(x, {\xi\over \kappa}) \leq E_2 + \varepsilon\ .
\ee
In particular the symbol of $\chi(\tilde P)$ is  constant one in
(\ref{II.5.3}), and does not show up in the calculations below.

\medskip
Plugging (\ref{II.4}) back in (\ref{II.2}) we get:

\begin{lemma}
If the support of $\check\varphi$ is small enough, (\ref{II.1}) is equal,
modulo $O(\hbar^\infty)$, to a finite sum of integrals of the form
\be\label{II.6}
2\pi \cdot \h^{-n} \int e^{i\h^{-1}[S(t,x,p) - x\cdot p + tE]}\
\check{\varphi}(t)\ \alpha(x,x,t\,;\, \h^{-1}p,\h^{-1})\ dx\,dt\,dp\ .
\ee

\medskip
\noindent
The estimates are locally uniform as $E$ ranges over regular values of
$H(x,p)$.
\end{lemma}

\noindent
(The extra $2\pi $ in (\ref{II.6}) comes from
$\int _0^{2\pi } \, \calK |_{\theta = \theta '}
\, d\theta $).

\medskip
We seek to apply the method of stationary phase to the integral (\ref{II.6}),
under the Main Hypothesis of \S 1.  We'll analyze next the critical points
of the phase.  Recall that we are working under the assumption that the
support of $\check{\varphi}$ is small, and so our reasoning will be for
$t$ in a neighborhood of zero which may not be the same at each occurrence.

\begin{lemma}
The critical points of $S(t,x,p) - x\cdot p + tE$ (as a function of $(t,x,p)$)
are precisely the solutions to the following equations:
\be\label{II.6.1}
H(x,p)\;=\;E\ ,\quad \mbox{and}\quad \phi_t(x,p)\;=\;(x,p)\ .
\ee
\end{lemma}

The proof is an easy consequence of the Hamilton-Jacobi equation that $S$
satisfies, and of (\ref{II.5.1}).  If $E\neq E_c$, then for small $t$
the only solutions to (\ref{II.6.1}) are $t=0$ and $(x,p)$ arbitrary
on $H^{-1}(E)$.  If $E=E_c$, the set of critical points of the phase is the
union
\be\label{II.6.2}
\{\; t=0,\ (x,p)\in H^{-1}(E_c)\;\}\cup\{\; t\;\mbox{arbitrary}\,,\;
(x,p)\in\Theta\;\}\ .
\ee

\medskip\noindent
We will correspondingly smoothly cut-off the integrand in (\ref{II.6}),
and break the integral in two parts, $\Upsilon_1$ and $\Upsilon_2$,
where $\Upsilon_1$ is the integral (\ref{II.6}) localized in the
$(x,p)$ variables to some neighborhood $\calU$ of $\Theta$ in phase space.
That is, define
\be\label{II.6.3}
\Upsilon_1 (\h, E)\ :=\
\h^{-n} \int e^{i\h^{-1}[S(t,x,p) - x\cdot p + tE]}\
\check{\varphi}(t)\ \rho(x,p)\,
\alpha(x,x,t\,;\, \h^{-1}p,\h^{-1})\ dx\,dt\,dp\ ,
\ee
where $\rho$ is a smooth function supported near $\Theta$ and identically
one in a smaller neighborhood of $\Theta$.  Then define $\Upsilon_2$ to be
the difference (\ref{II.6}) minus $\Upsilon_1$.  The only critical points
of the phase that contribute to the asymptotic behavior of $\Upsilon_2$
are of the form $t=0$ and $(x,p)\in H^{-1}(E)$	but {\em away} from
$\Theta$.   Since, by assumption, $\Theta$ is the set of all
critical points of $H$, this means that the critical points away from
$\Theta$ form a non-degenerate manifold, and hence {\em the asymptotics of
$\Upsilon_2$ can be handled by ordinary stationary phase}.  In fact,
the analysis of \cite{PU2} applies, and shows that $\Upsilon_2$ has a classical
asymptotic expansion of the form
\be\label{II.6.4}
\Upsilon_2\ \sim\ \h^{-n+1}\,\sum_{j=0}^\infty\; \h^j\, a_j\ ,
\ee
locally uniformly in $E$, with
\be
a_0\ =\ (2\pi )^{-n+1} \; \check{\varphi }(0) \int _{\Sigma _{E}} \rho
d\mbox{LVol}
\ee
(compare with (\ref{I.4})).
\bigskip
Next we concentrate on $\Upsilon_1$, which contains the new phenomena.
Let
\be\label{II.7}
W(t,x,p) = S(t,x,p) - x\cdot p\ ,
\ee
which is the non-trivial part of the phase in (\ref{II.6}).  Since
$W|_{t=0}=0$, there is a smooth function $F(t,x,p)$ such that
\be\label{II.8}
S(t,x,p) - x\cdot p\ =\ t\,F(t,x,p)\ ,
\ee
and hence the phase in (\ref{II.6}) is
\be\label{II.8.1}
t\,\left(\, F(t,x,p) + E\,\right)\ .
\ee
Moreover, the Hamilton-Jacobi equation implies that
\be\label{II.9}
F|_{t=0}\ =\ -H\ ,
\ee
while property (\ref{II.5.1}) implies that
\be\label{II.10}
d_{(x,p)}F\;=\;0\quad\Leftrightarrow\quad \forall t\ \phi_t(x,p)\;=\;(x,p)
\quad\Leftrightarrow\quad (x,p)\in\Theta\ .
\ee
We will now apply the Morse lemma with parameters to the function $F$.
Reasoning locally near a point of $\Theta$,
begin by introducing coordinates $(z',z'')$ in phase space
in such a way that $\Theta$ is
the submanifold $\{z'' = 0\}$.  Let $a = (t,z')$, which we regard
as parameters in the function $F = F(a,z'')$.  By (\ref{II.9}) and the
Main Hypothesis, the Hessian
of $F|_{a=(0, z')}$ at $z''=0$ is non-degenerate.  Hence, by Lemma 1.2.2. of
\cite{Duis}, there exists a change of variables,
\be\label{II.10.1}
(a,z'')\mapsto (a,y)\ ,\quad y=y(a,z'')=y(t, z', z'')
\ee
such that
\be\label{II.10.2}
F(a,z'')\ =\ F(t,z',0) + {1\over 2}\,\langle d^2_{z''}F_{(t,z', 0)}y , y
\rangle\ .
\ee
The neighborhood $\calU$ of $\Theta$ to which we have localized
$\Upsilon_1$ is chosen as the $(x, p)$-projection of the common domain
of these changes of coordinates, the latter being a suitable small
neighborhood of $\Theta \times\{ 0\}$. Note that here we may need to further
shrink the support of $\hat{\varphi }$.

\begin{lemma}  For all $t$ and all $(x,p)\in\Theta$,
\be\label{II.10.2.1}
S(t,x,p)\ =\ -t\,E_c + x\cdot p\ .
\ee
\end{lemma}
\begin{proof}
This easily follows from an explicit formula for $S$, in terms of the
Hamilton flow of $H$: we refer to D. Robert's book, \cite{R}, formula (39)
page 213.
\end{proof}
\begin{corollary}\label{II.cor}  For all $t$ and all $z'$,
\[
F(t,z',0)\ =\ - E_c\ .
\]
\end{corollary}


The quadratic forms appearing in (\ref{II.10.2}) depend on $(t,z')$.
However, by the second part of the Main Hypothesis, they can be
smoothly diagonalized: there exists a smooth family of orthogonal
matrices, $R(t,z')$, such that
\be\label{II.10.2.2}
R\,\left( d^2_{z''}F(t,z',0)\right)\,R^{-1}\ =\
\mbox{diag}\ (\lambda _1(t, z'), \cdots , \lambda _N(t, z') ).
%\left(
%\begin{array}{ccc}
%\lambda_1(t,z_1) & \cdots & 0 \\
%0 & \cdots & \lambda_N(t,z_1)
%\end{array}\right)\ .
\ee
Hence if we let $u = Ry$ and
$w_j\ :=\ |\lambda_j(t,z')|^{-1} u_j$ the phase becomes, after perhaps
a permutation of the $w_j$'s:
\be\label{II.10.3}
t\,(E-E_c) + {t\over 2}\,\left(w_1^2 + \cdots + w_{\nu}^2 - w_{\nu +1}^2 -
\cdots -w_N^2\right)\ .
\ee
We will write this as
\be\label{II.10.4}
t\,\left( E-E_c -  {1\over 2}\,\langle Qw , w \rangle\right)\  .
\ee
The change of coordinates introduces of course a Jacobian
\be \label{II.10.2.4}
|Du/ Dw| \ = \ |\det  d^2_{z''}F_{(t,z',0)} |
\ee
which will complicate the computation of the coefficients in the expansions of
theorem \ref{ONE}.
Since $F|_{t=0} = -H$, we see that the Jacobian
(\ref{II.10.2.4}), at $t=0$, equals the absolute value of the determinant of
the normal Hessian of $H$; however, we will also need the derivatives (up till
large order in $t$) of this Jacobian.
Note that the signature of $Q$ equals the
signature of the normal Hessian of $H$, with $\nu $ the number of its negative
eigenvalues.
\medskip
We summarize:

\begin{proposition}\label{II.summary}
Under the previous assumptions,
if the support of $\check{\varphi}$ is in a small neighborhood of
zero, the quantity
\[
\sum_{j=1}^\infty \ \varphi \left[{E_j(\h) - E\over\h}\right]
\]
is, modulo a classical asymptotic series of the form (\ref{II.6.4}),
equal to a finite sum of integrals of the form
\be\label{II.11}
\h^{-n} \int e^{i\,\h^{-1}\,t\,(E-E_c -{1\over 2}\langle Qw , w \rangle )} \
\check{\varphi}(t)\,\beta(t,w\,;\, \h^{-1})\, dt\,dw\ ,
\ee
where
\be\label{II.11.1}
\beta\ =\ 2\pi \int\; \rho\,\alpha|_{x=y}\, J\, dz_1
\ee
$J$ being the Jacobian of the change of variables $(t,x,p)\mapsto (t,z',w)$.
\end{proposition}





\section{A Generalized Stationary Phase Formula}

\medskip
In this section we will make a detailed study of the asymptotic behavior of
the class of oscillatory integrals with cubic phases of the form:
\be\label{3.1}
I(\kappa) = I_{a,Q}(\kappa) = \int_{\bbR^{N+1}}a(t,z)\,
e^{i\kappa t\langle Qz, z \rangle }\,
dt\, dz\ ,
\kappa \rightarrow \infty ,
\ee
which arose in the previous section.
Here $\langle Qz, z \rangle$
is a non-degenerate real quadratic from on $\bbR^N$, and $a \in
C_c^\infty(\bbR^{N+1})$.
More generally, we could have taken as amplitude a
classical symbol $a(t, z; \kappa )$ of order $0$, with fiber
variable $\kappa \in \bbR$, compactly supported in
the space variables $(t, z)$.

The main work in analyzing (\ref{3.1}) is for indefinite $Q$: the analysis in
the definite case is much easier and will be postponed till section 3.4.
We will first look in section 3.1 below at the
case of an indefinite form on $\bbR^2$.
The case of a general $Q$ with both indices
of inertia odd can immediately be reduced to this special case.
If one of the
indices of inertia of $Q$ is even one needs to do some additional work, which
occupies section 3.3. Finally, in section 3.5 we prove our main Theorem 1.1.

%The main results of this section are Theorems \ref{III.4}
%(for $N = 2$), \ref{III.6}
%(odd indices of inertia), Theorem \ref{III.9} and Corollary \ref{III.9a}
%(one even index of inertia) and finally Theorem
%\ref{III.10} (definite $Q$).

\medskip
Before starting our analysis, we want to make some remarks on
previously known results and general background, to put
this section in some perspective.
A large number of authors have studied over the years the asymptotic
behavior of a general oscillatory integral
\be
\label{3.1.5}
I_{a,f}(\kappa) \; = \; \int_{\bbR^n} a(y)\, e^{i\kappa f(y)}dy, \; \kappa
\rightarrow \infty,
\ee
with analytic phases $f$ and compactly supported smooth amplitudes $a$. We
mention the work of Arnold, Bernstein, Malgrange and of Varchenko,
and refer to the book \cite{AGV} for more
information and detailed references.
Using Hironaka's theorem on the resolution of singularities
one can very generally prove the existence of an asymptotic expansion
\be\label{3.2}
I_{a,f}(\kappa) \sim \sum_{\alpha \in A}\; \sum_{p \geq0}\; \sum_{k=0}^{K(f)}
C_{\alpha, p, k}[a]\, \kappa^{-\alpha-p}\, (\log \kappa )^k, \
\kappa \rightarrow \infty
\ee
cf. \cite{AGV}. Here $A$ will be some finite set of strictly positive
rational numbers, determined by $f$.  This general result does not,
however, provide information about either the principal exponents
$\beta(f) := -\min{A}$ and $K(f)$, or about the distribution
coefficients $C_{\alpha, p, k}[\cdot ]$.

Varchenko \cite{V} has shown how to
read off these principal exponents from the Newton
polyhedrae of $f$, for a large class of phases $f$; cf. also \cite{AGV},
chapters 6 to 8.  Unfortunately, his work does not
apply to the phases $tQ(z)$ we are interested in here. Moreover, Varchenko's results only
provide scant information about the distributions $C_{\alpha,p,k}$
in (\ref{3.2}).
These have to be known rather explicitly for the
applications we have in mind, since in general one will only be able to
detect the presence of a critical manifold from the lower order terms
in the expansion of $\Upsilon _{\h}$.

While it perhaps might be possible to extend the treatment of
\cite{V}, \cite{AGV} to the integrals (\ref{3.1}), we will use here
a more elementary and direct method, by which we will obtain a complete
expansion of these integrals which is as explicit as the classical
stationary phase expansion for quadratic phases.

After completion of this section, J. Duistermaat informed us
that it should also be possible to derive the asymptotics of
(\ref{3.1}) from the results of \cite{T}, which studies the
singularities of fundamental solutions of the operator $Q(D)$.

\subsection{The 2-dimensional case.}

We begin by analyzing (\ref{3.1}) if $N = 2$ and $Q$ is indefinite.
After a linear change of coordinates we may assume that $\langle Qz, z
\rangle = z_1^2-z_2^2$. Making the additional change of variables  $u =
z_1+z_2$, $v = z_1-z_2$, we obtain an integral of the form
\be\label{3.3}
I(\kappa) = \int_{\bbR^3} a(t,u,v)\,e^{i\kappa tuv} dt du dv\ .
\ee
We will first restrict ourselves to amplitudes of the form
\be\label{3.4}
a(t,u,v) = \varphi(t)f(u)g(v)\ , \quad \varphi,f,g \in C_c^\infty (\bbR).
\ee
The asymptotics for a general amplitude can easily be reduced to this
case by the following simple lemma (alternatively, one could rewrite
the proofs below for general $a$, at the cost of complicating the
notations):
\begin{lemma} \label{III.0.5}
Suppose that one has an asymptotic expansion of (\ref{3.1.5}) for product type
amplitudes $a(x) = a_1 (x_1) \cdots a_n (x_n) $:
\be
\label{3.4.5}
I_{a, f}(\kappa ) - \sum _{j=1}^{K-1} C_j [a] \kappa ^{n_j} =
\calO (\kappa ^{n_K} || a ||_{C^{L(K)}}),
\ee
with $K \in \bb N$ arbitrary, $L(K) \in \bbN$ depending on $K$,
$C_j \in \calD'(\bbR^n)$ and
$n_1 > n_2 > \cdots > n_j \to -\infty $. Then the same expansion (\ref{3.4.5})
holds for general amplitudes in $C^{\infty }_c(\bbR^n)$.
\end{lemma}
\begin{proof}
Clearly the expansion (\ref{3.4.5}) remains valid for finite sums of
amplitudes of product type. Now $C^{\infty }_c(\bbR) \otimes \cdots
\otimes C^{\infty }_c(\bbR)$ ($n$-fold tensor product) is dense in
$C^{\infty }_c(\bbR^n)$.
Hence we can approximate an arbitrary amplitude $b \in C^{\infty }_c(\bbR)$
in any $C^L$-norm
by such a finite sum $\tilde{a}$ of product amplitudes, up to an arbitrarily
small error $\varepsilon > 0$. If we take $L$ sufficiently large, write
$b = \tilde{a} + (b - \tilde{a}) $ and apply (\ref{3.4.5}), we can conclude
that
the RHS of
(\ref{3.4.5}) with $a$ replaced by $b$ can be estimated by
$ C \cdot \left(
\kappa^{n_K} \, || b ||_{C^{L(K)}} \, + \,
\varepsilon \cdot (1 + \sum _{j=1}^K \kappa ^{n_j})\right)$.
Since $\varepsilon $ can be taken
arbitrarily small, this proves the lemma.
\end{proof}

