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\noindent
DESY 95--090 \hfill ISSN 0418 -- 9833\\
May 1995
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\centerline{{\LARGE The Trace Formula and the Distribution of Eigenvalues of}}
\medskip
\centerline{{\LARGE Schr\"odinger Operators on Manifolds all of whose}}
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\centerline{{\LARGE Geodesics are closed}}
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\bigskip
\centerline{{\large Roman Schubert}}
\bigskip
\centerline{{\large II. Institut f\"ur Theoretische Physik}}
\smallskip
\centerline{{\large Universit\"at Hamburg}}
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\centerline{{\large Luruper Chaussee 149}}
\smallskip
\centerline{{\large 22761 Hamburg}}
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\begin{abstract}
We investigate the behaviour of the remainder term
$R(E)$ in the Weyl formula
$$
\# \{n|E_n\le E\}=
\frac{\mbox{Vol}(M)}{(4\pi )^{d/2}\, \Gamma(d/2+1)}\, E^{d/2}+R(E)
$$
for the eigenvalues $E_n$ of a Schr\"odinger operator on
a d-dimensional compact Riemannian manifold all of whose
geodesics are closed.
We show that $R(E)$ is of the form $E^{(d-1)/2}\,\Theta(\sqrt{E})$,
where $\Theta(x)$ is an almost periodic function of Besicovitch class $B^2$
which has
a limit distribution whose density is a box-shaped function.
This is in agreement with a recent conjecture of Steiner \cite{S,ABS}.
Furthermore we derive a trace formula and study higher order
terms in the asymptotics of the coefficients related to the periodic orbits.
The periodicity of the geodesic flow leads to a very simple structure
of the trace formula which is the reason why the limit
distribution can be computed explicitly.
\end{abstract}
\end{titlepage}
\newpage
\section{Introduction}
In the theory of quantum chaos \cite{G} there has been intensive
investigations of the statistical properties of the energy
levels of quantum mechanical systems. Special emphasis has
been on the relation to the classical limit $\hbar \to 0$.
It has been found out that the
statistical properties of the energy levels strongly depend
on the ergodic properties of the associated classical system.
Short range correlations in the spectrum are in case of classically
integrable systems well
described by a Poissonian distribution, whereas
for classically chaotic system the distributions
obtained from random matrix theory fit quite well.
But these models fail to describe long range correlations,
and also for short range correlation
exceptions are known. The so called arithmetic systems,
which are classically chaotic, possess some statistical
features very similar to integrable systems \cite{BSS}.
A statistical measure which gives a clear distinction between
classically chaotic and classically integrable systems was
missing until now.
Inspired by number theoretical results \cite{HB} there has been
recently a number of investigations of a global statistical
measure in case of integrable systems \cite{BCDL,B1,B2,B3,KMS,BKS}.
It is conjectured in \cite{S}, and numerically tested for some
chaotic systems in \cite{ABS}, that this measure provide
a classification
of quantum mechanical systems according to their classical limit.
To describe it we introduce some notation.
Let $(M,g)$ be a compact, d-dimensional
Riemannian manifold, with metric $g$, and $\Delta_g $ its
Laplace-Beltrami operator, defined by
$\Delta_g f=\mbox{div}_g(\mbox{grad}_g f)$.
Denote by $E_0 \le E_1 \le E_2 \dots\, $ the eigenvalues of $-\Delta_g $,
counted with multiplicities, and by
$N(E)=\# \{j|E_j \le E\}$ the counting-function.
The asymptotic behaviour of $N(E)$ for $E \to \infty$ is given by the famous
Weyl law \cite{H},
%
\begin{equation}
N(E) = \frac{\mbox{Vol}(M)}{(4\pi )^{d/2}\,\Gamma(d/2+1)}\,
E^{d/2} +O(E^{(d-1)/2})\quad .
\end{equation}
%
The remainder term is in general a very wild function
and difficult to study, but for certain two-dimensional integrable systems
the authors of the above cited articles were able to show that
$N(E)$ can be decomposed as follows:
%
\begin{equation}\label{Liouv}
N(E) = \frac{1}{4\pi }\mbox{Vol}(M) E+E^{1/4}\Theta (\sqrt{E})
\end{equation}
%
where $\Theta (x)$ is an almost periodic function of Besicovitch class $B^1$
(i.e., the mean value of $|\Theta (x)|$ exists, see below for
more information on $B^1$)),
whose frequencies are the lengths of the closed geodesics.