We return to (\ref{3.3},), (\ref{3.4}).  Performing the $t$-integral,
(\ref{3.3}) becomes
\newcommand{\hphi}{\hat{\varphi}}
\[
\int_{\bbR}\hphi (-\kappa uv)\,f(u)\,g(v)\, dudv
\]
which equals
\be\label{3.5}
I(\kappa) = \int \hphi (-\kappa \rho)
\left( \int_{\{ uv = \rho \}} f(u)\, g(v)\,
{du \over \vert u \vert} \right) \, d\rho \ ,
\ee
since $u^{-1}du \wedge d(uv) = du \wedge dv$; $u^{-1}du$ is the
Gelfand-Leray form of the function  $uv$ (cf. \cite{AGV}).  It is now
natural to use convolution on the multiplicative group $\bbR _{> 0}$:
\be\label{3.6}
(f  \ast  g)(\rho) = \int_0^\infty f(u) \,g({\rho \over u}) {du \over u}\ .
\ee
Writing $f_-(u) = f(-u)\; ,\; g_-(v) = g(-v)$, (\ref{3.5}) equals
\begin{eqnarray}
\nonumber
\lefteqn{I(k) = {1 \over \kappa} \left\{
\int_0^\infty \hphi (-\rho) [(f \ast g) \,+
(f_- \ast g_-)]
({\rho \over \kappa }) d\rho \, +\right.}\\
& & \left. \int_0^\infty \hphi (\rho)[(f
\ast g_-) \ + \ (f_- \ast g)] ({\rho \over \kappa}) d\rho  \right\}\ .
\label{3.7}
\end{eqnarray}
Hence it suffices to determine the asymptotics of (\ref{3.6}) for $\rho
\rightarrow 0^{+}$.  This we will do using the Mellin transform
$f \to \calM\,(f)$ of $f$. Recall that if
$f \in C_0^\infty ([0, \infty))$ and $\mbox{Re}\, s > 0$,
\[
F(s) \,= \,\calM \,(f)(s) \ := \ \int^\infty_0 f(u)u^{s-1}du\ ,
\]
Also recall that
\[
\calM \,[f \ast g] \ = \calM \, [f] \,\calM\, [g]
\]
and that
\be\label{3.8}
f(u) \, = \, {1 \over {2\pi i}} \ \int^{\sigma+i\infty}_{\sigma-i\infty}
F(s)\, u^{-s}ds, \quad \sigma > 0
\ee
(the inversion formula).  The following lemma is classical:
\begin{lemma} \label{III.1}
Let $f \in C^\infty_c ([0,\infty))$.  Then
$F(s)  := \calM \,[f](s)$ extends to a
meromorphic function on $\bbC\setminus \{ 0,-1,-2, \cdots \}$.	Moreover,
$F(s)$ has simple poles at $s = -k, \  k \in \bbN$, with residues
$f^{(k)}(0)/k!$ .   Finally,
\[
F(\sigma + i\tau)u^{-(\sigma + i\tau)}\
\]
is in the Schwarz space
$\calS(\bbR)$ as function of $\tau$ when $\sigma \notin  -\bbN$.
\end{lemma}
\begin{proof}
Repeated integration by parts shows that
\be\label{3.9}
F(s) \,= \,{(-1)^{k+1} \over s(s+1)...(s+k)}\; \int_0^\infty
f^{(k+1)}(u)\,u^{s+k}\,du, \quad \mbox{Re} s > 0
\ee
from which the first half of the lemma follows.  To prove the final
statement, observe that  $F(\sigma + i\tau)$, as of a function of $\tau$,
is one over
some polynomial in $\tau$ times the Fourier transform of $f^{(k+1)}
(e^y)e^{y(\sigma+k+1)}$, which is a rapidly decreasing function of $y$ as
long as $\sigma > - k - 1$.
\end{proof}

\medskip
In the sequel the following distributions $\Lambda_j(\cdot )$ will appear:
define constants $\gamma _j$ by
\be\label{3.10}
\gamma _j \; = \; \sum^j_{l = 1} {1 \over l}
\ee
and let
\be\label{3.11}
\Lambda_j(f) \ = \ -  \int^\infty_0 f^{(j+1)}(u)\log u \  du +
\gamma _j\,f^{(j)}(0)\ .
\ee
\begin{lemma}
Let  $f \in C_0^\infty([0,\infty))$, $F(s) = \calM \,[f](s)$.  The constant
term in the Laurent expansion of  $F$ around $s = -j$  equals
\[
\lim_{s \rightarrow -j}
\left(F(s) - {f^{(j)}(0) \over {j!}} {1 \over {s+j}}\right)\
=\  {1 \over {j!}}\, \Lambda_j(f)\ .
\]
\end{lemma}
\begin{proof}
By (\ref{3.9}),
\[
F(s) - {f^{(j)}(0) \over (s+j)j!} \ =\ \int^\infty_0 \, f^{(j+1)}(u) \,
{H(s+j,u) \over s+j} \,du,
\]
with
\[
H(z,u) \, = {(-1)^{j+1}\,u^z \over (z-1) \cdots (z-j)} + {1 \over j!}\ .
\]
Note that $H(0,u) = 0$, and that the coefficient of $z$ in the power series
expansion of $H(z,u)$  around  $z = 0$	is
\[
- {1 \over {j!}} (\log u \, + \gamma _j).
\]
This proves the lemma.
\end{proof}

\medskip
The following proposition is the main step in our analysis of $I(k)$.
\begin{proposition} \label{III.3}
Let $f,g \in C^\infty_0([0,\infty))$. Then
\begin{eqnarray}\label{3.12}
\nonumber\lefteqn{(f \ast g)(\rho) \sim }\\
& & \sum^\infty_{j=0} {1 \over (j!)^2} \left(
f^{(j)}(0)\, g^{(j)}(0) \rho^j\log \rho^{-1} +
\ \left( \Lambda_j(f) g^{(j)}(0) +
f^{(j)}(0) \Lambda_j(g)\right) \rho^j
\right)\ ,
\end{eqnarray}
as $\rho \rightarrow 0^+$.
\end{proposition}
\begin{proof}
By Mellin's inversion formula (\ref{3.8}), if
$F = \calM [f]\;,\; G = \calM [g]$, then
\be\label{3.13}
(f \ast g)(\rho) \, = \, (2\pi i)^{-1}
\int^{\sigma + \infty }_{ \sigma - \infty} F(s)G(s)
\rho^{-s}\,ds\;, \ \sigma > 0.
\ee
If we shift the path of integration to
$\mbox{Re} s = -K + \varepsilon\; (K \in \bbN \;,\;
0 < \varepsilon < 1$), using lemma \ref{III.1},
we obtain
\[
(f \ast g)(\rho) = \sum^K_{j=0} \mbox{Res}_{s=-j} (F(s)G(s)\rho^{-s}) +
O(\rho^{K+\varepsilon})
\]
Since $\rho^{-s} = \rho^j - (s+j) \rho^j \log (\rho )+ O((s+j)^2)$, Lemma 3.3 implies
that
\begin{eqnarray*}
\lefteqn{\mbox{Res}_{s=-j}(FG\rho^{-s}) = } \\
& &=\  (j!)^{-2} \left( f^{(j)}(0) g^{(j)}(0) \rho^j\log
\rho^{-1} \ + \ (\Lambda_j(f) g^{(j)}(0) + f^{(j)}(0)
\Lambda_j(g)) \rho^j \right),
\end{eqnarray*}
which completes the proof of (\ref{3.12}).
\end{proof}

\medskip
We now return to (\ref{3.7}).  Substitute the expansions (\ref{3.12})
and rearrange terms.  Then we obtain, after some computations,
\be\label{3.14}
I(\kappa) \sim \sum^\infty_{j=0} c_{j,1}\, \kappa^{-1-j} \log \kappa + c_{j,0}
\, \kappa^{-1-j} \  = \
\ee
\[
=\ \sum^\infty_{j=0} \, \sum_{l =0,1} \, c_{j,l} \,
\kappa^{-1-j}\, (\log \kappa)^l,
\]
where
\be\label{3.15}
(j!)^{2} \, c_{j,1} = (4\pi)\, i^j\, f^{(j)}(0)\,
g^{(j)}(0)\, \varphi^{(j)}(0)
\ee
and
\[
(j!)^2 \, c_{j,0} \, = \, -2f^{(j)}(0) g^{(j)}(0) \int_{\bbR} \, \hphi
(- \rho)\,  \rho ^j \log \vert \rho \vert\, d\rho
\]
\newcommand{\sig}{\mbox{sgn}}
\be\label{3.16}
-2 \pi i^j \varphi^{(j)}(0)\left[\, g^{(j)}(0) \int_{\bbR} f^{(j+1)} \sig u\log
\vert u \vert du \, + \, f^{(j)}(0) \int_{\bbR} g^{(j+1)}(u) \sig u\log
\vert u \vert du\,\right]
\ee
\[
\ +\ 8\pi\, i^j\, \gamma_j\, \varphi^{(j)}(0)\, f^{(j)}(0)\, g^{(j)}(0)\ .
\]
To write this in a more symmetric form, observe that since the Fourier
transform of $\rho^{-1}\log \vert \rho \vert$ is equal to
\be \label{3.16a}
\hat{(\rho ^{-1} \log \vert \rho \vert)} (x) =
\pi i\,(\log \vert x \vert + \gamma)\,\sig x
\ee
$\gamma$ being Euler's constant (cf. \cite{J}, theorem 48 on page 103),
%(cf. D.S.Jones, Generalized Functions,
%McGraw-Hill 1966], Thm. 48 on page 103.)
\[
2 \int \hphi(-\rho) \rho^j \log \vert \rho \vert d\rho \, = \, (-1)^j\, 2\,
\int \rho^{j+1}\,\hphi(\rho)\, \rho^{-1}\log \vert \rho \vert\, d\rho \, =
\]
\[
=\ 2\pi i^j \int \varphi^{(j+1)}(x)(\log \vert x \vert + \gamma)\sig
x\,dx
\]
\be\label{3.17}
=\ 2\pi i^j \int \varphi^{(j+1)}(x)\log \vert x \vert \, \sig x \, dx \, -
\, 4\pi i^j \gamma \varphi^{(j)}(0)\ .
\ee
To lighten the notation a little bit, let us introduce the tempered
distribution $\Gamma \in \calS^\prime (\bbR)$, defined by
\be\label{3.17a}
\Gamma (\psi) \, = \, \int_{\bbR} \psi^\prime (x)\log \vert x \vert
\, \sig x \, dx
\ee
We now combine (\ref{3.14})-(\ref{3.17}),
and finally replace the amplitude (\ref{3.4}) by a
general amplitude $a \in C^\infty_c(\bbR^3)$, using lemma \ref{III.0.5}.
The result is the following
asymptotic expansion: we
denote by $i_t,i_u,i_v$ the immersions $i_t(x) =
(x,0,0),\, i_u(x) = (0,x,0)$, etc.
\begin{theorem} \label{III.4}
Let
\be \label{3.18a}
I(\kappa)\ =\ \int_{\bbR^3}\; a(t,u,v)\,e^{i\kappa tuv} \, dt du dv, \ \ u \in
C^\infty_c(\bbR^3).
\ee
Then
\be\label{3.18}
I(\kappa) \sim \sum^\infty_{j=0}\; \sum_{l=0,1} \ C_{j,l}[a]\,
\kappa^{-1-j}\, (\log \kappa)^l\ ,
\ee
where
\be\label{3.19}
C_{j,1}[a]\ =\	4\pi\, (j!)^{-2}\,i^j\,(\partial_t \partial_u \partial_v)^j
a \, (0)
\ee
and
\be\label{3.20}
C_{j,0}[a] \,=\,
\ee
\[
-2\pi (j!)^{-2} i^j \left\{
\Gamma(i^\ast_t(\partial_t\partial_u\partial_v)^j a)
\, + \, \Gamma(i^\ast_u(\partial_t \partial_u \partial_v)^j a)
\, + \, \Gamma(i^\ast_v(\partial_t \partial_u \partial_v)^j a)
\right\} \,+\,
\]
\[
+\ 4\pi (j!)^{-2}i^j(\gamma + 2\gamma _j) (\partial_t \partial_u
\partial_v)^j a (0).
\]
\end{theorem}

\noindent
Here
\[
\Gamma(i^\ast_t(\partial_t \partial_u \partial_v)^j(a)) = \int_{\bbR}
(\partial^{j+1}_t \partial^j_u \partial^j_v a)\,(x,0,0)\,\log\vert x \vert\sig
x \,dx\ ,\mbox{etc.}
\]
Note the, of course necessary, symmetry of the formulas with respect to
permutations of  $u, v$ and $t$.

\medskip
In the Tauberian part of the proof of the modified Weyl law in the case
$\mbox{codim} \ \Theta = 2$ we will need the following observation:

\begin{corollary} \label{III.5}
Let
\[
I_E(\kappa) \, = \, \int a(t,u,v)\, e^{i \kappa t(uv-E)}\, du dt dv.
\]
Then
\be\label{3.21}
\kappa\, I_E(\kappa) \, = \, 4 \pi\log(\kappa - E)\, a(0) + O(1)\ , \ \kappa
\rightarrow \infty .
\ee
the  $O(1)$-error being uniform both in  $E \rightarrow 0$  and $\kappa
\rightarrow \infty$.
\end{corollary}
\begin{proof}
Specializing, as before, to amplitudes (\ref{3.4}), we have
\[
\kappa\, I_E(\kappa) \, = \, \int \hphi(-\rho) \int_{\{uv-E=\rho/ \kappa\}}
f(u)g(v) {du \over \vert u \vert}
\]
Proceeding as before, we have determine the asymptotic behavior for
$\rho \to 0$ of
\[
\int_{uv=\rho + E} f(u)g(v) {du \over u} \, = \, (f \ast g)(\rho + E) \, =
\, (2 \pi i)^{-1} \int^{\varepsilon + i \infty}_{\varepsilon - i \infty} F(s)
G(s) (\rho + E)^{-s}ds\; ,
\]
where $0 < \varepsilon < 1$.
Shift the path of integration to $\varepsilon - 1$ (assuming $\varepsilon <1$)
and note that the residue in  $0$  is
\[
\log(\rho + E)^{-1} f(0)g(0) \, + \, [\Lambda_0 (f)g(0) + f(0) \Lambda_0(g)]
\, = \,\log(\rho + E)^{-1} f(0)g(0) \, + \, O(1).
\]
The line integral over	$Re\, s = \varepsilon - 1$ is bounded by
\[
(\rho + E)^{1 - \varepsilon} \,\Vert F(\varepsilon - 1 + i \cdot) \Vert_2\,
\Vert G(\varepsilon - 1 + i \cdot) \Vert_2 \ ,
\]
which is $O(1)$ as $(\rho, E) \rightarrow 0$. This proves the corollary.
\end{proof}

\subsection{Case 1: Odd indices of inertia}

We next turn to $I_{a, Q}$ for general symmetric $Q$.  We will first
concentrate, in the following two sections, on the case of indefinite
$Q$'s, and postpone the definite case to section 3.4.  Let $\vert Q
\vert$ be the absolute value of $Q$ (defined by functional calculus
of symmetric matrices).
Then for some $\nu \in \bbN, \ \nu \geq 1$, after perhaps a permutation
of the coordinates,
\be\label{3.22}
\langle Q(\vert Q \vert ^{-1/2} z), \vert Q \vert^{-1/2}z \rangle =
\vert z^\prime \vert^2 - \vert z^{\prime \prime}\vert^2
\ ,\ z=(z^\prime, z^{\prime\prime})\in \bbR^{\nu } \times \bbR^{N-\nu }\ .
\ee
If both indices of inertia
$\nu $ and $N - \nu $ of $Q$ are $>1$, we can introduce polar coordinates
\[
z^\prime = r \zeta, \  z^{\prime \prime} = s \eta\ , \zeta \in S_{\nu -1}\,,\;
\eta \in S_{N-\nu -1}\,,\;  r,s > 0 \ ,
\]
and $I = I_{a, Q}$ can then be written as
\be\label{3.23}
I(\kappa) = \int_{\bbR} \int^{\infty}_0 \int^\infty_0
\alpha(t,r,s)e^{i \kappa t(r^2-s^2)}dr ds dt
\ee
with
\begin{eqnarray}
\nonumber
\lefteqn{\alpha(t,r,s) = \vert\det Q \vert^{-1/2}r^{\nu-1}\, s^{N-\nu -1}\,}
\cdot \\
 & & \int_{S_{\nu -1}}
\int_{S_{N-\nu -1}} a(t, \vert Q \vert^{-1/2} (r \zeta , s \eta)) \,
d\sigma_{\nu -1}(\zeta) d\sigma_{N-\nu -1}(\eta) \; .
\label{3.24}
\end{eqnarray}

Here the $\sigma's$ denote the rotation invariant measures on the unit spheres
$S_{\nu -1}$, $S_{N-\nu -1}$, with the usual normalization.

If $\nu = 1$ or $N - \nu = 1$, we have to slightly adapt the formula
for $\alpha $. If, for example, $N - \nu = 1$ but $\nu > 1$
then we have to replace (\ref{3.24}) by
\be \label{3.24'}
\alpha(t,r,s) = \sum_{\pm } \, \vert\det Q \vert^{-1/2} r^{\nu-1}\,
\int_{S_{\nu -1}}  \, a(t, \vert Q \vert^{-1/2} (r \zeta , \pm s))
d\sigma_{\nu -1}(\zeta),
\ee
while in the important special case $\nu = N - \nu = 1$,
\be \label{3.24''}
\alpha(t,r,s) = \sum _{\pm } \, \sum _{\pm } \, \vert\det Q \vert^{-1/2}
a(t, \vert Q \vert^{-1/2} (\pm r, \pm s)) .
\ee
(In this last case we of course just have an integral like (\ref{3.23})
but extended over $\bbR^3$, to which the results of the previous section
immediately apply).

\medskip \noindent
Now suppose that $\alpha(t,r,s)$ in (\ref{3.23}), extends to
an even $C^\infty$-function of both r and of s on all of $\bbR^2$.
Then the integral (\ref{3.23}) equals $1 / 4$
times the same integral extended over $\bbR^3$ and the change of
variables $u = r+s, v=r-s$, reduces this last integral to
(\ref{3.18a}). We now can apply theorem \ref{III.4}, which leads, after
some computations, to the following result:
\begin{theorem} \label{III.6}
Suppose that $\alpha (t, r, s) \in C^{\infty }_c(\bbR \times [0, \infty)^2)$
extends to an even $C^{\infty }$-function in both variables $r$ and $s$
separately.  Then $I(\kappa )$
defined by (\ref{3.23}) has an asymptotic expansion
\be \label{3.25}
I_{a,Q}(\kappa) \sim \sum^\infty_{j=0} \, \sum_{l=0,1}	c_{j,l}\,
\kappa^{-1-j}\,(\log \kappa)^l\,, \; \kappa \rightarrow \infty\ ,
\ee
with
\be\label{3.25a}
2^{2j}\, i^{-j}\, (j!)^2\,  c_{j,1} \  = \ {\pi \over 2}\,
(\partial_t( \partial^2_r - \partial^2_s))^j  \alpha (0, 0, 0)
\ee
and
\be\label{3.25b}
2^{2j}\, i^{-j}\, (j!)^2\, c_{j,0} \ = \
- {\pi \over 4} \int_{\bbR}
(\partial_t( \partial^2_r - \partial^2_s))^j
\partial _t \alpha (t,0,0)
\sig  t\log \vert t \vert \, dt \,
\ee
\[
- {\pi \over 2} \int^\infty_0
(\partial_t( \partial^2_r - \partial^2_s))^j
(\partial_r\alpha  +  \partial_s \alpha ) (0, {r \over 2}, {r \over 2})
\log r\, dr
\]
\[
+ {\pi \over 2}\,(\gamma + 2\gamma _j)\,
(\partial_t( \partial^2_r - \partial^2_s))^j
\alpha (0, 0, 0)
\]
where, as before,
$\gamma $ is Euler's constant and $\gamma _j$ is given by (\ref{3.10}).
\end{theorem}

\medskip
This theorem applies in particular to $I_{a, Q}$
when $Q$ has both indices of inertia odd, with $\alpha $ given by
(\ref{3.24}) - (\ref{3.24''}).