The first term in (\ref{Liouv}) describes the mean behaviour of
$N(E)$, while
the second term generally oscillates around zero with an amplitude
growing like $E^{1/4}$. So $\Theta (x)$ describes the normalized fluctuations
of $N(E)$ around the mean behaviour, and to get a measure of this
fluctuations one can ask for the limit distribution of $\Theta (x)$.
The second main result of the above cited papers is that
the function $\Theta (x)$ has a limit distribution,
i.e., there exists a non-negative function $p(x)$, such that for every bounded
continuous function $g(x)$, and any density of a probability distribution
$\rho(x) $, concentrated on $[0,1]$,
%
\begin{equation}\label{lim}
\lim_{T\to \infty} \frac{1}{T} \int_0^T g(\Theta (x))\rho (x) dx=
\int_{-\infty}^{\infty} g(x)p(x)dx\quad .
\end{equation}
%
The density $p(x)$ of the limit distribution is an entire
function of $x$, possessing the following asymptotic behaviour
on the real line,
%
\begin{equation}\label{nog}
\ln (p(x)) \sim -C_{\pm}x^4 \qquad x \to \pm \infty, \quad C_{\pm}>0 \quad .
\end{equation}
%
Several numerical studies have shown that $p(x)$ can look quite
different for different integrable systems \cite{B3}. But for classically
chaotic systems
numerical tests suggest that there is a universal behavior,
$p(x)$ was always found to be in excellent agreement with a normalized
Gaussian distribution, in
contrast to (\ref{nog}).
The conjecture has been formulated \cite{S,ABS}, that the limit
distribution of the normalized remainder term of $N(E)$ is
always a Gaussian distribution for chaotic systems, while for integrable
systems it is non-universal and not Gaussian, and so
provides a distinction between classically integrable
and classically chaotic systems.
This conjecture is well supported by the so far known
analytical and numerical results \cite{ABS,ASS,BSS1,BSS},
including the results of this paper.
In this note we will study the limit distribution
for a class of Hamilton operators on compact manifolds,
which are characterized by the property that the flow generated
by their principal symbol is periodic. To be more precise
$P$ should be a classical pseudo-differential
operator of order two, positive, elliptic and self-adjoint,
with the property that
the Hamiltonian flow generated by its principal symbol is simply
periodic.
To simplify the notation, we will state most of the results only for
the case when $(M,g)$ is a compact Riemannian manifold and
$P=-\Delta_g +V(x)$ is a positive
Schr\"odinger operator on $M$ (in natural units
$\hbar =2m=1$),
where $\Delta_g$ is the Laplace Beltrami operator associated
with $g$
and $V(x)$ is a smooth ($C^{\infty}$), and therefore bounded, potential on $M$.
The principal symbol of $P$ is just the Hamiltonian
$H(x,\xi)=\sum g^{ij}(x)\xi_i \xi_j$ which generates the geodesic
flow on $M$, and the periodicity of the flow means that all geodesics
are closed. An example for such a manifold is the sphere $S^2$,
and a potential system is for instance the spherical pendulum.
Further examples will be given in section 2.
Note that the principal symbol is generally not the Hamiltonian given
by the classical limit $\hbar \to 0$, but differs from it by the potential energy.
This displays the fact that we will use high energy asymptotics, rather
than the limit $\hbar \to 0$, and, because of the bounded potential, the high
energy limit is ruled by the kinetic energy.
Our main result is:
\begin{Satz}
Let $(M,g)$ be a compact d-dimensional Riemannian manifold with simply
periodic geodesic flow of period $2\pi$. $P=-\Delta_g +V$ a positive
Schr\"odinger
operator on $M$, where $\Delta_g$ is the Laplacian associated with $g$,
and $V$ is
a smooth function on $M$.
Then $N(E)$ has the following representation
\begin{equation}\label{N}
N(E)=\frac{\mbox{\em Vol}(M)}{(4\pi)^{d/2}\,\Gamma(d/2+1)} \,
E^{d/2}+E^{(d-1)/2}\,\Theta (\sqrt{E}).