\subsection{Case 2: At least one even index of inertia.}

If $\nu $ or $N - \nu $ is even, or if both are, the nature of the
asymptotic expansion of $I_{a, Q}$ changes: it will no longer contain
log-terms. Instead, half-integer powers of $\kappa $ will appear. We
will derive this from a more general result, namely an asymptotic
expansion of (\ref{3.23}) for arbitrary $\alpha \in C^{\infty }_c (\bbR
\times [0, \infty )^2)$: {\em this} expansion will in general contain
both logarithms and half-integer powers.  However, if $\alpha $ is odd
in one of the two variables the coefficients of the log terms will
disappear.  Similarly, if $\alpha $ is even in both $r$ and in $s$ the
coefficients of all half-integer powers will vanish. Note that the
$\alpha $'s defined by (\ref{3.24}) - (\ref{3.24''})
will have some definite parity both in
$r$ and in $s$.

\medskip

The difference between these cases can heuristically be understood as follows:
suppose for example that $\alpha $ is odd in $r$. Then we can write
$\alpha = r \alpha '(r, s, t)$ with $\alpha '$ even in $r$.
If we now make the change of variables $\rho = r^2$ in (\ref{3.24}) and
next introduce a new coordinate $w = \rho - s^2$, the phase becomes just
$t \cdot w$, which is a non-degenerate phase function which can
in principle be handled by
classical stationary phase. The set of critical points now will just be
$\{t=0, \rho = s^2 \} = \{t = 0, r = \pm s \}$. Contrast this with the
critical set of $t(r^2 - s^2)$, which additionally includes
$\bbR \times \{ r = s = 0 \}$. However, if one tries to actually carry out
this approach, one finds that the new amplitude will contain singularities
at $w = 0$, coming from the fact that our domain of integration contains
a cusp at $0$. Rather than dealing directly with this complication we
have preferred to follow the slightly more roundabout route of
analyzing (\ref{3.23}) for arbitrary $\alpha $ without parity restrictions.

\medskip

Let therefore $\alpha $ in $C^{\infty }_c (\bbR \times [0, \infty )^2)$ be
arbitrary. If as before we make the change of variables
$u = r+s, v = r-s$, (\ref{3.23}) becomes
\be\label{3.27}
I(\kappa) \, = {1 \over 2} \int_{u \geq \vert v \vert } \int_{\bbR} \alpha
\left(t, {u+v \over 2}, {u-v \over 2}\right)\,e^{i\kappa tuv}\, dt du dv\ .
\ee
Specializing again to amplitudes of the form
$\alpha \left(t, {u+v \over 2}, {u-v \over 2}\right) =
\varphi (t) f(u) g(v)$ we can write this as
\be\label{3.28}
{1 \over 2 \kappa }
\int_0^\infty d\rho \ \hphi(-\rho) \int^\infty_{(\rho / \kappa)^{1 \over
2}} {du \over u} \ f(u)g({\rho \over\kappa u})	\ +
\ee
\[
{1 \over 2\kappa } \int^\infty_0  d\rho \
\hphi(\rho) \int^\infty_{(\rho / \kappa)^{1 \over 2}} {du \over u} \
f(u)g_- ({\rho \over\kappa u}),
\]
where $g_-(v) := g(-v)$. We therefore need a half-sided version of Proposition
\ref{III.3}:

\begin{proposition} \label{III.7}
There exist certain (computable) universal constants $A_{jk}$ such that if
we let
\be \label{3.28a}
q_j(f,g) \, = \,
\sum _{k=0}^j A_{j,k} \left( f^{(k)}(0) g^{(j-k)}(0) \, - \,
g^{(k)}(0) f^{(j-k)}(0) \right)
\ee
then, for $\rho \rightarrow 0+ $,
\be\label{3.29}
\int^{\infty}_{\sqrt{\rho}} f(u)g({\rho \over u}) {{du} \over u} \sim \
\ee
\[
\sum^\infty_{j=0} \, {1 \over 2} (j!)^{-2} f^{(j)}(0)g^{(j)}(0)
\,\rho^j\,\log\rho^{-1} \, +
\, (j!)^{-2} g^{(j)}(0) \Lambda_j(f) \, \rho^j \, + \,
q_j (f, g)) \, \rho^{j/2}
\]
\end{proposition}

\medskip
\noindent
{\bf Remark.} The precise value of the constants $A_{jk}$ won't be important
for the sequel.
Note that $q_j(f, g)$ is anti-symmetric in $f$ and $g$, and that
\be\label{3.30}
\int^\infty_{\sqrt{\rho}} f(u) g({\rho \over u}) {du \over u} \, +
\int^\infty_{\sqrt{\rho}} g(u) f({\rho \over u}) {du \over u} \, =
\int^\infty_0  f(u) g({\rho \over u}) {du \over u}
\ee
Hence, in view of the anti-symmetry of the $q_j$, Proposition \ref{III.7}
implies Proposition \ref{III.3}.
We can in fact use \ref{III.3} to short-circuit the
proof of \ref{III.7} (which is completely different) at a certain point.

\medskip
\noindent
\begin{proof}
First expand $g(\rho/ u)$ in the left hand side of
(\ref{3.29}) in a power series in $\rho/ u$. Then
\be\label{3.31}
\int^\infty_{\sqrt{\rho}} f(u)\, g({\rho \over u}) {du \over u} \, \sim \,
\sum^\infty_{j=0} \,
{g^{(j)}(0) \over j!} \, \rho^j \int^\infty_{\sqrt{\rho}} f(u)u^{-(j+1)}du\ .
\ee
The error-term after breaking off the series at $j = K$ can be estimated
by
\[
C \, \Vert g^{(K+1)}\Vert_\infty\, \Vert f \Vert_1\, \rho^{{K+1} \over 2}
\]
so this will indeed lead to an asymptotic expansion.
Next, integrating by parts,
\be\label{3.32}
\int^\infty_{\sqrt{\rho}} f(u)\, u^{-(j+1)}\, {{du} \over u} =
\sum^{j-1}_{k=0} {(j-k-1)! \over {j!}} f^{(k)} ({\sqrt{\rho}}) \rho^{{k-j}
\over 2} -
\ee
\[
- {f^{(j)}(\sqrt{\rho}) \over {j!}} \log \sqrt{\rho} - {1 \over {j!}}
\int^\infty_{\sqrt{\rho}} f^{(j+1)}(u)\log u \, du.
\]
Expand each of the $f^{(k)}({\sqrt{\rho}})$ in a power series in
${\sqrt{\rho}}$, and replace the integral on the right hand side of (\ref{3.32})
by
\be\label{3.33}
- \, \int^\infty_0 f^{(j+1)}(u)\log u \, du \, + \int^{\sqrt{\rho}}_0 f^{(j+1)}
(u)\log u \, du \; \sim \,
\ee
\[
- \, \int^\infty_0 f^{(j+1)}(u)\log u \, du \,
+ \, \sum^\infty_{l = 0} {f^{(j+1+l)}(0) \over {l !} }
\left({{\rho^{(l+1)/2}} \over {l +1}}\log {\sqrt{\rho}} -{\rho^{(l+1)/2}
\over (l + 1)^2} \right)\ .
\]
Now substitute (\ref{3.32}) and (\ref{3.33})
into (\ref{3.31}) and rearrange terms.	The
coefficient of $\rho^j\log {\sqrt{\rho}}$ turns out to be a telescoping series,
and the end result is an asymptotic expansion
\be\label{3.34}
\sum_{j \geq 0} -(j!)^{-2} g^{(j)}(0)f^{(j)}(0) \rho^j\log {\sqrt{\rho}} \ -
\ee
\[
\sum_{j \geq 0} (j!)^{-2}g^{(j)}(0)
(\int _0^{\infty } f^{(j+1)}(u)\log u \, du) \, \rho^j
\ +
\]
\[
\sum_ {n \geq 0} \, \left( \sum_{k+l+j=n, \, k \leq j-1} \,
{(j-k-1)! \over (j!)^2 l !}\,
g^{(j)}(0) f^{(k+l)}(0)\, - \,
\sum_{2j+l+1=n} \,
{f^{(j + l + 1)}(0)g^{(j)}(0) \over (j!)^2(l + 1)(l + 1)!}
\right)\, \rho^{n \over 2}
\]
To finish the proof, we look at the coefficient of $\rho ^{n/2}$
in (\ref{3.34}).
A term	$f^{(p)}(0)g^{(p)}(0)$ will occur
if either $j = k+l = p$, which
implies that $n = j+k+l = 2p$  even, or if $j + l + 1 = j = p$,  which is
not possible since $l \geq 0$.	Hence the sum $\sum_{n \geq 0}$
in (\ref{3.34}) can be
rewritten as (changing the summation index $n$ into $j$)
\be\label{3.35}
\sum_j \,  (\sum_{k \leq j-1}{(j-k-1)! \over (j!)^2 (j-k)!})
(g^{(j)}(0)f^{(j)}(0)) \cdot \rho ^j \ +
\sum _{j \geq 0} q_j(f, g) \rho ^{j/2}
\ee
for suitable $q_j$.
Since the coefficient of $g^{(j)}(0) f^{(j)}(0)$ in (\ref{3.35}) is exactly
$(j!)^{-2} \gamma _j$, the antisymmetry of the $q_j$
follows from (\ref{3.30}) and
Proposition 3.4. Finally, since $q_j$ depends on products of derivatives
$f^{(a)}(0) g^{(b)}(0)$ with $a + b = j$, it has to have the form
(\ref{3.28a}).	The precise values of the $A_{j, k}$
follow from a computation which we omit.
\end{proof}

\medskip
\noindent
%{\bf Remark.} The $Q_j$ and $q-j$ can clearly be computed from (\ref{3.34}),
%(\ref{3.35}),
%but we did not attempt to find a general formula.  The first ones are
%\[
%q_0(f, g) = 0
%\]
%\[
%q_1(f; g)\ =\ g'(0)f(0) - g(0)f'(0)
%\]
%\[
%q_2(f, g) = {1 \over 4} (g^{(2)}(0)f(0) - f^{(2)}(0)g(0)
%\]
%\[
%q_3(f, g) =
%{1 \over 3\cdot 3!} (g^{(3)}(0)f(0) - f^{(3)}(0)g(0))\, +
%\,{1 \over 2} (g^{(2)}(0)f'(0) - f^{(2)}(0)g'(0))\ .
%\]
%
Note, for later use, that $q_0 = 0$ and that
\be \label{3.35a}
q_1(f; g)\ =\ g'(0)f(0) - g(0)f'(0)
\ee

\medskip \noindent
We can now substitute (\ref{3.29}) in (\ref{3.28}) to obtain the asymptotic
expansion of (\ref{3.27}) and hence of (\ref{3.23}).
We will leave the details to the reader, and will just note the following points.
First, at a certain stage of the computation one again has to use
(\ref{3.16a}). Next, in order to have a compact expression for the coefficients
of the half-integer powers in the asymptotic expansion below, let us introduce
the distributions:
\be
q_j \, = \, (-1)^j \, \sum _{k=1}^j \,
A_{j,k} \left( \partial _u^k \partial _v^{j-k} -
\partial _v^k \partial _u^{j-k} \right) \delta (u, v),
\ee
supported at the origin, and also the distributions
\be \label{3.35b}
F_j^{\pm } \ = \ \Gamma ({j \over 2} + 1) \
e^{\mp i \pi (j + 2) / 4} \ (t \mp i0) ^{-j/2 - 1};
\ee
$F^{\pm }_j$ is the Fourier transform of $\rho _{\pm }^{j/2}$, where
$\rho _{\pm } := \mbox{max} \, (\pm \rho , 0)$, cf. \cite{Ho}.
With these notations we have the following result:
\begin{theorem} \label{III.9} Let
\[
I(\kappa ) \ = \ \int_{} \,
\alpha(t, r, s) e^{i \kappa t (r^2 - s^2)} dt dr ds,
\]
$\alpha $ in $C^{\infty }_c (\bbR \times [0, \infty )^2)$. Then
\be\label{3.38}
I(\kappa) \sim \left(
\begin{array}{c}
\mbox{asymptotic expansion (\ref{3.25})}\\
\mbox{of Theorem \ref{III.6}}
\end{array}
\right)
\, + \, \sum_{j \geq 1} b_j \kappa^{-1-j/2}\,
\ee
with
\be \label{3.38a}
b_j = \langle F_j^{-}(t) \otimes q_j(u, v) + F_j^{+}(t) \otimes
q_j(u, -v), \ \alpha(t, {{u+v} \over 2}, {{u-v} \over 2}) \rangle
\ee
\end{theorem}

Since any odd function differentiated an even number of times and
evaluated at $0$ gives $0$, we immediately obtain the following corollary:

\begin{corollary} \label{III.9a}
Suppose $Q$ has at least one even index of inertia.
Then $I = I_{a, Q}$ has an asymptotic expansion
\be \label{3.38b}
I(\kappa ) \ \sim \ \sum _{j = 0}^{\infty } \ c_j \kappa ^{-1-j/2}
\ee
in integer and half-integer powers of $\kappa $, with coefficients which can
be read off from (\ref{3.24}) - (\ref{3.24''}),
(\ref{3.25b}) and (\ref{3.38a}).
\end{corollary}

\medskip \noindent
{\bf Remark.} From (\ref{3.35a}), (\ref{3.35b}) and (\ref{3.38}) we read off
that for example the coefficient of $\kappa ^{-3/2}$ is:
\[
-\langle F_1^- (t), \partial _s \alpha (t, 0, 0) \rangle
-\langle F_1^+ (t), \partial _r \alpha (t, 0, 0) \rangle,
\]
so half-integer powers will in general always occur, unless of course
$\alpha $ is even in $r$ and $s$. In the latter case all coefficients $b_j$
in (\ref{3.38}) will vanish.


\medskip
\noindent
{\bf Remark.}
If $N > 2$, the
first non-zero term in the asymptotic expansions (\ref{3.25}), (\ref{3.38})
of $I_{a, Q}$ is:
\be \label{3.38'}
-{\pi \over \kappa} \int^\infty_0 \partial_r [\alpha(0, {r
\over 2}, {r \over 2})]\log r dr = {\pi \over \kappa}
\int^\infty_0 \alpha(0,r,r) {dr \over r},
\ee
the integration by parts being allowed since $\alpha(0) = 0$.	We claim that
this is precisely the coefficient in the Weyl law:
\be \label{3.38''}
{2\pi \over \kappa} \int_{\{ Q=0 \}} a(0,z)\, dL_Q(z),
\ee
where $dL_Q$ is the
Liouville measure on $\{ Q=0 \}$, which is a well-defined locally finite
measure if $N > 2$.
One obtains (\ref{3.38''}) formally
by introducing the local coordinate $q = Q(z)$ in
(\ref{3.1}) (strictly speaking not allowed at $z=0$)
and applying the classical
stationary phase formula to the resulting 2-dimensional $(t,q)$ integral.
The equality of (\ref{3.38'}) and (\ref{3.38''}) follows from an easy
computation: if we
assume wlog that $Q(z) = \vert z' \vert^2 - \vert z'' \vert^2
= r^2 - s^2$, then
\[
dL_Q = \pm {1\over 2} {r^ks^{N-k-1} d\sigma_{k-1} \wedge
d\sigma_{N-k-1}\wedge ds - r^{k-1}s^{N-k}d\sigma_{k-1}\wedge
d\sigma_{N-k-1} \wedge dr \over (r^2-s^2)},
\]
(with
$
d\sigma_{k-1} = d\sigma_{k-1}(\zeta ), \ d\sigma_{N-k-1} =
d\sigma_{N-k-1}(\eta )
$), since
$dL_Q \wedge dQ $ equals the volume form on $\bbR^N$. Hence
\[
\int_{\{ Q=0 \}} a(0,z) \lambda_Q(z) = {1 \over 2} \int_{\{ r=s \}}
{r\alpha(0,r,s)ds - s\alpha(0,r,s)\over (r^2-s^2)}dr =
{1 \over 2} \int^\infty_0 \alpha(0,r,r) {dr \over r}.
\]


\subsection{Case 3: $Q$ definite.}

Finally we consider the easier case of definite $Q$.  Assume for
example that $Q$ is negative definite.	Replacing $a(z, t)$ by
$|\det Q|^{-1/2} a( |Q|^{-1/2}z, t)$ we may assume that
$\langle Qz, z \rangle = -\vert z \vert ^2$.
If we specialize again to
amplitudes of the form $\varphi(t) a(z)$, (\ref{3.1}) equals
\be\label{3.40}
I(\kappa ) =
{1 \over \kappa^{N/2}}
\int^\infty_0 \hphi(r^2)\, \alpha({r \over \kappa^{1/2}})
\,r^{N-1} dr \,
\ee
where
\be\label{3.41}
\alpha(r) = \int_{S_{N-1}}a(r \zeta)\, d\sigma_{N-1}(\zeta)\ .
\ee
if $N > 1$, and
\[
\alpha (r) = a(r) + a(-r)
\]
if $N = 1$.
Now
expand $\alpha $ in a power series at $r = 0$ and substitute this series
in (\ref{3.40}). In this way we obtain an asymptotic expansion
\[
I(\kappa ) \sim
\sum_{j \geq 0} \, c_j \, \kappa ^{-(N + j)/2}
\]
with distribution coefficients $c_j$ given by
\[
c_j = c_j[a \varphi ] = {{\alpha ^{(j)}(0)} \over j!}
\int _0^{\infty } r^{j + N -1} \hphi(r^2) \ dr.
\]
\[
= {1 \over 2} {{\alpha ^{(j)}(0)} \over j!} \
\langle \rho _{+}^{ {N + j\over 2} -1}, \hphi (\rho ) \rangle
\]
If we finally use (\ref{3.35b}) again we arrive at the following result:

\begin{theorem} \label{III.10}
Suppose that $Q$ is negative definite. Then
\be
\label{3.42}
I_{a, Q}(\kappa ) \sim
\sum_{j \geq 0} \, c_j \, \kappa ^{-(N + j)/2} ,
\ee
with
\be
\label{3.43}
c_j =
{1 \over 2} {{\Gamma ((j + N)/2)} \over {\Gamma (j+1)}}
e^{- i \pi (j+N)/4} \
\langle
(t - i0)^{-(j+N)/2}, \partial _r ^j \alpha (t, 0) \rangle;
\ee
and $\alpha (t, \rho ) = \vert \det\ Q \vert ^{-1/2}
\int _{S_{N-1}} a(t, \vert Q \vert ^{-1/2} (r \zeta)) \ d\sigma _{N-1}$
(if $N > 1$).
\end{theorem}

In particular, if $N = 2$, the expansion starts with a term of order
$1 / \kappa $, with coefficient
\be
\label{3.44}
{\pi \over 2} \alpha (0, 0) \ -
\ {i \over 2} \int _{\bbR} { {\alpha (t, 0)} \over t} dt ,
\ee
by the well-known identity
$(t - i0)^{-1} = \mbox{PV}(1/t) + i \pi \delta (t)$.