\end{equation}
Here $\Theta (x)$
is an almost periodic function of Besicovitch class $B^2$
with the Fourier series
\begin{equation}\label{Fser}
\Theta (x)=\frac{ib_1}{2\pi}
\sum_{\tiny\begin{array}{c}k\not= 0\\k\in {\bf Z}\end{array}}
\frac{e^{i\frac{\pi}{2}k\nu}}{k}e^{-2\pi ikx}\quad \mbox{\em mod}\,B^2 \,\, ,
\end{equation}
where $b_1=\frac{d\,\mbox{\scriptsize{Vol}}\, (M)}{(4\pi)^{d/2}\Gamma(d/2+1)}$,
and $\nu$ is the
common Maslov index {\em\cite{UZ,D-G}}
of the primitive periodic orbits.
$\Theta(x)$ has a limit distribution whose density is a box-shaped function:
\begin{equation}\label{step}\displaystyle
p(x)=\left\{\begin{array}{r@{\qquad}l}\displaystyle
0& -\frac{b_1}{2}\ge x \\
\frac{1}{b_1}& -\frac{b_1}{2}\le x \le \frac{b_1}{2} \\
0 & \quad \frac{b_1}{2} \le x \quad .
\end{array} \right.
\end{equation}
\end{Satz}
Some remarks are in order:
{\bf Remark 1}. The space $B^p$ is defined as follows. Denote by
$||.||_{B^p}$ $(p\ge 1)$ the seminorm
$$
||f||_{B^p}:=\left( \lim_{T\to\infty}\frac{1}{T}\int_0^T|f(x)|^p dx
\right)^{1/p}.
$$
A function $f(x)$ belongs to a Besicovitch class $B^p$ if
it can be approximated, in the above seminorm, by trigonometric
polynomials. One can show that this is equivalent to the existence
of a formal Fourier series $\sum_{k\in{\bf Z}}a_k e^{i\lambda_k x}$,
with $a_k\in{\bf C}, \lambda_k \in{\bf R}$, such that
$$
\lim_{N\to\infty}||f(x)-\sum_{|k|k_0$ the number of eigenvalues
in $I_k$ is
$m_k =R(k+\frac{\nu}{4}) $.
\end{Satz}
Denote by
\begin{equation}\label{cluster}
\mu_k(E) =\frac{1}{m_k}\sum_{E_i\in I_k}\delta(E-(E_i-\lambda_k^2))
\end{equation}
the normalized spectral-measure of the k'th eigenvalue-cluster.
Weinstein has shown \cite{W3} that for a Zoll Laplacian there is a
unitary operator $U$ such that
$$
U\Delta_g U^{-1}=\Delta_0 +A \quad ,
$$
where $\Delta_0$ is the canonical Laplacian on the sphere,
and $A$ is a pseudo-differential operator of order zero.
Let $a_{V}(x,\xi)$ be the principal symbol of $A+UVU^{-1}$ and
denote by
$$
\overline{a_{V}}(x,\xi)=\frac{1}{2\pi}\int_0^{2\pi}a_{V}(\Phi^t(x,\xi))dt
$$
the average of $a_{V}(x,\xi)$ over the geodesic flow $\Phi^t$,
generated by the canonical metric of $S^d$. In case of
$\Delta_g=\Delta_0$, $a_V$ clearly reduces to $a_V=V$, and this
construction applies also to any other CROSS.
The asymptotic distribution of the eigenvalues in the cluster can be
described now.
\begin{Satz}{\em\cite{W3,CV1}}
There exists a sequence of distributions $\{\beta_i\}_{i\in {\bf N}}$,
supported in $[-K,K]$,
such that for every $\varphi \in C^{\infty}({\bf R})$;
\begin{equation}
\langle \mu_k ,\varphi \rangle \sim \langle \beta_0 ,\varphi \rangle +
\langle \beta_1 ,\varphi \rangle\frac{1}{k^2} +
\langle \beta_2 ,\varphi \rangle\frac{1}{k^4} +
\cdots \quad k\to \infty \; ,
\end{equation}
and
\begin{equation}\label{aver}
\langle \beta_0 ,\varphi \rangle =\frac{1}{\int\!\! \int_{H(x,\xi)<1}dx\, d\xi }
\int\!\! \int_{H(x,\xi)<1} \varphi(\overline{a_{V}}(x,\xi))dx\, d\xi \quad .
\end{equation}
Here $H(x,\xi)=\sum g^{ij}(x)\xi_i \xi_j$ is the Hamilton function
which generates the canonical geodesic flow, and by
$\langle \beta ,\varphi \rangle$ we denote the application of the
distribution $\beta$ to the function $\varphi$.
\end{Satz}
\medskip
{\bf Remark}. The above theorems are, with slight modifications, also
valid in the more general pseudo-differential operator setting
(see \cite{H}, chapter 29.2). One can show that for every pseudo-differential
operator $P$, satisfying our assumptions, there is a pseudo-differential
operator $Q$ of order zero, such that the square-roots of the eigenvalues
of $P+Q$ are an arithmetic sequence ${\lambda}$. Then (\ref{aver}) is valid
with $a_V$ replaced by the principal symbol $q$ of $Q$.
$b_1$ is given by $\frac{d}{(2\pi)^d}\int\!\int_{p<1}dxd\xi$, where $p$
is the principal symbol of $P$, and $dxd\xi$ is the canonical measure in
phase space. In Theorem (2.3) additionally
terms of odd order can occur in the asymptotic expansion.
\bigskip
%We give a list of the relevant data for some CROSSes (taken from \cite{Be,CV1}%)
%in the table below.
\begin{table}
\begin{center}
$$
\begin{array}{r|c|c|c|c|c}
M &\dim M &\nu &E_k &\lambda_k & b_1 \\ \hline
S^n & n &2(n-1) &k(k+n-1) & k+\frac{n-1}{2}& \frac{2}{(n-1)!} \\
%P^n{\bf R} & n &n-1 &k(k+\frac{n-1}{2})& k+\frac{n-1}{4}& \\
P^n{\bf C} & 2n &2n &k(k+n) & k+\frac{n}{2}& \frac{2}{n!(n-1)!} \\
P^n{\bf H} & 4n &4n+2 &k(k+2n+1)& k+\frac{2n+1}{2}&\frac{2}{(2n-1)!(2n+1)!} \\
P^2{\bf Ca}& 16 &22 &k(k+11) & k+\frac{11}{2} &\frac{2\,*\, 39}{15!} \\
\end{array}
$$
\caption[t]{The relevant data for some CROSSes, from \cite{Be,CV1}.}
{The Maslov indices, the eigenvalues, the corresponding arithmetic sequence,
and the coefficient of the leading order term in the polynomial which describes the multiplicities.}
\end{center}
\end{table}
\section{The trace formula}
The trace formula is a relation between the quantum mechanical
energy-spectrum and the periodic orbits of the classical
system. From \cite{D-G,UZ} follows that for $SC_{2\pi}$ manifolds
the trace formula reads:
%
$$
\sum_{n=0}^{\infty}\delta(p-\sqrt{E_n})=\sum_{k\in {\bf Z}}\alpha_k(p)
e^{i\frac{\pi}{2}\nu k}e^{-2\pi ikp}.
$$
%
The left hand side is the spectral momentum density $d(p)$. The
sum on the right hand side is a sum over all families
of periodic orbits, $\nu $ is the Maslov index of the primitive
orbits, and for the functions $\alpha_k(p) $ only the asymptotic
behavior is known:
$$
\alpha_k (p) = \frac{\mbox{Vol}(M)}{(4\pi)^{d/2}\,\Gamma(d/2+1)}\,p^{d-1}
+O(p^{d-2}), \quad p \to +\infty ,
$$
and
$$
\alpha_k (p) = O(p^{-\infty}), \quad p \to -\infty \quad .
$$
There is no dependence on the potential in the leading order,
but this is not very
surprising because the potential is bounded and therefore
the high energy limit is dominated by the kinetic
energy, i.e., by the metric. But what is more surprising is that,
because the volume is invariant under Zoll deformations \cite{W1},
the leading order in the trace formula does not distinguish between
Zoll metrics on $S^n $.
We will choose a different approach to the trace formula, based on
the information from the last section, which will give us also
the higher order aymptotics for $\alpha_k(p) $, where the differences
between different Zoll metrics, and the influence of a potential
will be visible.
Our starting point is an exact trace formula for the CROSSes.
%%%%%%%%%%%%%%%%%%%%
\begin{prop} Let $(M,g)$ be a CROSS and $P_g =-\Delta_g +c$ the Laplacian
shifted by a constant $c$ such that the
eigenvalues become
$\lambda_n^2 =(n+\nu /4)^2$, $n=1,2,3,\dots$,
with multiplicities $R(\lambda_n)$.
$R(t)$ is the polynomial from theorem $2.2$, and $\nu$ is the common Maslov
index of the primitive periodic orbits.