\subsection{Proof of Theorem 1.1}

We will now finish the proof of Theorem 1.1, which is almost immediate now.
Part B and the first part of A follow from Proposition \ref{II.summary} and
the stationary phase results above.  To compute $j_0$, we have to apply
theorem \ref{III.6} with an $\alpha $ given by (\ref{3.24}) - (\ref{3.24''}),
where we have to take $a $ equal to the amplitude $\check {\varphi } \beta $ of
proposition \ref{II.summary} and where $Q$ is also as in \ref{II.summary}.
Note that $|\mbox{det}\, Q| = 2^{-N}$ and that $\nu $ will now be the number of
{\em negative } eigenvalues of $Q$, because of the minus sign in the phase
of (\ref{II.11}).
Thus $\alpha $ is of the form
\be\label{proof.1}
\alpha(t, r, s)\ =\ r^p\, s^q\, F(t, r, s)\
\ee
with $p = \nu -1$ and $q = N - \nu -1$, both even by assumption,
and we are interested in the smallest
value of $j$, $j_0$, for which
\be \label{proof.1a}
(\partial_r^2 - \partial_s^2)^j (r^p s^q F) \neq 0 \ \mbox{at} \ r = s = 0.
\ee
By the binomial theorem, (\ref{proof.1a}) equals
a sum over $l$, from $l=0$ to $l=j$ of
\be\label{proof.2}
{{j!} \over {l! (j-l)!}} \
\partial_s^{2(j-l)}\,[s^q\,\partial_r^{2l}\,(r^p\,F)]\ .
\ee
We are interested in the $l$'s for which this is not zero
at $(r,s)=(0,0)$.  By Leibnitz's rule, (\ref{proof.2}) will be zero
at $(0, 0)$ if $p>2l$ or if $q>2(j-l)$.
Hence (\ref{proof.2}) will be non-zero at the origin only if
\be\label{proof.3}
p\leq 2l \leq 2j-q\ .
\ee
It is clear that the smallest value $j_0$ of $j$ for which there exists
an $l \leq j$ such that (\ref{proof.3}) holds is $j_0 = (p + q)/2 = N/2 - 1$,
for which $l = p/2 = (\nu -1)/2$. This completes the proof of Theorem 1.1.

\medskip \noindent
Finally we compute the coefficient $c_{j_0,1}$, as preparation for the proof
of theorem \ref{TWO} in the next section. By (\ref{3.25a}) and (\ref{proof.2}),
\be \label{proof.3a}
c_{j_0, 1} = {\pi \over 2} 2^{-(p+q)} i^{{{p+q} \over 2}}
{ {p! \, q!} \over {\left( {{p+q} \over 2} \right) !
\left( {p \over 2} \right) !  \left( {q \over 2} \right) !}} \,
\partial _t^{(p+q)/2} \alpha (t, 0, 0).
\ee
Now recall that $\Sigma _{k-1} =$ (surface measure of the unit-sphere in
$\bbR^k$) if $k \geq 2$ and $\Sigma _0 = 2$ if {k = 1}. Hence
$\alpha (t, 0, 0) = 2^{N/2} \, \Sigma _{\nu -1} \Sigma _{N - \nu -1} \,
\check{\varphi }(t) \beta (t, 0)$. If we next use the duplication formula
for the $\Gamma $-function:
\[
\Gamma (2z) = \pi ^{-1/2} 2^{2z-1} \Gamma (z) \Gamma (z + {1 \over 2}),
\]
together with $\Gamma (x+1) = x!$, and remember that $p = \nu -1$ and
$q = N - \nu -1$, then (\ref{proof.3a}) can be written as


\begin{eqnarray}
\nonumber
\lefteqn{c_{N/2 - 1, 1} = (2i)^{N/2 - 1} \cdot
{{\Gamma \left( {{\nu } \over 2} \right) \,
\Gamma \left( {{N - \nu } \over 2} \right)} \over
{\Gamma \left( {N \over 2} \right)}} \cdot } \cdot \\
 & & \Sigma _{\nu -1} \cdot
\Sigma _{N - \nu - 1} \cdot
\partial _t^{N/2 - 1} \left( \check{\varphi } \beta \right) (0, 0).
\label{proof.3b}
\end{eqnarray}



\bigskip
To finish this section we look at the main term in the
expansion of $\Gamma _1$ (cf. (\ref{II.6.3})) when
$N = \mbox{codim}\, \Theta = 2$ and the normal Hessian of $H$ positive definite
on $\Theta $, also for the proof of theorem \ref{TWO}. In this case
$\Theta $ is in itself already a component of $\Sigma _{E_c}$ and we choose
the cut-off $\rho $ in (\ref{II.6.3}) such that
$\mbox{supp}\, \rho \cap \left( \Sigma _{E_c} \setminus \Theta \right) =
\emptyset $. Then by theorem \ref{III.10} (and in particular formula
(\ref{3.44}),
$\Upsilon _{1, \hbar } = c_0 \hbar ^{n-1} + \calO (\hbar ^{n - 1/2})$
with
\be \label{proof.5}
c_0 = \left( \pi (\check{\varphi } \beta )(0, 0) \, - \,
i \int _{\bbR } \, {{\check{\varphi }(t) \beta (t, 0)} \over t} \, dt
\right).
\ee

\input{critical2.tex}
ENDBODY

\newcommand{\ve}{\lambda}
\newcommand{\supp}{\mbox{supp}}
\renewcommand{\hphi}{\check{\varphi}}

\section{Near a Critical Energy Level}

Having established the nature of the asymptotic expansion of the sums
(\ref{I.3}), we now turn to the problem of computing some of the
coefficients appearing in it.  This will be done using ideas from the
symbolic calculus of singular Lagrangian distributions, in the sense of
Guillemin and Uhlmann, \cite{GUh}, which will play a basic r\^ole
in this chapter. We are greatly indebted to Gunther Uhlmann for
explaining aspects of it to us.


\subsection{The Action-Energy Distribution $\Upsilon (s,E)$.}

The approach in this section is based on a careful study of the
singularities of the following distribution:
\be\label{4.2}
\Upsilon (s, E)\ =\ \sum_{k=1}^\infty\ \left(\;
\sum_j \vp\left( k[E_j(1/k)-E]\right)\;\right)\; e^{iks}\ .
\ee
In \cite{PU2}, we proved:

\begin{lemma}\label{44.11}
The series (\ref{4.2}) 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{4.3}
\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{lemma}

In the present setting, the set defined by
conditions (\ref{4.3}) is the union $\Lambda_0\cup\Lambda_1$ of
two Lagrangian submanifolds of $T^*(S^1\times\bbR)\setminus 0$: let
\be\label{4.4}
\begin{array}{ccl}
\Lambda_0 & = &  \{\ (s,E ;\kappa,\ve)\ ;\ (s,E)\;=\;(0,E_c)\; ,\;
|\ve/\kappa|<K\ \} \\
\Lambda_1 & = & N^*\{\ s=0\;\}

\end{array}
\ee
(we identify as usual $S^1$ with $\bbR / 2\pi \bbZ$).
Then the precise statement is:

\begin{lemma}\label{44.22}
There exist $K>0$ and $\delta>0$ such that, if $\supp(\hphi)\subset
[-K,K]$ then
\[
\wf (\Upsilon)\cap \{\; |E-E_c|<\delta\;\}\subset
\Lambda_0\cup\Lambda_1\ .
\]
\end{lemma}

The proof is a straightforward analysis of the conditions (\ref{4.3}):
$\Lambda_1$ arises from every point in $H^{-1}(E_c - \delta , E_c + \delta )$
and the period $t=0$, while $\Lambda_0$ arises from the critical
points, which are of course periodic of all periods and
for which the associated action
$s$ is zero.  Notice that $\Lambda_0$ is a conic subset of the cotangent
space to the plane at $(s=0, E=E_c)$, and that the slope of a ray in
$\Lambda_0$ has an interpretation of a time, $t$, as period of
every point in $\Theta$.

\medskip
The analysis of \S 2 yields the following oscillatory
integral representation for $\Upsilon$:

\begin{lemma}\label{44.33}
Let $\supp(\hphi)$ be small enough, and let $\chi \in C^{\infty }(\bbR)$ be a
cut-off function such that $\mbox{supp} \, \chi \subset (0, \infty )$,
$\chi (\kappa ) \equiv 1$ if $\kappa > 1/2$. Then,
modulo a distribution conormal to $\{s=0\}$,
\be\label{4.5}
\Upsilon (s,E)\ =\ \int\ e^{i\kappa\,[t(E-H_Q(w)) + s]}\,
\kappa^n\, \hphi (t)\, \chi (\kappa ) \beta (t,w;\kappa)\, dw
\,dt\,d\kappa\ ,
\ee
where $H_Q(w) = E_c + {1\over 2}\langle Qw,w\rangle$ and $\beta $ is a classical
symbol of order $0$ with fiber variable $\kappa \in \bbR $.
\end{lemma}
Next, we will show that this expression implies that $\Upsilon$ is a singular
Lagrangian distribution in the sense of Guillemin, Melrose and Uhlmann,
more precisely, in the class
$I^{-1, (1-N)/2}(\bbR^2 ;\Lambda_0, \Lambda_1)$ of \cite{GUh}.


\subsection{$\Upsilon$ as a singular Lagrangian distribution.}

We will slightly change notation in this sub-section, in order to
facilitate comparison with the formulas of \cite{GUh}, where the letter
$s$ is used in a different meaning. We will therefore write $x$ instead of $s$ and
$y$ instead of $E - E_c$. The dual variables to $(x, y)$ will be denoted by
$(\kappa , \lambda )$, as before.
\begin{proposition}\label{44aa11.1}
Let $a(t,w;\kappa),\; (t,w;\kappa)\in\bbR\times\bbR^N\times\bbR $,
be a classical symbol of order $m$, compactly supported
in the space variables $(t,w)$.  Let $Q$ be a non-singular
$N\times N$ symmetric matrix.  Then the oscillatory integral
\[
\Upsilon(x,y)\ =\ \int\; e^{i\kappa[\,t(y-\langle Qw,w\rangle/2 ) +x]} \;
a(t,w;\kappa) dt\,dw\,d\kappa
\]
is a singular Lagrangian distribution in the space
$I^{m-1,(1-N)/2}(\bbR^2 ;\Lambda_0, \Lambda_1)$ of \cite{GUh}, where
\[
\Lambda_0\ =\ N^*\{x=0\,,\,y=0\}\ \quad\mbox{and}\quad
\Lambda_1\ =\ N^*\{x=0\}\ .
\]
\end{proposition}

It will follow that, microlocally on $\Lambda_j\setminus
\Lambda_0\cap\Lambda_1$, $\Upsilon$ is a Lagrangian distribution whose
symbol becomes singular as one approaches the intersection.  We will
describe its symbol in corollary \ref{4a.VII} below.

We break the proof of the proposition in a couple of lemmas.

\begin{lemma}\label{44aa11.2}
Let $a(t,w;\kappa)$ be a symbol of order $m$, compactly supported
in the space variables.  Assume further that it vanishes if $|\kappa|\leq 1$.
Define
\[
b(w; \lambda,\kappa)\ :=\ a(\lambda/\kappa\,,\,w\,;\,\kappa)\ .
\]
Then $b$ is a symbol of order $m$, with fiber variables
$\zeta = (\lambda, \kappa)$. Furthermore, $b$ is a classical
symbol if $a$ is.
\end{lemma}
\begin{proof}
First notice that, since the support of $a$ is in the set
$\{|t|\leq C\,,\, |\kappa|\geq 1\}$ for some $C$, $b$ is not only well-defined
but it is supported in a conic set where
\be\label{4a.1.1bis}
|\kappa| \leq \vert (\lambda,\kappa )\vert \ \leq \sqrt{1+C^2}\,|\kappa|\
\ee
Hence large $\vert (\lambda,\kappa )\vert $ estimates are equivalent to large
$|\kappa|$ estimates.  The asymptotics below refer to this regime.
It is now straightforward to show that $b \in S^m$, using the relations
\[
\partial _{\lambda } b = {1 \over {\kappa }} \partial _t a, \ \ \
\partial _{\kappa } b = \partial _{\kappa } a \, - \,
{{\lambda } \over {\kappa ^2}} \partial _t a\; .
\]
Finally, if $a$ is a classical symbol with asymptotic expansion
\be\label{4a.1.1}
a(t,w;\kappa)\ \sim\ \sum_{j=0}^\infty\; a_j(t,w)\, \kappa^{m-j},
\ \ \ \kappa \to \infty ,
\ee
valid in the sense
of classical symbols,
then
\be\label{4a.1.2}
b(w; \lambda,\kappa) \sim\ \sum_{j=0}^\infty\; a_j(\lambda/\kappa,w)\,
\kappa^{m-j}\ , \ \ \ \kappa \to \infty
\ee
(and similarly for $\kappa \to -\infty $),
as it is easily checked that this asymptotic series can be differentiated
term by term.
\end{proof}

Notice that the leading order term in the classical expansion for $b$
on $\{ \kappa > 0 \}$ is
\be\label{4a.1.3}
b_0(w; \lambda,\kappa)\  =\  a_o(\lambda/\kappa,w)\, \kappa^m\ ,
\ee
which is indeed homogeneous of degree $m$ in $(\lambda,\kappa)$.

\begin{lemma}\label{44aa11.3}
Let $b(w; \zeta)\in C^\infty(U)$,
$U\subset\bbR^N\times(\bbR^M\setminus\{ 0\})$ conic,
be a classical symbol of order $m$ compactly supported in $w$, and define
\be\label{4a.1.3.1}
u(\zeta , \sigma)\ =\ \int\; e^{-i\sigma\langle Qw,w\rangle/2}\;
b(w; \zeta)\; dw\ .
\ee
Then $u$ is a classical product-type symbol in the pair of fiber variables
$(\zeta,\sigma)$, with bi-order $(m, -N/2)$, and leading symbol
on $\{ \sigma > 0 \}$
\be\label{4a.1.3.4}
u_{0,0}(\zeta, \sigma)\ =\ \left( {2\pi\over\sigma}\right)^{N/2}\;
{e^{-\pi i \mbox{\tiny sgn}Q/4}\over |\det Q|^{1/2}}\;b_0(0; \zeta)\ ,
\ee
where $b_0$ is the leading symbol of $b$.
\end{lemma}
\begin{proof}
We recall that a function $u(z,\zeta,\sigma)$ is a product-type symbol
with bidegree $(p,q)$ iff one has estimates of the form
\be\label{4a.1.4}
|\,\partial^\alpha_z\,\partial^\beta_\zeta\,\partial^\gamma_\sigma\, u|
\; \leq \; C\,(1+|\zeta|)^{p-|\beta|}\, (1+|\sigma|)^{q-\gamma}
\ee
uniformly in $z,\zeta,\sigma$.	One denotes the space of such symbols by
$S^{p,q}$.

\smallskip\noindent
{\sc Scholium.}  {\em Let $b(w,\zeta)$ be a non-necessarily classical symbol
of order $m$, compactly supported in $w$.  Then $u$, defined by
(\ref{4a.1.3.1}), is in $S^{m,-N/2}$.  Moreover, as $\sigma\to\infty$
\be\label{4a.1.4.1}
u(\zeta,\sigma)\ \sim\ \left( {2\pi\over\sigma}\right)^{N/2}\;
{e^{-\pi i \mbox{\tiny sgn}Q/4}\over |\det Q|^{1/2}}\;
\sum_{k=0}^\infty\ {1\over k!}\,(R^kb)(0,\zeta)\; \sigma^{-k}
\ee
where
\[
R\ =\ {i\over 2}\; \langle\,Q^{-1}\partial_w , \partial_w\,\rangle\ ,
\]
and the asymptotics is symbolic
in the sense that the difference between $u$ and the sum of the first
$K$ terms on the right-hand side is in $S^{m,-N/2-K-1}$.}
\par\noindent
{\sc Proof of the Scholium.}  By the method of stationary phase,
e.g. Lemma 7.7.3 in \cite{Ho}, we get that
the absolute value of the difference between the LHS and the sum of the first
$K$ terms is bounded by
\be\label{4a.6}
{ C\over \sigma^{N/2+K+1}}\sum_{|\alpha|\leq 2K+[N/2]} \| D^\alpha_w b \|_{L^2}
\ .
\ee
Since $\| D^\alpha_w b \|_{L^2}\leq C_\alpha|(1+\zeta|)^m$, we get that if
$\sigma > 1$,
\begin{eqnarray}
\nonumber
\lefteqn{\left|\, u -  \left( {2\pi\over\sigma}\right)^{N/2}\;
{e^{-\pi i \mbox{\tiny sgn}Q/4}\over |\det Q|^{1/2}}\;
\sum_{k=0}^K\ {1\over k!}\,(R^kb)(0,\zeta)\; \sigma^{-k}\right| \leq}\\
& & {C\over\sigma^{N/2+K+1}}\;( 1+|\zeta|)^m\ .
\label{4a.7}
\end{eqnarray}
This is
(\ref{4a.1.4}) for the difference $u-\sum^K$ with $\gamma=0,\;\beta=0$.
To obtain the estimate for the partial derivatives
$|\partial^\beta_\zeta\,\partial^\gamma_\sigma\, (u-\sum^K)|$,
we repeat the previous argument with the symbol of order $m-|\beta|$,
\be\label{4a.7.1}
\left(-i\langle Qw,w\rangle\right)^\gamma\; \partial^\beta_\zeta b (w,\zeta)\ .
\ee
Under the transform (\ref{4a.1.3.1}), this symbol corresponds to
$\partial^\beta_\zeta\partial^\gamma_\sigma u$.  Since (\ref{4a.7.1})
vanishes at $w=0$ to order $2\gamma$, the large $\sigma$ expansion of
$\partial^\beta_\zeta\partial^\gamma_\sigma u$ is of order $-N/2-\gamma $.
This shows that $u$ is a product type symbol.
Finally, to prove (\ref{4a.1.4.1}) in the full symbolic sense use the fact
that the stationary phase expansion can be differentiated term by term, and
that we have uniform large $\zeta $ estimates for the derivatives of $b$.