Then there is the following periodic orbit sum, valid in $\cal{S}'({\bf R})$,
for the spectral momentum density:
\begin{equation}\label{exact trace}
\sum_{n=1}^{\infty}R(\lambda_n)\delta (p-\lambda_n)=
R^{+}(p)\sum_{k\in {\bf Z}}e^{i\frac{\pi}{2}\nu k} e^{-2\pi ikp},
\end{equation}
where $R^{+}(p)$ denotes a smooth ($C^{\infty}$) function with
$R^{+}(p)=R(p)$ for $p\ge\nu/4+1=\lambda_1$, and
$ R^{+}(p)=0$ for $p\le \nu/4$.
\end{prop}
%%%%%%%%%%%%%%%%%%%%
{\em Proof}. With the Poisson identity \cite{H}
$$ \sum_{n \in {\bf Z}} \delta(p-n)=\sum_{k \in {\bf Z}}e^{-2\pi ikp}$$
we get
\begin{eqnarray*}
\sum_{n=1}^{\infty}R(\lambda_n)\delta(p-\lambda_n)&=&
R(p)\sum_{n=1}^{\infty}\delta(p-(n+\frac{\nu}{4}))\\
&=&R^{+}(p)\sum_{n\in {\bf Z}}\delta(p-\frac{\nu}{4}-n)
=R^{+}(p)\sum_{k\in {\bf Z}}e^{i\frac{\pi}{2}\nu k}e^{-2\pi ikp}\quad .
\end{eqnarray*}
\hfill $\Box $\\
%%%%%%%%%%%%%%%%%%%
{\bf Remark}. In the following calculations it will be important that
$R^{+}(p)$ is smooth and vanishes for negative $p$. To make such
a choice for $R^{+}(p)$ possible is the reason that we let $n$ start
with $1$ instead of $0$, so that the eigenvalues of $P_g$ are strictly
positive.
\medskip
Using the trace formula for CROSSes, and the information from the
preceding section we will now derive a trace formula for general
$SC_{2\pi}$ manifolds. The idea is to construct, for a given
$SC_{2\pi}$ manifold, an operator ${\bf A}$ which maps $\delta(p-\lambda_n)$
to $\frac{1}{R(\lambda_n)}\sum_{E_i \in I_n}\delta(p-\sqrt{E_n})$, and so
the sum
$d_0(p)=\sum R(\lambda_n)\delta(p-\lambda_n)$ to the spectral momentum
density $d(p)=\sum \delta(p-\sqrt{E_n})$. Applying this
operator to (\ref{exact trace}) will lead to the trace formula.
\medskip
Let $\chi(s)$ be a function with $\chi \in C^{\infty}_0 ({\bf R})$,
supp$\, \chi \in [-1,1]$, $\chi (0)=1$, and $\sum_{l=0}^{\infty}\chi (s-l)=1$
for $s>0$.
We define a distribution $\mu (s)$ , depending on a parameter s, by
\begin{equation}\label{mu}
\mu (s,E):=\sum_{l=0}^{\infty}\chi (s-l)\mu_l (E) \quad ,
\end{equation}
where $\mu_l$ is the normalized spectral measure of the l'th
eigenvalue cluster (\ref{cluster}).
Then we have $\mu(k,E)=\mu_k (E)$, and from theorem 2.3 follows that
for every $\varphi \in C^{\infty}$
\begin{equation}
\langle \mu (s), \varphi \rangle \sim \sum_{i=0}^{\infty}
\langle \beta_i ,\varphi \rangle \frac{1}{s^{2i}}\quad s\to \infty\quad .
\end{equation}
This allows us to write for the spectral density
$D(E)=\sum R(\lambda_n)\mu_n(E-\lambda_n^2)$ of any $SC_{2\pi}$ manifold
%
$$
D(E)=\int_{-\infty}^{\infty} D_0(s)\mu(\sqrt{s}-\nu/4,E-s)ds \quad ,
$$
%
with $D_0(E)=\sum R(\lambda_n)\delta(E-\lambda_n^2)$, as a short calculation shows.