\hfill $\nabla$

\smallskip
Let us finally show that if $b$ is classical $u$ is also a {\em classical}
product-type symbol in the sense of definition 3.1(iii) of \cite{GUh}.
Let
\[
b(w; \zeta)\ \sim\ \sum_{j=0}^\infty b_j(w; \zeta)
\]
be the classical symbol expansion of $b$,
where $b_j$ is homogeneous in $\zeta$ of degree $m-j$.	Define
\be\label{4a.2}
u_j(\zeta,\sigma)\ =\ \int\; e^{-i\sigma\langle Qw,w\rangle}\;
b_j(w; \zeta)\; dw\ .
\ee
Then $u_j$ is also homogeneous in $\zeta$ of degree $m-j$.  Moreover
\be\label{4a.3}
u(\zeta , \sigma)\ \sim\ \sum_{j=0}^\infty u_j(\zeta,\sigma)
\ee
in the following sense: for each $M$,
\be\label{4a.4}
u - \sum_{j=0}^M u_j\ \in\ S^{m-M-1,-N/2}\ ,
\ee
by the first part of the Scholium applied to $b-\sum_{j=0}^Mb_j$.

\smallskip
Applying the second part of the Scholium to $u_j$, we get for each $j$ an
expansion
\be\label{4a.5}
u_j(\zeta,\sigma)\ \sim\
\sum_{k=0}^\infty\ u_{j,k}
\ee
as $\sigma\to\infty$, where
\be\label{4a.7bis}
u_{j,k}\ = \ \left( {2\pi\over\sigma}\right)^{N/2}\;
{e^{ -i \pi \mbox{\tiny sgn}Q/4}\over {|\det Q|^{1/2}}}\;
{1 \over {k!}}\:
(R^kb_j)(0,\zeta)\; \sigma^{-k}
\ee
is bi-homogeneous in $(\zeta,\sigma)$ of bi-degree $(m-j,N/2-k)$. The
meaning of (\ref{4a.5}) is that for all $K$,
\[
u_j - \sum_{k=0}^K\,u_{j,k}\  \in S^{m-j, -N/2-K-1}\ .
\]
Together, (\ref{4a.3}) and (\ref{4a.5}) signify precisely that
$u(\zeta, \sigma )$ is a classical product-type symbol.
\end{proof}

\medskip\noindent
{\em Proof of Proposition (\ref{44aa11.1}).}
We first introduce a cut-off at $\kappa = 0$:
let $\rho \in C^{\infty }_c(\bbR)$ such that $\rho (\kappa ) = 0$ for
$|\kappa | \leq 1/2$, $\rho (\kappa ) = 1$ for $|\kappa | \geq 1$.
Then
\[
\Upsilon(x,y)\ =\ \int\; e^{i\kappa[\,t(y-\langle Qw,w\rangle/2 ) +x]} \;
a(t,w;\kappa)\;\rho(\kappa) dt\,dw\,d\kappa\quad \mbox{mod}\; C^\infty\ .
\]
Make the change of variables $(t,w,\kappa)\ \mapsto\ (\lambda,w,\kappa)$
where $\lambda = t\kappa$.  Then
\[
\Upsilon\ =\ \int\; e^{i[\lambda(y-\langle Qw,w\rangle /2)+\kappa x]}
\; a(\lambda/\kappa,w;\kappa)\, \rho(\kappa)
{d\lambda \over {|\kappa |} }\,dw\,d\kappa\quad \mbox{mod}\; C^\infty .
\]
By Lemma \ref{44aa11.2}, the amplitude $b(w;\lambda,\kappa)$
of this oscillatory integral is a classical
symbol of order $m-1$ in the fiber variables $\zeta = (\kappa,\lambda)$.
By the Fourier inversion formula we can write, in the sense of
oscillatory integrals,
that modulo $C^{\infty } $
\be \label{4aa.6}
\Upsilon\ =\ (2 \pi )^{-1} \int\;
e^{i[-\sigma\langle Qw,w\rangle /2 + \lambda y+\kappa x + s(\sigma-\lambda)]}
\ee
\[
\ \ \ \ a(\lambda/\kappa,w;\kappa)\, \rho(\kappa)
{d\lambda \over {|\kappa |} }\,dw\,d\kappa\,d\sigma \,ds\,d\lambda.
\]
Performing the $dw$ integral we get, with the notation of
Lemma \ref{44aa11.3},
\be\label{4a.7.2}
\Upsilon\ =\ (2\pi )^{-1} \int\;
e^{i[\lambda(y-s) +\kappa x + s\sigma]}
\; u(\kappa,\lambda;\sigma)\, d\kappa\,d\lambda\,d\sigma \,ds
\quad \mbox{mod}\; C^\infty .
\ee
This basically shows that $\Upsilon $ is in the desired class: compare with
formula (3.9) of \cite{GUh}.
To obtain complete agreement with the latter
we only have to introduce a compact cut-off $\chi (s)$ in $s$, such that
$\chi (s) \equiv 1$ in a neighborhood of $0$. To justify
this, note that the integral (\ref{4a.7.2}) with a factor $(1 - \chi )$
introduced in the integrand will give a function in $C^{\infty }_c(\bbR^2)$,
by a straightforward integration by parts argument, using
$(1/i\sigma ) \partial _s$
and $(1/is) \partial _{\lambda }$ (which is well-defined on
$\mbox{supp}\,(1 - \chi )$).  In the sequel we will for
for simplicity just ignore this compact cut-off in $s$, and work
with (\ref{4a.7.2}).
\nobreak\par\hfill$\Box$

\medskip
Combining (\ref{4a.1.3}) and (\ref{4a.1.3.4}), we see that
the leading symbol of the amplitude in (\ref{4a.7.2})
on $\{ \kappa > 0, \sigma > 0 \}$ is
\be\label{4a.7.3}
u_{0,0}(\kappa,\lambda ; \sigma) = {1 \over {2\pi }}\,
\left( {2\pi\over\sigma}\right)^{N/2}
{e^{-\pi i \mbox{\tiny{sgn}}Q/4}\over |\det Q|^{1/2}}\;
a_0(t=\lambda/\kappa, w=0)\,\kappa^{m-1}\ ,
\ee
while (c.f.(\ref{4a.2}))
\be\label{4a.7.4}
u_0(\kappa,\lambda ; \sigma)\ =\ {1 \over {2\pi }} \, \kappa^{m-1}\;\int\;
e^{-i \sigma \langle Qw,w\rangle/2}\,
a_0(t=\lambda/\kappa, w)\, dw\	\ (\kappa > 0).
\ee

\medskip
\begin{corollary} \label{4a.VII}
(cf. \cite{GUh}, Proposition 3.4.)\par\noindent
1.- Microlocally on $\Lambda_0\setminus (\Lambda_0\cap\Lambda_1)$, $\Upsilon$
is in the class $I^{m-N/2+1}(\bbR^2,\Lambda_0)$.  Moreover, its principal symbol
at the covector $(x=0,y=0; \kappa > 0,\lambda > 0)$ is
\be\label{4a.8}
S_{\Lambda_0}(\kappa,\lambda)\;=\; {1 \over {2\pi }}\;
\left({2\pi\over\lambda}\right)^{N/2} \kappa^{m-1}  \, \cdot
\ee
\[
\cdot \,
{e^{-\pi i \mbox{\tiny sgn}Q/4}\over |\det Q|^{1/2}}\;a_0(t= \lambda/\kappa ;0)
\; |d\lambda d\kappa|^{1/2}\ .
\]
(Note that $\lambda \neq 0$ on
$\Lambda_0\setminus (\Lambda_0\cap\Lambda_1)$ so this is well-defined.)
\par\noindent
2.-  Microlocally on $\Lambda_1\setminus (\Lambda_0\cap\Lambda_1)$, $\Upsilon$
is in the class $I^m(\bbR^2,\Lambda_1)$.  Moreover, its principal symbol
at the covector $(x=0,y; \kappa =0 ,\lambda)$ is
\be\label{4a.9}
S_{\Lambda_1}(y, \kappa )\ =\
\,\kappa^{m-1}\;
\left(\int_{\langle Qw,w\rangle /2 = y}\, a_0(t=0, w)\, dL_y(w)\right)\;
|dy d\kappa|^{1/2}\ ,
\ee
where $dL_y$ is the Liouville measure on the level set
$\{ w; \langle Qw,w\rangle /2 = y\}$ (note that $L_y$ is well-defined since
$y \neq 0$ on $\Lambda_1\setminus (\Lambda_0\cap\Lambda_1)$).
\end{corollary}
\begin{proof}
This is a direct application of \cite{GUh}, Proposition 3.4, according to
which the symbol of $\Upsilon$ on $\Lambda_0$
is obtained from $u_{0,0}$, c.f. (\ref{4a.7.3}) by replacing
$\sigma$ by $\lambda$; this yields (\ref{4a.8}).
The symbol on $\Lambda_1$ equals
\be\label{4a.10}
S_{\Lambda_1}(y,\kappa )\ =\
\left(\int u_0(\kappa,\lambda =0;\sigma )\, e^{iy\sigma}\, d\sigma \right)\;
|dy d\kappa|^{1/2}\ .
\ee
Replacing $u_0$ by (\ref{4a.7.4}) yields (\ref{4a.9}), by the defining
property of $L_y$.
\end{proof}

Note that from (\ref{4a.8}),
\be\label{4a.10.1}
{1 \over {2\pi } } \,
a(t,w=0) = \left( {t\over 2\pi}\right)^{N/2}\;
|\det Q|^{1/2}	e^{i \pi \mbox{\tiny sgn}Q/4}\; S_{\Lambda_0}(1,t)\,
|d\lambda d\kappa|^{-1/2}\; .
\ee

\medskip \noindent
{\bf Remark.}
Notice that if $\hat{S}_{\Lambda _1} (\sigma , \kappa )$
denotes the Fourier transform of
$S_{\Lambda _1}(y, \kappa )$ with respect to $y$, then by (\ref{4a.10}),
(\ref{4a.7.4}) and
the stationary phase expansion (\ref{4a.1.4.1}) (ignoring half-densities):
\be\label{4a.11}
a_0(0,0)\ =\
 \lim_{\sigma\to\infty}\ \left( {\sigma\over 2\pi}\right)^{N/2}\,
\kappa ^{-(m-1)}
|\det Q|^{1/2} e^{i \pi \mbox{\tiny sgn}Q/4}\;
\hat{S}_{\Lambda_1} (\sigma, \kappa)\ ,
\ee
which determines the growth of $\hat{S}_{\Lambda _1} $ as $\sigma \to \infty $.
On the other hand, by (\ref{4a.10.1}),
$S_{\Lambda_0}$ blows up as $\lambda\to 0$, in such a way
that
\be\label{4a.12}
{1 \over {2\pi }} \, a_0(0,0)\ =\ { 1 \over {(2\pi)^{N/2}}}\, |\det Q|^{1/2}
\,e^{i \pi \mbox{\tiny sgn}Q/4}\; \lim_{\lambda\to 0}\,
\left[\, \lambda^{N/2}\, S_{\Lambda_0}(1,\lambda)
\right]\ .
\ee
Together, (\ref{4a.11}) and (\ref{4a.12}) give an example of the compatibility
condition on the symbols $S_{\Lambda _0}, \ S_{\Lambda _1}$ of
$I^{p,l}(\bbR^2, \Lambda_0, \Lambda_1)$, see \cite{GUh} Proposition 5.5.


%here starts the old 4b.tex

\subsection{The symbols of $\Upsilon$.}

We return to the notations of section 4.1. Proposition \ref{44aa11.1}
can be applied to the distribution $\Upsilon_1(s,E)$, given by
the right hand side of (\ref{4.5}). Note that $m = n$,
that, by the change of variables \S 2, $\det Q = \pm 1$ here,
and that the signature of $Q$,
which we will denote by {\em sgn}, is that of the Hessian of $H$
on $\Theta $ (which, we recall, is connected). The amplitude $a$ of the
previous section will now again be the amplitude
$\kappa ^n \check{\varphi } \chi \beta $
of proposition \ref{44.33}.
We will now describe the symbols of
$\Upsilon_1$ along each of the $\Lambda_j$ in terms of the Hamiltonian and
its flow.

\begin{proposition} \label{4B.1}
Away from $E=E_c$, $\Upsilon_1 (s,E)$ is conormal to $\{ s=0 \}$.
Its symbol at the covector $(s=0, E; \kappa, \lambda =0)$ is
\be\label{4b.1}
S_{\Lambda_1}(E,\kappa)\ =\  (2\pi )^{-(n-1)} \,
\kappa^{n-1}\, \check{\varphi} (0)\, \int_{\Sigma_E}\,\rho\, dL_E\;
| dy\, d\kappa|^{1/2}\ ,
\ee
where $dL_E$ denotes the Liouville density on $\Sigma_E$.
\end{proposition}

\medskip \noindent
\begin{proof}
In \cite{PU2} we proved that if $E$ is away from $E_c$, then
$\Upsilon$ itself is a conormal distribution of the desired sort with symbol
\be\label{4b.2}
(2\pi )^{-(n-1)} \,
\kappa^{n-1}\, \check{\varphi } (0)\, \mbox{LVol}\,(\Sigma_E)\;
| dy\, d\kappa|^{1/2}\ .
\ee
The same proof applies to $\Upsilon_1$, which is $\Upsilon$ with the amplitude
cut-off by $\rho$.
\end{proof}

\medskip \noindent
This proposition, in combination with (\ref{4a.11})
gives a formula for the value of $\beta _0$ in $t = 0, w = 0$:
\begin{corollary}\label{4b4b.1}
\be\label{zero}
\beta _0(0, 0)\ =\
{e^{i \pi \mbox{\tiny sgn}/4}\over (2\pi)^{n - 1 + N/2}}\;
\lim_{\sigma\to\infty}\, e^{i\sigma E_c}\,
\sigma^{N/2}\, \int\, e^{-i\sigma H(x,p)}\,
\rho(x,p)\, dx\, dp\ .
\ee
\end{corollary}

Note that the right hand side can be evaluated by classical stationary
phase, under the Main Hypothesis on $H$.
This corollary, in combination with (\ref{proof.3b}), gives a formula for
the coefficient $c_{0, 1}$ of theorem \ref{ONE} in case $N = 2$.
If $N > 2$ we need the derivatives with respect to $t$ of $\beta $
in $(0, 0)$. These cannot be obtained from $S_{\Lambda _1}$, since putting
$\lambda = 0$ in the right hand side of (\ref{4a.9}) corresponds to putting
$t = 0$ in $\beta $. We will now obtain these derivatives through a study
of the symbol $S_{\Lambda _0}$ of $\Upsilon $ on
$\Lambda _0 \setminus (\Lambda _0 \cap \Lambda _1)$, based upon results
already obtained in \cite{GU}.

Recall that  $\{ f_t \}_t$ denotes the Hamilton flow of $H$. If
$0 < |t| < \eta $ with $\eta $ sufficiently small, then $t$ will not be a
period of either the flow on $\Sigma _{E_c} \setminus \Theta $ nor of the
linearized flows $d(f_t)_z $ on $T_z(T^*M ) / T_z(\Theta )$
for $z \in \Theta $. The latter half of this statement is equivalent to
saying that $\Theta $ is a clean fixed point manifold of $f_t $. Recall
the canonical measure $d\mu _t$ associated to $\Theta $ and $f_t $ in this
situation (cf. the appendix ). In the remainder of this subsection we
will prove the following proposition:

\begin{proposition}\label{betazero}
Let $0 < |t| < \eta $ for sufficiently small $\eta $. Then
\be\label{4bb.6}
\beta_0(t,0)\ =\
(2\pi)^{-(n-1)}\ \left( |t|^{N/2}\;
\int_\Theta\; d\mu_t \right) .
\ee
\end{proposition}

\begin{proof}
First, since $\check{\varphi }$ has compact support, it follows from
\ref{44.22} that $WF(\Upsilon ) \cap N^* \{E = E_c \} = \emptyset$.
Hence we can {\em restrict} $\Upsilon$ to the line $E=E_c$, thereby
obtaining a distribution in the single variable $s$,
\be\label{4bb.1}
\Upsilon_{E_c}(s)\ =\ \sum_{k=1}^\infty\; \varphi\left(
k[E_j(1/k)-E_c]\right)\; e^{iks}\ .
\ee
This distribution was already studied in \cite{GU} for a restricted class
of $\varphi $'s: let $J$ be an open subset of $(0, \infty )$ not containing
any of the periods of either the flow on $\Sigma _{E_c}$ or the
linearized flows along $\Theta $.
Then by applying Theorem 5.6 of \cite{GU} to the operator
$\tilde{P}$ of \S 2 (and using that $\mu _t = \mu _{-t}$) we have the following:
\begin{proposition} \label{thm5.6}
Under the Main Hypothesis, if $\mbox{supp}\ \check{\varphi } \subset J$, then
the distribution $\Upsilon_{E_c}(s)$ is conormal to $s = 0$,
with principal symbol at $(s, \kappa ) = (0, 1)$ equal to the constant
\be\label{4bb.2}
(2\pi)^{N/2 - n}\; e^{i\pi m/4}\; \int_J\left(
\int_\Theta\, d\mu_t\,\right)\;\check{\varphi}(t)\, dt
\ee
where $m$ is some integer (Maslov factor).
\end{proposition}

\medskip
Now it follows from the results of the previous subsection that if
$\mbox{supp}\, \check{\varphi }  \subset (0 , \eta ) $
with $\eta $ sufficiently small, then
$\Upsilon =
\Upsilon _{\varphi }$ is
a Lagrangean distribution associated to
$\Lambda _0 \setminus (\Lambda _0 \cap \Lambda _1)$:
\[
\Upsilon (s, E) \ = \
\int \, u(\kappa, \lambda; \lambda)\,
e^{i (\kappa s + \lambda (E-E_c))}\,
d\kappa d\lambda
\]
with symbol $u $ supported in a cone $\{ \kappa > 0, C^{-1} \leq
\lambda / \kappa \leq C \}$, since $ a = \hat{\varphi } \chi \beta
\equiv 0$ for $t = \lambda / \kappa $ in a neighborhood of $0$.
Now we can also compute the principal symbol of
$\Upsilon _{E_c}$ as the push-forward of the principal symbol of
$\Upsilon(s,E)$ on $\Lambda_0$ along the map
$(\kappa, \lambda ) \to \kappa $, namely as
\be \label{4bb5}
\int u_{0, 0}(\kappa ,\lambda; \lambda) \, d\lambda \, = \,
\ee
\[
{1 \over {2\pi }} \,
e^{-i\pi\mbox{\tiny sgn}/4} \, \kappa ^{n-1} \,
\int \,
\left( {{2\pi } \over {\lambda }} \right)^{N/2}
\check{\varphi }(\lambda / \kappa)
\beta _0 (\lambda / \kappa , 0) \, d\lambda
\]
\[
= \,
(2\pi )^{N/2 -1} \, \kappa ^{n - N/2} \,
e^{-i\pi\mbox{\tiny sgn}/4} \,
\int _{\bbR} \, t^{-N/2} \check{\varphi }(t) \beta _0(t, 0) \, dt.
\]

\medskip \noindent
At $\kappa = 1$ this has to be equal to (\ref{4bb.2}) for all
$\check{\varphi }$ supported in $(0  , \eta ) $.  This implies
formula (\ref{4bb.6}) for $t > 0$
up to the factor $\exp(i\pi(\mu+\mbox{sgn})/4)$.
However, since $\beta_0$ is continuous (in fact smooth) at $t=0$,
comparing this result with the formula
(\ref{zero}) (which is positive) we see that
necessarily $\exp(i\pi(\mu+\mbox{sgn})/4) =1$.
Proposition \ref{betazero} follows for $t > 0$. To obtain the formula for
$t < 0$ we repeat the argument, noting the following changes: if
$J \subset (-\infty , 0)$ in Proposition \ref{thm5.6}, the Maslov factor
$e^{i\pi m/4}$ in front gets replaced by it's complex conjugate. Similary,
(\ref{4a.7.3}) has to be replaced by it's conjugate if we replace $\sigma $ by $-\sigma $ ($\sigma > 0$).
\end{proof}

\subsection{Proof of Theorem 1.2}

Parts (i) and (ii) of Theorem \ref{TWO} are immediate consequences of
proposition \ref{betazero} and formulas (\ref{proof.3b}) and
(\ref{proof.5}) of section 3.

To prove part (iii) first note that by
Proposition \ref{4B.1} and the second part of corollary \ref{4a.VII}
(in particular formula (\ref{4a.9}) if $y \neq 0$,
\be \label{4c.1}
\int _{\langle Qw, w \rangle / 2} \, \beta(0, w) \, dL_{Q, y}(w) \ = \
\int _{\Sigma _{E_c + y}} \, \rho \, dL_{E_c + y} \ ,
\ee
the measures being the respective Liouville densities on
$\{ \langle Qw, w \rangle / 2 \, = \, y \} $ and on $\Sigma _{E_c + y} $.
Now if $N > 2$ we can take the limit for $y \to 0$, since then both
$dL_{Q, 0} $ and $dL_{E_c} $ are well-defined finite measures on
$\{ \langle Qw, w \rangle \, = \, 0 \}$ and $\Sigma _{E_c}$, respectively.
If we combine the resulting formula with the observation made in the second
remark after theorem \ref{III.9a}, theorem \ref{TWO}, (iii) follows.




\section{Small $\hbar$ and small $|E-E_c|$ Estimates}
\newcommand{\phihat}{\check{\varphi }}
\newcommand{\calh}{{\frak h}}

In the preceding sections we established an asymptotic
expansion of $\Upsilon_{\h,E}(\varphi)$ for $E=E_c$ critical, which
complemented the trace formula for a regular $E$.  The problem arises
to try to understand how the transition from the regular regime to
a critical one takes place, in particular when $N=2$ and the normal
Hessian is indefinite (since in that case the top order of the
asymptotics changes).  In this section we prove a first (very modest)
result in this direction, by showing that the top order estimate in the
expansion of Theorem 1.1 holds uniformly for $E$ in a neighborhood of
size $\h$ of $E_c$.

\medskip
The starting point of our analysis is again Proposition \ref{II.summary}.
If we put $t=\lambda\h$ in formula (\ref{II.11}), then
\be\label{f1}
\Upsilon_{\h, E}\;=\; \h^{-(n-1)}\, \int\, e^{i\lambda(E-E_c)}\,
u(\lambda, \h, \lambda)\, d\lambda
\ee
where
\be\label{f2}
u(\lambda,\h; \sigma)\;=\; \phihat (\lambda\h)\,\int_{\small{\bbR^N}}\,
\beta (\lambda\h, w; \h^{-1})\, e^{-i\sigma\langle Qw,w\rangle/2}\, dw\,.
\ee
(Compare with the notations of \S 4.2, and in particular Lemma \ref{44aa11.3}.)
The integral (\ref{f1}) converges absolutely for $\h > 0$, since
$\phihat$ has compact support.	Be Lemma \ref{44aa11.3},
$u(\lambda,\h;\lambda)$ is a symbol in $\lambda$ of order
$-N/2$.  This immediately justifies the following Proposition, which will be
needed in the proof of Theorem 1.3 in the next section.

\begin{proposition}\label{F1}
If $N>2$, then
\[
\Upsilon_{\h, E}\;=\; \calO (\h^{-(n-1)})
\]
uniformly for $E$ in a neighborhood of $E_c$.
\end{proposition}

\medskip
In the remainder of this section we concentrate on the case $N=2$.
We first break up the integral (\ref{f1}) in three pieces, using a
fixed cut-off function $\chi = \chi(\lambda)\in C^\infty (\bbR)$
such that
\begin{enumerate}
\item $\chi (r) = 0\quad \forall \,r\,, |r| \leq 1$ and
$\chi (r) = 1\quad \forall \,r\,, |r| \geq 2$.
\item $\chi$ is an even function.
\end{enumerate}

Obviously,
\[
\int\, e^{i\lambda(E-E_c)}\, \left( 1-\chi (\lambda) \right)
u(\lambda, \h, \lambda)\, d\lambda\;=\; \calO (1)
\]
uniformly in $\h$ and $E$, since the integrand is bounded and
compactly supported.  Hence, if we introduce the integrals
\be\label{f3}
I^{\pm}(\h, E)\;=\; \int_{(0,\pm\infty)}
e^{i\lambda(E-E_c)}\, \chi (\lambda)\, u(\lambda, \h, \lambda)\, d\lambda
\ee
then
\be\label{f4}
\h^{n-1}\,\Upsilon_{\h, E}\;=\; I^{+}+I^{-}+ \calO (1)\,.
\ee
Next, in turn, we will analyze $I^{+}$ and $I^{-}$.  First, by Lemma
\ref{44aa11.3} with $N=2$,
\[
u(\lambda, \h; \lambda)\;=\; {C^{+}\over\lambda}\, f(\lambda\h) +
\calO (\lambda^{-3/2})\; , \quad \lambda\to \infty
\]
where we have written
\[
f(t)\;=\; \phihat (t)\, \beta(t,0;\h^{-1})\quad\mbox{and}\quad
C^{+}\;=\; 2\pi\, e^{i\pi\mbox{\tiny sgn}Q/4}
\]
(remembering that $|\det Q| = 1$).  Hence
\be\label{f5}
I^{+}\;=\; C^{+}\, \int e^{i\lambda(E-E_c)}\, f(\lambda\h)\,
{\chi(\lambda)\over\lambda}\, d\lambda + \calO (1)\,,
\ee
the $\calO (1)$ term being uniform in $\h$ and $E$.  If we integrate
(\ref{f5}) by parts, starting from $\lambda^{-1} = (\log \lambda)'$,
and then change variables $t=\lambda\h$ again, we see that
\be\label{f6}
I^{+}\;= \; -J_1-J_2 + \calO (1)
\ee
with
\be\label{f7}
J_1\;=\; C^{+}\,\int_0^\infty  e^{it(E-E_c)/\h}\, f'(t)\,
\chi(t/\h)\, \log(t/\h)\, dt\,,
\ee
and
\be\label{f8}
J_2\;=\; i\h^{-1} (E-E_c)\, C^{+}\,\int_0^\infty  e^{it(E-E_c)/\h}\,
f(t)\, \chi(t/\h)\, \log(t/\h)\, dt\, ,
\ee
and the error term in (\ref{f6}) is in fact
\[
C^{+}\,\int_0^\infty\, e^{i\lambda(E-E_c)}\, f(\lambda\h)\, \chi'(\lambda)
\, \log\lambda\, d\lambda
\]
which is trivially $\calO (1)$ in $E$ and $\h$ since $\chi'$ is compactly
supported.  It will be convenient to introduce the notation
\be\label{f9}
\omega\;=\; {E-E_c\over\h}\;.
\ee
We will obtain estimates as $\h\to 0$ and $E\to E_c$ simultaneously as long
as $\omega$ remains bounded.

\medskip
First we estimate $J_1$.  In (\ref{f7}), write $\log(\h^{-1}t) =
\log(\h^{-1}) + \log (t)$, and note that $f'(t)\log (t)$ is integrable,
so that
\[
J_1\;=\; C^{+}\, \log(\h^{-1})\, \int_0^\infty\, e^{i\omega t}\, f'(t)\,
\chi(t/\h)\, dt + \calO (1)\,.
\]
Now if $\h\to 0$, then $\chi(t/\h)\to 1$ pointwise on $(0,\infty)$.
If we let $\calh$ denote the characteristic function of $(0,\infty)$ then,
since
\[
\int\, |1 - \chi(t/\h) |\, dt\;=\; \calO (\h^{-1})\,,
\]
we have that
\[
J_1\;=\; (2\pi)\, C^{+}\, \log (\h^{-1})\, (f'\calh)\check{}(\omega ) +
\calO (1)\,,
\]
where the $\check{}$ denotes the inverse Fourier transform: If $g$ is
a function on $\bbR$, $g\check{}(\omega)=(2\pi)^{-1}\int g(x) e^{ix\omega}dx$.

\medskip
Next we treat $J_2$ in a similar way:  Since $f(t)\log(t)$ is integrable,
\[
J_2\;=\; (2\pi)\, C^{+}\, \log (\h^{-1})\, i\omega (f\calh)\check{}(\omega ) +
\calO (|\omega|\log(\h^{-1})/\h) + \calO (|\omega|)\,.
\]
Combining these two estimates, (\ref{f6}) implies:

\begin{lemma}\label{F2}
$I^{+}$ can be estimated as follows:
\be\label{f10}
I^{+}\;=\; -(2\pi)\, C^{+}\, \log(\h^{-1})\,
\left( (f'\calh)\check{}(\omega) + i\omega (f\calh)\check{}(\omega ) \right) +
\ee
\[
+ \ \calO (|\omega|+1)\,,
\]
where $\omega$ is defined by (\ref{f9}) and $\calh$ is the characteristic
function of $(0,\infty)$.
\end{lemma}

\bigskip
We next look at $I^{-}$,which can be easily reduced to $I^{+}$ by the
stationary phase expansion of Lemma \ref{44aa11.3} {\em but now
for} $\lambda\to -\infty$.  We obtain that, modulo $\calO (1)$
\[
I^{-}\;=\; \overline{C^{+}}\,\int_{-\infty}^0\, e^{i\lambda(E-E_c)}\,
f(\lambda\h)\,{\chi(\lambda)\over -\lambda}\, d\lambda\,.
\]
\[
=\; \overline{C^{+}}\; \int_0^\infty\, e^{-i\lambda (E-E_c)}\,
f_{-}(\lambda\h)\, {\chi(\lambda)\over\lambda}\, d\lambda\, ,
\]
where $f_{-}(t) := f(-t)$, and where we used the fact that $\chi$
is even.  This is of the same form as (\ref{f5}), with $E-E_c$ replaced
by $E_c-E$.  Hence by Lemma \ref{F2},
\be\label{f11}
I^{-}\;=\; -(2\pi)\, \overline{C^{+}}\, \log(\h^{-1})\,
\left(\, (f_{-}'\calh)\check{}(-\omega) -i\omega(f_{-}\calh)\check{}
(-\omega)\,\right)
\ee
\[
+ \ \calO (|\omega|+1)\, .
\]
Adding (\ref{f10}) and (\ref{f11}), and using that $f_{-}' = -(f')_{-}$,
we obtain, by (\ref{f4}), the following

\begin{lemma}\label{F3}
If $\omega = (E-E_c)/\h$, then, as $\h\to 0$
\be\label{f12}
\h^{(n-1)}\Upsilon_{\h, E}\; =\; -2\pi\,\log(\h^{-1})\,
\left\{\, C^{+}(f'\calh)\check{} (\omega) -
\overline{C^{+}}((f')_{-}\calh)\check{}(-\omega)
\right.
\ee
\[
\left. + i\omega \left[ C^{+}(f\calh)\check{} (\omega) -
\overline{C^{+}}(f_{-}\calh)\check{}(-\omega)\right]
\right\} + O(|\omega|+1)\, .
\]
\end{lemma}

\medskip
It now becomes important to distinguish between the case of a definite
$Q$, for which $C^{+}$ is purely imaginary,
\be\label{f13}
C^{+}\;=\; 2 \pi e^{\pm i\pi/2}\;=\; \pm 2\pi i\quad  (Q\,\mbox{positive or negative
definite})
\ee
and that of $Q$ indefinite, for which
\be\label{f14}
C^{+}\;=\; 2\pi \quad (Q\,\mbox{indefinite})\,.
\ee
We will also use the following elementary observation:	If $g$ is
an integrable function on the line, then
\be\label{f15}
(g\calh)\check{}(\omega) + (g_{-}\calh)\check{} (-\omega)\;=\;
\check{g} (\omega)\,,
\ee
and
\be\label{f16}
(g\calh)\check{}(\omega) - (g_{-}\calh)\check{} (-\omega)\;=\;
{ i \over \pi }\, \calH(\check{g})(\omega)
\ee
where $\calH$ denotes the Hilbert transform (convolution with the
principal value of $1/t$ on $\bbR$).

\medskip
Now if $Q$ is definite, then $\overline{C^{+}} = -C^{+}$, and Lemma
\ref{F3} implies that
\[
\Upsilon_{\h, E_c+\omega\h}\;=\; \calO ( (|\omega | + 1)\h^{-(n-1)})\,,
\]
using (\ref{f15}) and the fact that $(f')\check{}(\omega) = -i\omega
\check{f}(\omega)$ .  In case $Q$ is indefinite, Lemma \ref{F3} yields a
more interesting result:

\begin{theorem}\label{F4}
Suppose that $N=2$ and that the normal Hessian is indefinite.  Then, with
$\omega\in\bbR$ fixed, as $\h\to 0$
\be\label{f17}
\Upsilon_{\h, E_c+\omega\h}\;=\; 4\pi \log(\h^{-1}) \phihat (0)\,
\beta_0(0,0) + \calO (|\omega| + 1)\, .
\ee
\end{theorem}

(Compare with (\ref{proof.3a}), (\ref{proof.3b}) for $N = 2$.)

\medskip \noindent
\begin{proof}
If we combine Lemma \ref{F3} with (\ref{f14}) and (\ref{f16}), we
obtain a one-term expansion of the type (\ref{f17}), with the coefficient
of $\log(\h^{-1})$ equal to
\[
-4\pi\left\{ i\calH(\check{f'})(\omega) -
\omega\calH(\check{f})(\omega)\right\}\; =
\]
\[
=\; -4\pi \left\{ \calH(u\check{f}(u))(\omega) -
\omega\calH(\check{f})(\omega)\right\}\; =
\]
\[
=\; 4\pi\, \int\check{f} (u)\, du\;=\; 4\pi f(0)\, .
\]
Remembering that $f(t) = \phihat (t) \beta(t,0,\h^{-1})$ and replacing
$\beta$ by $\beta_0 + \calO (\h)$, the Theorem follows.
\end{proof}


\section{A Tauberian lemma}

We will first prove a general Tauberian lemma,
with the help of which we  then will prove Theorems \ref{THREE} and
\ref{FOUR}. This lemma is a straightforward generalization of the one
in section 3 of \cite{BU}
(cf. in particular lemma 3.4 and the proof of theorem 3.2 in {\em loc. cit.})
and we will be sketchy.
Also, with an eye toward future applications, we state the lemma in
somewhat greater generality than actually needed here.

\subsection{Tauber with weights}

We will consider expressions of the type  (\ref{I.8.a}):
\be \label{5.1}
\Upsilon ^w_{E, \hbar }(\varphi ) \, = \,
\sum _j \, w_j(\hbar ) \, \varphi \left(
{{E_j(\hbar ) - E} \over \hbar} \right) \ .
\ee
As before, $\varphi $ will range over the set of Schwarz functions on the line
with compactly supported Fourier transform.
We will make the following assumptions on
$w_j(\h )$, $E_j(\h )$, $E$ and on $\Upsilon^w_{\h }$ itself:

\medskip \noindent
\begin{assumptions}\label{5.I}
\rm{ (i) There exists a positive function $\omega(\h)$, defined on an interval
$(0, \h _0)$, and a finite measure $c_0 $ on $\bbR$ such that for all $\vp $:
\be \label{5.3}
\Upsilon _{E, \h }^w (\vp ) = c_0(\vp ) \omega(\h) + o(\omega(\h)), \
\h \to 0
\ee
(both $c_0 $ and $\omega $ depending on $E$, in general).

\medskip \noindent
(ii) There exists a $k \in Z$ such that $\h ^k = \calO (\omega(\h )), \
\h \to 0$.

\medskip \noindent
(iii) There exists an $\varepsilon > 0$ such that for all $\vp $
one can find a constant $C_{\vp }$ such that for all
$E' \in [E - \varepsilon , E + \varepsilon  ]$ :
\be \label{4}
|\Upsilon ^w_{E', \h}(\vp )| \leq C_{\vp } \omega (\h ),
\ee
locally uniform in $E'$.