If we use $d(p)=2p^{+}D(p^2)$, where $p^+$ is defined similar to
$R^+ (p)$, we get an expression for $d(p)$
$$
d(p)=2p^{+}\int d_0(t)\mu(t-\nu/4,p^2-t^2)dt \quad .
$$
So we have constructed the operator ${\bf A}$ with Schwartz kernel
$K_{\bf A} (p,t)=2p^{+}\mu(t-\nu/4,p^2-t^2)$. Applying ${\bf A}$
to both sides of equation (\ref{exact trace}) leads to
%
\begin{equation}\label{pre trace}
d(p)=2p^{+}\sum_{k\in \bf{Z}}e^{i\frac{\pi}{2}\nu k}
\int R^{+}(t)\mu(t-\nu/4,p^2-t^2)e^{-2\pi ikt}dt\quad ,
\end{equation}
%
and for $\alpha_k (p)$ we get
$$
\alpha_k (p)=e^{2\pi ikp}{\bf A}(R^+ e^{-2\pi ik(.)})\quad ,
$$
so we have arrived at the trace formula. Note that
$d_0(p)$ must not come from a CROSS, here we use (\ref{exact trace})
just as a summation formula for certain distributions.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
Let $(M,g)$ be a $SC_{2\pi}$ manifold, and $P=-\Delta_g +V$ a positive
Schr\"odinger operator with spectral
momentum measure $d(p)$, then
\begin{equation}
d(p)=\sum_{k \in {\bf Z}}\alpha_k(p)\,e^{i\frac{\pi}{2}\nu k}\,
e^{-2\pi ikp}\quad ,
\end{equation}
as elements of $\cal{S}'(\bf{R})$, with
\begin{equation}\label{alpha}
\alpha_k(p)=p^{+}\int_{-\infty}^{p^2} \frac{R^{+}(\sqrt{p^2-s})}{\sqrt{p^2-s}}
\,\mu(\sqrt{p^2-s}-\nu/4,s)\,e^{-2\pi ik(\sqrt{p^2-s}-p)}\,ds\quad .
\end{equation}
\end{prop}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\medskip
{\em Proof}. From (\ref{pre trace}) we get
\begin{equation}
\alpha_k(p)=2p^{+}\int_0^{\infty} R^{+}(t)\,
\mu(t-\nu/4,p^2-t^2)\, e^{-2\pi ik(t-p)}\,dt
\end{equation}
and (\ref{alpha}) follows by a change of variables.
There only remains to justify the
interchange of summation and integration, i.e., we have to show
that the right hand side is well defined as a distribution.
Let $\varphi \in \cal{S}(\bf{R})$, then
$$
\int d(p) \varphi(p) dp =\sum_{k\in \bf{Z}}e^{i\frac{\pi}{2}\nu k}
\int \alpha_k(p) \varphi(p) e^{-2\pi ikp}dp
$$
but
\begin{eqnarray*}
\int \alpha_k(p) \varphi(p) e^{-2\pi ikp}dp&=&\int\!\! \int \varphi(p)\,2p^{+}
\mu(t-\nu/4,p^2-t^2)\, dp\,\, R^{+}(t)\,e^{-2\pi ikt}\,dt\\
&=&\int\left\{ \int \varphi(\sqrt{q+t^2}\,)\,\frac{(\sqrt{q+t^2})^{+}}{\sqrt{q+t^2}}
\,\mu(t-\nu/4,q)\,R^{+}(t)\, dq \right\} \, e^{-2\pi ikt}\,dt\\
&=&O(k^{-\infty})
\end{eqnarray*}
because the inner integral is a Schwartz function of $t$
(here it enters that $R^+ $ is smooth) and so its Fourier
transform is again a Schwartz function.
So the sum exists.
\hfill $\Box $\\
%%%%%%%%%%%%%%%%%%%%%%%%%
We will now study $\alpha_k(p)$ more closely, but first we look at an
example which displays already typical properties.\\
{\bf Example}. Let $M$ be a CROSS, and $P=P_g+C$ where $C$ is a
constant,
and $P_g$ is the shifted Laplacian defined in proposition 1. Then
$E_n=\lambda_n^2+C$ and $ \mu_k=\delta(E-C)$, so
$\mu(s,E)=\delta(E-C)$ and this leads to
$$
\alpha_k(p)=p^+ \frac{R^+(\sqrt{p^2-C})}{\sqrt{p^2-C}}
e^{-2\pi ik(\sqrt{p^2-C}-p)}
$$
for $p^2>C$, and $0$ else. If one fixes $k$ and let
$p\to \infty$ then $\alpha_k (p)$ has
a regular asymptotics
$$
\alpha_k (p)= b_1 p^{d-1}+O(p^{d-2})\quad ,
$$
if one takes the limit $p,k\to \infty$ with $k/p \to \gamma =$const., then
$$
\alpha_k (p)= ( b_1 p^{d-1}+O(p^{d-2} ))\, e^{\pi i \gamma c}\quad ,
$$
and for $p$ fixed and $k\to \infty$ $\alpha_k (p)$ is oscillating.