\medskip \noindent
(iv) The $w_j(\h )$ are non-negative and bounded: there exists a constant
$C \geq 0$ such that for all $j$ and all $\h, 0 < \h < \h _0$:
\be \label{5}
0 \leq w_j(\h ) \leq C
\ee

\medskip \noindent
(v) The eigenvalues $E_j(\h )$ satisfy the following rough estimate:
for each $C_1$ there
exist constants $C_2, N_0$ such that
\be \label{6}
\# \{j: E_j(\h) \leq C_1 + k \h \} \leq C_2 (\h ^{-1} k)^{N_0} \ .
\ee
}
\end{assumptions}

\bigskip \noindent We will write in the sequel
\be \label{5.6a}
x_j \, = \, x_j(\hbar) \, \equiv \, {{E_j(\hbar ) - E} \over {\hbar }} \ .
\ee

\medskip \noindent
Define the weighted counting function by
\be
N_{E,c}^w(\h ) = \sum _{|x_j(\h)| \leq c} w_j(\h )
\ee
We note the following trivial consequence of (iii):

\begin{lemma} \label{5.II}
Let $c > 0$. Then there exists a constant $C = C$ such that
for all $E' \in [E - \epsilon, E + \epsilon]$
\be
N^w_{E',c} \leq C \omega (\h)
\ee
\end{lemma}

The following is our Tauberian lemma:

\begin{theorem} \label{5.III}
If the assumptions \ref{5.I} are satisfied, and
if $\chi _{[-c, c]}$ is the characteristic function of the interval
$[-c, c]$, then
\be \label{7a}
N_{E,c}^w(\h ) = c_0(\chi _{[-c, c]}) \omega (\h) + o(\omega (\h)), \
\h \to 0
\ee
\end{theorem}
\begin{proof} Fix $c > 0$, and let $\chi = \chi _{[-c, c]}$.
Let $\psi $ be a non-negative Schwartz function with compactly supported Fourier
transform $\hat{\psi }$
such that $\hat{\psi }(0) = 1$ and define for $\delta > 0$,
\be
\vp _{\delta} = \psi _{\delta } * \chi, \ \ \ \psi _{\delta }(x) =
\delta ^{-1} \psi (\delta ^{-1} x)
\ee
If $\eta > 0$, the rapid decraese of $\psi $ implies that
$ \min _{|x| \leq c-\eta } \ \vp _{\delta } \geq
1 - C_N (\delta \eta ^{-1})^N $
and hence
\be \label{7}
(1 - C \delta \eta ^{-1}) N^w_{E, c-\eta}(\h ) \leq
\Upsilon^w_{E, \h }(\vp _{\delta })
\ee
Next, since $\vp _{\delta } \leq 1$ and $w_j(\h ) \geq 0$,
\be
\Upsilon^w_{E, \h }(\vp _{\delta }) \leq N^w_{E, c+\eta}(\h ) +
\sum _{|x_j | \geq c+\eta } w_j(\h ) \vp _{\delta }(x_j),
\ee
$x_j = x_j(\h )$ defined by (\ref{5.6a}). For $k \in N$ let
\be
I_k = \{j: c+ \eta + k \leq |x_j(\h )| \leq \eta + k + 1 \}
\ee
We split the sum on the right as
\be \label{8}
\sum _{k \leq [\varepsilon \h ^{-1}]} \ \sum _{j \in I_k} \ \ \ \ + \ \ \
\sum _{k > [\varepsilon \h ^{-1}]} \ \sum _{j \in I_k} \equiv \ I \ + \ II
\ee
with $\varepsilon $ as in \ref{5.I}(iii).
We make the following two observations: first,
\be \label{9}
j \in I_k \Rightarrow |E_j(\h ) - (E \pm k\h )| \leq c' \h
\ee
where $c' = c + \eta + 1$. Next,
\be \label{10}
c + \eta + k \leq |x| \leq c + \eta + k + 1 \Rightarrow
|\vp _{\delta}(x)| \leq C_N (\delta \eta ^{-1})^N (k+1)^{-N};
\ee
this is again a consequence of the rapid decrease of $\psi $.
To estimate the first sum in (\ref{8}), note that
by (\ref{9}), $k \leq [\varepsilon \h ^{-1}]$
implies $E \pm k\h \in [E - \varepsilon, E + \varepsilon]$. Hence, by
lemma \ref{5.II} and (\ref{10}),
\be \label{11}
| I | \ \leq \ C_N (\delta \eta ^{-1})^N \omega(\h )
\ee
Next, by (\ref{9}) and \ref{5.I}(v),
$\# I_k \leq C \h ^{-N_0} k^{N_0}$.
Using (\ref{10}) again, together with \ref{5.I}(ii) it follows that
if $N$ is sufficiently
large, $II$ also satisfies the estimate (\ref{11}).

\medskip \noindent
In conclusion, we have shown that for sufficiently large $N$,
\be \label{12}
(1 - C \delta \eta ^{-1}) N^w_{E, c-\eta}(\h ) \leq
\Upsilon^w_{E, \h }(\vp _{\delta }) \leq
N^w_{E, c+\eta }(\h ) + C_N (\delta \eta ^{-1})^N \omega(\h )
\ee
To conclude the proof, divide all terms in (\ref{12}) by $\omega (\h )$, first
let $\h \to 0$ and then $\delta \to 0$, to obtain
\be
\limsup _{\h \to 0} \omega(\h )^{-1} N^w_{E, c- \eta }(\h ) \leq
c_0(\chi _{[-c, c]}) \leq
\liminf _{\h \to 0} \omega(\h )^{-1} N^w_{E, c+ \eta }(\h )
\ee
Finally, replace $c$ by $c \pm \eta $ and let $\eta \to 0$.
\end{proof}

\medskip \noindent {\bf Remarks:} (a) As the proof shows,
the requirement (iv) could have been weakened to:

\medskip \noindent {\em
(iv') There exist $C, K >0$ such that if $|E_j(\h ) - E| \leq 2 \epsilon$,
$\epsilon $ as in \ref{5.I}(iii), then $0 \leq w_j(\h ) \leq C$, and if
$|x_j(\h )| \leq k$, then $0 \leq w_j(\h) \leq C (\h ^{-1}k)^K$ .}

\medskip \noindent (b)
As noted in the introduction, one can extend the main results of this paper to
Schr\"odinger operators on $\bbR^n$. In that case we would only sum over
$\{j: \vert E_j(\h ) - E \vert \leq \h ^{1 - \epsilon} \}$ in (\ref{5.1}),
$\varepsilon \in (0, 1)$ arbitrary but fixed, and
restrict $E$ to $E < V_{\infty } :=
\liminf _{\vert x \vert \to \infty } V(x)$, $V$ being the potential.
We then can replace the assumption \ref{5.I}(v) by

\medskip \noindent {\em (v') If $C_1 < \liminf V(x)$ there exist constants
$C_2$ and $N_0$ such that
\be \label{6a}
\# \{j: E_j(\h) \leq C_1 \} \leq C_2 \h ^{-N_0} \ .
\ee
}
This turns out to be sufficient because of the restriction in the
summation over $j$ introduced just now.
The validity of (\ref{6a}) for Schr\"odinger operators follows from
rough estimates on the number of eigenvalues (e.g. by min-max)
which show that we can in fact take $N_0 = n$.

\subsection{Proof of theorem 1.3}

We want to apply theorem \ref{5.III} with $\omega (\h )$ equal to either
$\h ^{-(n-1)} \log \h^{-1}$ or
$\h ^{-(n-1)}$. Theorems \ref{ONE} and \ref{TWO} give
\ref{5.I}(i) if $\hat{\varphi }$ has sufficiently small support.
To obtain (\ref{5.3}) for arbitrary $\varphi $ we have to show that
if $0$ is not in $\mbox{supp}\, \hat{\varphi }$ then
\be \label{13}
\Upsilon _{\h } = \Upsilon _{E_c, \h }(\varphi ) \ = o(f(\h )),
\ee
$f(\h )$ being the leading order term in the expansion of theorem 1.1.
We split $\Upsilon _{\h }$ as $\Upsilon _{1, \h } + \Upsilon _{2, \h}$
as in section 2. Since we assumed that set of periodic points of the
Hamilton flow on
$\Sigma _{E_c} \setminus \Theta $ has
measure $0$,
it follows by known arguments (cf. \cite{DG})
that
\be \label{14}
\Upsilon _{2, \h} = o(\h ^{-(n-1)}).
\ee

Next, since by assumption $\Theta $ is a clean fixed point set of the flow
for all times $t \neq 0$, it follows easily using the techniques of
section 4.3 or of section 5 in \cite{GU} that
\be \label{15}
\Upsilon _{2, \h} = \calO (\h ^{-(n - {N \over 2})})
\ee
(compare formula (\ref{4bb5}) which shows that $\Upsilon _{E_c}$ has
symbol of order
$n - {N \over 2}$).
Together, (\ref{14}) and (\ref{15}) imply  (\ref{13}).

Assumption \ref{5.I}(ii) is trivially satisfied. Next, \ref{5.I}(iii) follows
from corollary \ref{III.5} if $N = 2$ and the Hessian is indefinite,
and is proposition 5.1 if $N > 2$.
Finally, \ref{5.I}(v) follows for example from known Weyl-type estimates on the
number of eigenvalues of $A_{\hbar }$ which can, for example, be found in
\cite{R}.
Hence all assumptions \ref{5.I} are satisfied. and Theorem \ref{THREE} is
an immediate consequence of Theorem \ref{5.III}.


\section{Eigenfunction Estimates}

\noindent
{\bf Proof of Theorem 1.4.}
Theorem 1.4 is proved in exactly the same way as Theorems 1.1 and 1.2.
One only needs to replace the operator
$\chi(\tilde{P})\exp(i(-tD_\theta\tilde{P}+(s+tE)D_\theta)$
of \S 2 with the operator
\be\label{uno}
A\,\chi(\tilde{P})\exp(i(-tD_\theta\tilde{P}+(s+tE)D_\theta)\, ,
\ee
where $A$ is a zeroth order $\Psi$DO on $M\times S^1$
such that $A$ restricted to the $k$-eigenspace of $D_\theta$
is $\mbox{Op}\,(a,\h)$ where $\h=1/k$.	(The definition of
$\mbox{Op}\,(a,\h)$ is precisely through the construction of such an $A$.)
The operator (\ref{uno}) is a Fourier integral operator of the
same kind as the one we have been working with (i.e. with $A=I$).
Thus all statements about phase functions and wave-front sets
hold unchanged for (\ref{uno}) and the distributions $\Upsilon$
constructed from it.  The only change is that in the symbol
of the operator, which gets multiplied by the symbol of $A$.
This has the effect of changing the formulae for the coefficients
$c_{j,l}$, as stated in Theorem 1.4.

\bigskip\noindent
{\bf Proof of Theorem 1.5.}
The proof of Theorem \ref{FOUR} parallels those of \cite{CdV2} and
\cite{HMR}, and therefore we will be rather sketchy.
The first ingredient in the proof is a positive quantization method.
The operator $\Op (a,\h)$ is,  for $\h = 1/m$
the restriction to
$\calH_m = L^2(M)\otimes \bbC e^{im\theta}\subset L^2(M\times S^1)$
of a zeroth order pseudodifferential operator on $M\times S^1$ with principal
symbol $a(x,\xi/\tau)$ away from a conic neighborhood of $\tau = 0$;
here $(x,\xi)\in T^*M$ while $(\theta, \tau)\in T^*S^1$. (This is
easily extended to a procedure for arbitrary $\h $)
Let us denote the latter operator by $a(x,D_\theta^{-1}D_x)$.
To define a positive quantization,  it suffices to quantize the symbol
$a(x,\xi/\tau)$ by Friedrich's method, obtaining thereby an operator denoted
$a^F(x,D_\theta^{-1}D_x)$, satisfying
\[
a\geq 0 \quad\Rightarrow\quad a^F(x,D_\theta^{-1}D_x)\geq 0\,.
\]
Obviously, by restricting $a^F(x,D_\theta^{-1}D_x)$ to $\calH_m$ we still
get a positive operator,
\[
\Op ^F(a,1/m)\;=\; a^F(x,D_\theta^{-1}D_x)|_{\calH_m}\,.
\]
Since the difference $a(x,D_\theta^{-1}D_x)- a^F(x,D_\theta^{-1}D_x)$
is of order $(-1)$, it is easy to check that
\[
< \psi_j^{\h }\,,\, [\Op (a,\h) - \Op ^F (a,\h)] \psi_j^{\h } >\; =\;
O(\h )\,.
\]
So it is enough to prove the Theorem for
$\langle \psi_j^{\h }\,,\, \Op^F (a,\h)\psi_j^{\h }\rangle$.
Write, using Theorem \ref{THREE'},
\[
\sum_{j\in J} \langle \psi_j^{\h}\,,\, \Op^F (a,\h)\psi_j^{\h}\rangle\;
\varphi\left(\,{E_j(\h) - E_c\over\h} \right)\,\sim\,
c_0(\varphi)\omega(\h) m(a) + o(\omega(\h))\,
\]
as $\h\to 0$, where we have introduced the notation
\[
J\, =\, J(E_c, c,\h) \,=\, \{\, j\, ;\, |E_j(\h)-E_c|<c\h\,\}\,.
\]
The formulae for $c_0$, $m$ and $\omega$ are given in Theorem \ref{THREE'},
here we simply note that $m$ is either a constant multiple of the Liouville
measure or of the measure
$d\mu_0$ on $\Theta $.	It follows that
\[
w_j(a,\h)\;\stackrel{def}{=}\;
\langle \phi_j^{\h}\,,\, \Op^F (a,\h)\phi_j^{\h}\rangle
\]
satisfy the hypotheses of the Tauberian Theorem \ref{5.III} provided
$a\geq 0$. Hence, given the hypotheses on the flow, theorem \ref{5.III}
implies as before that
\[
\sum_{j\in J} \langle \psi_j^{\h}\,,\, \Op^F (a,\h)\psi_j^{\h}\rangle
\, = \, 2c m(a) \omega (\h) + o(\omega (\h )) \, ,\;\h\to 0 \ .
\]
Hence, using Theorem \ref{THREE},
we conclude that if $a\geq 0$, then
\be\label{g.1}
\lim_{\h\to 0}\,{1\over N(\h)}\,
\sum_{j\in J} \langle \psi_j^{\h}\,,\, \Op^F (a,\h)\psi_j^{\h}\rangle
\,=\, { {m(a)} \over {m(1)} }\,.
\ee
The restriction $a\geq 0$ can then be lifted, as follows.
The fact that the quantization $\Op ^F$ is positive,
and the Riesz representation theorem, imply that the map
\[
a\mapsto \langle \phi_j^{\h}\,,\, \Op^F (a,\h)\phi_j^{\h}\rangle
\]
defines a positive Borel measure of mass $1$ on $T^*X$, which we
denote by $\calV^\h_j$, so that
\[
w_j(a,\h)\,=\,\langle \psi_j^{\h}\,,\, \Op^F (a,\h)\psi_j^{\h}\rangle\,=\,
\int_{T^*M} a\, d\calV^\h_j\,.
\]
In terms of these measures, (\ref{g.1}) says that the Borel probability
measures
\[
{1\over N(\h)}\,\sum_{j\in J} \calV^\h_j
\]
converge weak*
as $\h\to 0$ to the measure $m/m(1)$ on the positive smooth functions.
This statement extends to the positive continuous functions by a
density argument and the fact that we are dealing with probability measures.
Finally,  by linearity, we can conclude that {\em (\ref{g.1}) holds for
all continuous functions a}.


\medskip
The next step is to establish the estimate
\be\label{g.2}
\int_{T^*M} (a\circ\phi_t - a)\, d\calV^\h_j\,=\,
O_t(\h)\,,
\ee
where $\{ \phi_t\}$ is the classical Hamiltonian flow of the system.
(This step is superfluous if $N=2$, but for brevity of exposition
we treat both cases $N>2$ and $N=2$ at once.)
This means that in the semi-classical limit the measures $\calV^\h_j$
are invariant, and follows from the Egorov-type theorem
of the appendix in \cite{PU2}, proved
for a suitable microlocalization of  $e^{i t S_{\h}/\h}$.

\bigskip
Let us now fix $a\in\calS (T^*M)$.  Notice that it suffices to
prove the Theorem for $a+c$ with $c$ an arbitrary constant, and
so without loss of generality we may assume that $m(a)=0$.
We must prove that
\be\label{g.2a}
\lim_{\h\to 0} {1\over N(\h)} \sum_{j\in J} |w_j(a,\h)|\,=\,
0\,.
\ee
By (\ref{g.2}), for every $T>0$
\[
|w_j(a,\h)|\,=\,
\left|\,{1\over T}\int_{T^*M} \int_0^T a\circ\phi_t\, dt\,d\calV^\h_j
\,\right| + O(\h)\,,
\]
and hence
\be\label{g.3}
\sum_{j\in J} |w_j(a,\h)|\,=\,
\sum_{j\in J}
\left|\,{1\over T}\int_{T^*M} \int_0^T a\circ\phi_t\, dt\,d\calV^\h_j
\right| + O(\h)N(\h)\,.
\ee
By positivity of the $\calV$,
\[
\sum_{j\in J} |w_j(a,\h)|\,\leq\,
\sum_{j\in J}
\int_{T^*M}\left|\, {1\over T}\int_0^T a\circ\phi_t\, dt\,\right|\,
d\calV^\h_j
+ O(\h)N(\h)\,,
\]
and so, if we divide by $N(\h)$ and let $\h\to 0$ we obtain
\be\label{g.4}
\lim_{\h\to 0}\,{1\over N(\h)}
\sum_{j\in J} |w_j(a,\h)|\,\leq\,
\lim_{\h\to 0}\,{1\over N(\h)}
\sum_{j\in J}
\int_{T^*M}\left|\, {1\over T}\int_0^T a\circ\phi_t\, dt\,\right|\,
d\calV^\h_j
\ee
and the second term is
\be\label{g5}
{ 1 \over {m(1)} } \,
m\left(\left|\, {1\over T}\int_0^T a\circ\phi_t \, dt\right|\,\right)\,,
\ee
by the extension of (\ref{g.1}) to continuous functions.
Let us now examine each of the following cases:

\smallskip\noindent
(a) If $N>2$, (\ref{g5}) is
\[
{1\over \mbox{LVol}}\,
\int_{\Sigma_E} \left|\, {1\over T}\int_0^T a\circ\phi_t \, dt\right|\,
d\lambda\,,
\]
which tends to
\[
{1\over \mbox{LVol}(\Sigma_E)}\int_{\Sigma_E} a\, d\lambda\,=\,0
\]
as $T\to\infty$, if the flow on $\Sigma_{E_c}$ is ergodic.

\smallskip\noindent
(b) If $N=2$ and $a$ is constant equal to zero on $\Theta$, so is
(\ref{g5}) since in this case it equals
\[
{1\over \mu_0(\Theta)}\,
\int_{\Theta} \left|\, {1\over T}\int_0^T a\circ\phi_t \, dt\right|\,
d\mu_0\,=\,
\int_{\Theta} |a|\,d\mu_0\,.
\]

In both cases we have established (\ref{g.2a}), and the proof of
Theorem \ref{FOUR} is complete.


\appendix
\section{Appendix: The Density $\mu_t$.}
\newcommand{\coker}{\mbox{coker}\,}
\newcommand{\Image}{\mbox{Im}\,}

The purpose of this Appendix is to discuss the nature of the
measure $\mu_t$ on the critical manifold $\Theta$.
The former is a special case of the density associated to
a clean fixed-point manifold of a symplectomorphism, as defined by
Duistermaat and Guillemin in \cite{DG}, Lemma 4.3.  Here we will simply
amplify their definition.

\medskip
Fix once and for all a point $z\in\Theta$.  We will describe the
density $\mu_t$ on the tangent space $T_z\Theta$.  To simplify
notation, let
\[
W:= T_z\Theta\,\quad
V:= T_zX\, ,\quad\mbox{and}\quad F_t := d(f_t)_z\, .
\]
In the definition of $\mu_t$ the time $t$ is a parameter, which will mostly
be fixed and therefore omitted from the notation.
Since $\Theta$ is a clean fixed point set of $\{f_t\}$, for small $t$,
\be\label{An0}
W\;=\; \ker (I-F)\, .
\ee

\subsection{The Duistermaat-Guillemin construction}

We review Lemma 4.3 in \cite{DG}, regarding the symplectic linear
algebra setting introduced above: Let $V$ be a symplectic vector
space, $F:V\to V$ a linear symplectomorphism.  Write
\[
T\;=\; I-F\quad\mbox{and}\quad W = \ker (T)\,.
\]

\begin{lemma}\label{Ap1}
The image of $T$ is the symplectic orthogonal to $W$,
$W^{\bot}$.
\end{lemma}
\begin{proof}
Let $u\in W$ and $v\in V$. Then, if $\omega$ denotes the symplectic
form on $V$,
\[
\omega (u, (I-F)(v))\;= \; \omega(u,v) - \omega (F^{-1}(u), v)\;=\; 0\, .
\]
Hence $\Image (T)\subset W^{\bot}$, but by a dimension count they must
be equal.
\end{proof}
It follows that
\be\label{ap1}
\coker (T)\; =\; V/(W^{\bot}) \; \cong \; W^*\,,
\ee
where the isomorphism $V/(W^{\bot}) \; \cong \; W^*$ is induced
by the symplectic form.  Hence there is a natural isomorphism
\be\label{ap2}
|W|^{1/2}\otimes |\coker (T)|^{-1/2}\;\cong\; |W|^1\,.
\ee
Specifically, if $\alpha\in |W|^{1/2}$ and $\beta\in |\coker (T)|^{-1/2}$,
then $\alpha\otimes\beta$ is the density on $W$ which, to a given
ordered basis $e = (e_1,\ldots e_k)$ of $W$ associates the number
\be\label{ap3}
(\alpha\otimes\beta)\,(e)\;=\; \alpha (e)\cdot\beta(e^*)\,,
\ee
where $e^*$ is the basis of $W^*$ dual to $e$, and it is considered
as a basis of $\coker (T)$ via (\ref{ap1}).

\medskip
Consider next the exact sequence
\be\label{ap4}
0\to \ker (T) \to V \stackrel{T=I-F}{\to} \ V\to \coker (T)\to 0
\ee
which induces a trivialization
\be\label{ap5}
|\ker (T)|^{1/2}\otimes |V|^{-1/2}\otimes |V|^{1/2}\otimes |\coker (T)|^{-1/2}
\;\cong\;\bbR\,.
\ee
Since $|V|^{-1/2}\otimes |V|^{1/2}\cong\bbR$ naturally, we get that
\be\label{ap6}
|\ker |^{1/2}\otimes |\coker (T)|^{-1/2}\;\cong\;\bbR\,,
\ee
and hence, by (\ref{ap2}), we obtain an isomorphism
\be\label{ap7}
|\ker (T)|^1\;\cong\;\bbR\,.
\ee
The natural density $\mu$ on $W=\ker (T)$ is then the one associated to
$1\in\bbR$, under (\ref{ap7}).	We will call it the
{\em Duistermaat-Guillemin density.}

We next give a more pedestrian
description of this density,
when evaluated on an arbitrary basis $e=(e_1,\ldots ,e_k)$
of $W$.
Choose systems of vectors in $V$, as follows:

\begin{itemize}
\item Vectors $f = (f_1,\ldots ,f_k)$ basis of a complement of $W^{\bot}$ such
that
\be\label{ap7.1}
\omega(e_i , f_j)\;= \; \delta_{ij}\,.
\ee
\item Vectors $v=(v_1,\ldots ,v_{2n-k})$ basis of a complement of $W$
in $V$.
\end{itemize}

If we let $\nu \in |V|^{1/2}$ be an arbitrary half-density then, chasing
through the definition of the DG density, we have

\begin{lemma} \label{Ap1.0} The DG density evaluated on the basis $e$ gives
\be  \label{app7.a}
\mu (e) = { {\nu (v \wedge e)} \over {\nu (Tv \wedge f)} },
\ee
where we have let $e = \wedge _j e_j$, etc.
\end{lemma}

\noindent
{\bf Remark.}  Implicit in the statement is that (\ref{app7.a}) is independent
of the choice of $f, v, \nu $ as long as (\ref{ap7.1}) is satisfied.

\medskip
As an application of this lemma we now prove that the DG density
$\mu _t$ is invariant under time-reversal:

\begin{lemma} \label{Ap1.00}
If $\mu _t$ is the DG density on $W$ associated to the
symplectic flow $F_t$ then
\[
\mu _t = \mu _{-t} \ .
\]
\end{lemma}

\begin{proof} Let $e, \ v$ and $f$ be as in lemma \ref{Ap1.0}. Note that
$T_{-t} = - T_t F_{-t}$. Hence, observing that $F_{-t}v$ spans a complement
of $W$ and that $F_{-t} e_j = e_j$ since $F_t = I$ on $W$, lemma \ref{Ap1.0}
gives:
\[
\mu _{-t} (e) = { {\nu (v \wedge e)} \over {\nu (T_t F_{-t} v \wedge f)} }
\]
\[
= { {\nu (F_{-t}v \wedge e)} \over {\nu (T_t F_{-t} v \wedge f)} }
\cdot
{ {\nu (v \wedge e)} \over {\nu (F_{-t} (v \wedge e))} }
\]
\[
= \mu _t(e) \, ,
\]
since $\mbox{det} \, (F_{-t}) = 1$ (because $F_t$ is symplectic).
\end{proof}

\begin{example} \rm{ We compute the DG density $\mu _t$ if
the codimension of $W $ is 2.
By the symplectic classification of quadratic forms
(cf. e.g. \cite{Ho3}, theorem 21.5.3)
one then can find linear symplectic coordinates
$(x_1, \ldots, x_n, \xi _1, \ldots , \xi _n)$ on $V$ such that the Hessian
$d^2_zH $ at $z$ is
\[
{1 \over 2} \alpha ^2 (x_1^2 + \xi _1 ^2) \ \ \ \mbox{or} \ \ \
x_1^2 + x_2^2
\]
if it is positive semi-definite, and is
\[
\alpha x_1 \xi _1
\]
if it is indefinite (and therefore hyperbolic). Here $\alpha $ is some
positive number.
Making obvious choices of bases in lemma \ref{Ap1.0},
an easy computation shows that in these cases, respectively
\[
\mu _t = (2 - 2\cos \alpha t)^{-1/2} dx_2 \wedge \ldots \wedge dx_n \wedge
d\xi _2 \wedge \ldots \wedge d\xi _n \, ,
\]
\[
\mu _t = (2|t|)^{-1} dx_3 \wedge \ldots \wedge d\xi _n
\]
and
\[
\mu _t = (2 - 2\cosh (\alpha t) )^{-1/2} dx_2 \wedge \ldots \wedge dx_n \wedge
d\xi _2 \wedge \ldots \wedge d\xi _n \ .
\]
One can do similar calculations if $N > 2$, using the classification.
}
\end{example}


\subsection{A special case.}

In this section we will write down the density $\mu $ still more concretely
while making two simplifying assumptions, the first of which is
\be \label{iso}
\mbox{Assume}\ \Theta\ \mbox{is isotropic.}
\ee
This is certainly the case for critical manifolds of classical Hamiltonians
$H(q,p) = {1\over 2} |p|^2 + V(q)$, since then necessarily $\Theta\subset
\{ p=0\}$.

\medskip
We choose a basis $g = (g_1, \ldots , g_{2n-2k})$ of a complement of $W$ in
$W^{\bot}$ and bases $e$ and $f$ as in lemma \ref{Ap1.0}. If we choose for
$v$ the system of vectors $(g, f)$ and make a convenient choice of $\nu $,
this lemma then gives:

\begin{lemma}\label{Ap1.1}
If $T_{[f,g],[e,g]}$ denotes the matrix of the mapping $T=I-F$ between the
basis $(f,g)$ and $(e,g)$.  Then the DG density evaluated on $e$ gives
\be\label{ap7.2}
{1\over |\det T_{[f,g],[e,g]}|^{1/2}}\, .
\ee
\end{lemma}

Notice that $e,f,g$ is a basis of $V$.	The span of $g$ is a symplectic
subspace, realizing the symplectic normal space $W^{\bot}/W$, and $(e,g)$
is a basis of $W^{\bot}$.  The
span of $(f,g)$ is a realization of the normal space to $\Theta$ at $z$.
By Lemma \ref{Ap1}, the mapping $T=I-f$ maps the span of $(f,g)$ onto
$W^{\bot}$.

\bigskip
It would be desirable to have an invariant choice of the subspaces
$\mbox{span} (e,f)$ and $\mbox{span} (g)$.
Such exists under the following assumption:
\be \label{app8a}
\mbox{Assume that}\
\quad \ker (T^2) \; = \; \ker (T^3) \,.
\ee
This immediately implies
\be \label{ap8}
\forall l \in \bbZ ^+ \quad
\ker (T^2)\;=\; \ker (T^{2+l}) \, ,
\ee
as is easily proved by induction. Assumption (\ref{app8a})
is also verified by the linearization of $H(q,p) = {1\over 2}
|p|^2 + V(q)$.	We will assume both (\ref{iso}) and (\ref{app8a})
until the end of this subsection.

\begin{lemma}\label{Ap2}
Let $\overline{W} := \ker (T^2)$ and $S := \Image (T^2)$.  Then
\be\label{ap9}
V\;=\; \overline{W}\oplus S\,,
\ee
\be\label{ap10}
W\subset\overline{W}\quad \mbox{and}\quad
W^{\bot}\;=\; W\oplus S\,.
\ee
Furthermore both $\overline{W}$ and $S$ are symplectic subspaces invariant
under $F$.
\end{lemma}
\begin{proof}
We begin by making the trivial remarks that $\ker (T)\subset \ker (T^2)$
and that $\Image (T^2)\subset \Image (T)$, which translate into
\[
W\subset \overline{W}\quad\mbox{and}\quad S\subset W^{\bot}\,.
\]
Invariance of $S$ and $\overline{W}$ under $T$, and thus under $F$,
is immediate from the definitions.

\smallskip
To prove (\ref{ap9}), assume $v\in \overline{W}\cap S$.  Then $\exists u\in V$
such that $v=T^2(u)$ and $T^4(u)=0$.  By, (\ref{ap8}) $u\in\ker (T^2)$
and so $v=0$.  Hence $\overline{W}\cap S = \{ 0\}$ and (\ref{ap9}) follows.

\smallskip
We will next prove that
\be\label{ap11}
\overline{W}\cap W^{\bot}\;=\; W\,.
\ee
The inclusion $\supset$ is obvious, so take $v\in\overline{W}\cap W^{\bot}$.
Since $W^{\bot} = \Image (T)$, $\exists u\in V$ such that $v=T(u)$.
Since $v\in\overline{W}$, $0=T^2(v) = T^3(u)$, and again by
(\ref{ap8}) $u\in\overline{W}$ and so $0=T^2(u)=T(v)$, i.e. $v\in W$.
This proves (\ref{ap11}).

\smallskip
To prove the second part of (\ref{ap10}), notice that by (\ref{ap9})
$W\cap S =\{ 0\}$, and that the inclusion $\supset$ is trivial.  If
$v\in W^{\bot}$, then write by (\ref{ap9}) $v=\overline{w} +s$, but since
$S\subset W^{\bot}$, $\overline{w}\in \overline{W}\cap W^{\bot}= W$,
by (\ref{ap11}).  Thus $W^{\bot}= W\oplus S$, which also implies that
$S$ is symplectic.

\smallskip
It remains to be proved that $\overline{W}$ is symplectic.
Let $v\in \overline{W}\cap(\overline{W})^{\bot}$.
Since $W\subset\overline{W}$, $(\overline{W})^{\bot}\subset W^{\bot}$.
By (\ref{ap11}), $v\in W$.  Now pick $u\in V$ arbitrary; we will prove
that $\omega (v,u)=0$; it will follow that $v=0$ and the proof will be
complete.  Decompose $u=\overline{w}+s$ according to (\ref{ap9}).
Since $v\in \overline{W}^{\bot}$, $\omega (v,\overline{w})=0$, and
since $v\in W$ and $s\in S\subset W^{\bot}$, $\omega (v,s)=0$. Hence
$\omega (v,u) =0$.

\end{proof}

\begin{corollary}\label{Ap3}
The mapping $T$ induces an exact sequence
\[
0\to W \hookrightarrow \overline{W} \stackrel{T}{\to} W \to 0 \; .
\]
\end{corollary}
\begin{proof}
The only non-trivial statement is that $W \subset T(\overline{W})$.
Since $W\subset W^{\bot} = \Image (T)$, given $w\in W$ $\exists v\in V$
such that $w= T(v)$.  Decompose $v=\overline{w} + s$, following
(\ref{ap9}).  Since the decomposition is invariant and $W\subset\overline{W}$,
we must have $w=T(\overline{w})$ (and $T(s)=0$).
\end{proof}

\medskip
The density $\delta$ on $W$ induced by $F$ will now be described in
terms of the restriction of $T$ to $\overline{W}$ and to $S$.

\medskip
The direct sum relations of Lemma \ref{Ap2} imply that the dimension of
$\overline{W}$ is twice that of $W$.
Hence $W$ is a Lagrangian subspace of $\overline{W}$, and the
symplectic form induces an isomorphism
\be\label{ap12}
W^*\;\cong\; \overline{W}/W\, .
\ee
On the other hand, by Corollary \ref{Ap3}, the map $T$ induces an
isomorphism
\be\label{ap13}
\overline{T}\;: \overline{W}/W\to W\;.
\ee
Putting these two together, we get a map
\be\label{ap14}
D\, : \, W^* \to \overline{W}/W \stackrel{\overline{T}}{\to} W
\ee
which is a linear isomorphism.	We'll see that this defines a density on
$W$.  If $k=\mbox{dim}\,W$, consider the $k$-fold exterior product
\be\label{ap15}
\wedge^kD : \wedge^kW^* \to \wedge^k W\,.
\ee
We must introduce some notation.  If $e=(e_1,\ldots e_k)$, let
$\wedge e := e_1\wedge\cdots \wedge e_k$, and let $e^* = (e^*_1,\ldots,
e^*_k)$ be the basis of $W^*$ dual to $e$, that is defined by the conditions
\[
( e^*_i\,,\,e_j)\;=\; \delta_{ij}\,.
\]
Finally, recall that $\wedge^k W^* \cong (\wedge^k W)^*$.

\begin{lemma}\label{Ap4}
The formula
\[
\delta (e)\;= \; {1\over | (\wedge e^*\, ,\,\wedge^k D(\wedge e^*)) |^{1/2}}
\]
defines a density, $\delta$, on $W$.
\end{lemma}
\begin{proof}
Let $f$ be a different basis of $W$, and $M\in \mbox{GL}(k,\bbR)$ the matrix of
passage from $e$ to $f$.  Then $M^{-1}$ is the matrix from $e^*$ to $f^*$,
and so
\[
\wedge f^*\;= \; \det (M)^{-1} \cdot \wedge e^*\,,
\]
which implies that $\delta (f) = \det (M)\delta (e)$.
\end{proof}
To see the relation with the DG density of
Lemma \ref{Ap1.1}, let $e$ be a basis of $W$ and complete
it to a {\em symplectic} basis $e,f$ of $\overline{W}$.
Then (\ref{ap11}) implies that $f$ induces a linearly independend system
in $V / W^{\bot }$ and thus spans a complement of $W^{\bot }$.
The definition
of the map $D$ implies that
\be\label{ap15.1}
D(e^*_i)\;= \; T(f_i)\,,
\ee
and hence that $(\wedge e^*\, ,\,\wedge^k D(\wedge e^*))$ equals the
determinant of $T$ between the basis $f$ and $e$.

\medskip
Consider, on the other hand, the symplectic linear map induced by $F$
on $S$:
\be\label{ap16}
F^{\bot} : S\to S\, .
\ee
$(I-F^{\bot}) : S\to S$ is invertible since $S\cap W = \{ 0\}$.
Then:

\begin{proposition}\label{Ap5}
The Duistermaat-Guillemin density on $W$ is
\[
\mu\;=\; {\delta \over |\det (I-F^{\bot})|^{1/2}}\,.
\]
\end{proposition}

\subsection{Small $t$ Behaviour.}

Recall that we mean to apply the discussion of the previous subsections
to $F = F_t = d(f_t)_z$ for fixed $z\in\Theta$ and small $t$.  We
now bring back the time into the picture, and show that $|t|^{N/2}\mu_t$
is smooth at $t=0$.

Since $W = \ker (I - F_t)$ is independent of $t$, for $t$ small, we can pick
systems of vectors $e, \ f $ and $v$ as in lemma \ref{Ap1.0} which are
also independent of $t$.
Since $F_t = \exp (t A)$, for some infinitesimal symplectic linear
transformation $A$ on $V$,
\be\label{ap17}
T_t\;=\; I-F_t\;=\; t\,U_t
\ee
for a smooth family $U_t$ of linear maps, namely
\be\label{ap18}
U_t\;=\; \sum_{l=0}^\infty {t^l\over (l+1)!}\,A^{l+1}\,.
\ee
Hence if $\nu \in |V|^{1/2}$ is arbitrary then
$\nu (T_tv \wedge f) = |t|^{N/2} \nu (U_tv \wedge f)$, where
$N = \mbox{codim} \, W$ as before, and therefore
\[
|t|^{N/2} \mu _t (e) = { {\nu (v \wedge e) } \over {\nu (U_tv \wedge f)} } \, .
\]
Next, $t \to \left( \nu (U_tv \wedge f) / \nu (v \wedge e) \right)^4$ is an
analytic function of $t$ which is non-zero in $t = 0$, since
$U_0 = A$ has kernel precisely $W$. This last statement follows from the
well-known fact that
the infinitesimal generator $A$ of the linearized flow
is the linear transformation on $V$ dual via the symplectic form
to the Hessian:
\[
\omega( A(v), w)\;=\; d^2H(v,w) ,
\]
and $W$ is the radical of the Hessian, by the Main Hypothesis in section 1.
Hence

\begin{corollary}
$|t|^{N/2}\mu_t$ is a smooth $t$-dependent density for $t$ in a neighborhood
of $0$.
\end{corollary}




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