But notice that
$$
|\alpha_k (p)|= b_1 p^{d-1}+O(p^{d-2})\quad ,
$$
independent of $k$.
We will now show that the asymptotic behavior of general $\alpha_k (p)$
is similar.
%%%%%%%%%%%%%%%%%%
\begin{lemma}
$\alpha_k (p)$ has the following asymptotic properties:
\begin{enumerate}
\item[i)] Generally, for $p\to \infty$,
$$
\alpha_k (p) = b_1 \langle \beta_0 , e^{-2\pi ik(\sqrt{p^2-(.)}-p}\rangle
\, p^{d-1} +O(p^{d-2}) ,
$$
depending on how fast $k$ growth compared to $p$.
\item[ii)] For $p\to \infty$ and $k$ fixed
\begin{eqnarray*}
\alpha_k (p)= b_1\, p^{d-1}&+&i\pi kb_1\int s\beta_0(s)ds\, p^{d-2}
\\&-& \left[ \frac{1}{2}b_1 \pi^2 k^2\int\beta_0 s^2 ds +\frac{1}{2}b_1(d-2)
\int\beta_0 s ds-b_3\right] p^{d-3}+O(p^{d-4}) .
\end{eqnarray*}
\item[iii)]For $p,k \to \infty$, with $k/p\to \gamma=const.$
$$
\alpha_k (p)= b_1 \langle\beta_0 ,e^{i\pi\gamma (.)}\rangle p^{d-1}
+O(p^{d-3}).
$$
\end{enumerate}
Here all terms in ii) and iii) have been written down which
depend on $\beta_0$ only,
the next order terms contain contributions from $\beta_1$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\medskip
{\em Proof}. Inserting (\ref{mu}) in (\ref{alpha}) gives
%
\begin{eqnarray*}
\alpha_k (p)&=&\sum_{l=0}^{\infty}\frac{1}{m_l}\sum_{E_j \in I_l}p^+
\int\frac{R^{+}(\sqrt{p^2-s})}{\sqrt{p^2-s}}\chi (\sqrt{p^2-s}-\nu/4-l)
\delta(s-(E_j-\lambda_l^2))e^{-2\pi ik(\sqrt{p^2-s}-p)}ds\\
&=&\sum_{l=0}^{\infty}\frac{1}{m_l}\sum_{E_j \in I_l}p^+
\frac{R^{+}(\sqrt{p^2-(E_j-\lambda_l^2)})}{\sqrt{p^2-(E_j-\lambda_l^2)}}
\chi (\sqrt{p^2-(E_j-\lambda_l^2)}-\nu/4-l)
e^{-2\pi ik(\sqrt{p^2-(E_j-\lambda_l^2)}-p)}
\end{eqnarray*}
%
but $|E_j-\lambda_l^2|\le K$ for $E_j \in I_l$, so for $p$, $l$ large
%
$$
p^+
\frac{R^{+}(\sqrt{p^2-(E_j-\lambda_l^2)})}{\sqrt{p^2-(E_j-\lambda_l^2)}}
\chi (\sqrt{p^2-(E_j-\lambda_l^2)}-\nu/4-l) \sim b_1 p^{d-1}\chi(p-\nu/4-l)
+O(p^{d-2})
$$
%
and since, for $l\to\infty$,
$$
\frac{1}{m_l}\sum_{E_j \in I_l} e^{-2\pi ik(\sqrt{p^2-(E_j-\lambda_l^2)}-p)}
=\langle \mu_l,e^{-2\pi ik(\sqrt{p^2-(.)}-p)}\rangle =
\langle \beta_0,e^{-2\pi ik(\sqrt{p^2-(.)}-p)}\rangle +O(\frac{1}{l^2})
$$
we get
%
$$
\alpha_k (p) = b_1 p^{d-1}\langle \beta_0,e^{-2\pi ik(\sqrt{p^2-(.)}-p)}\rangle +O(p^{d-2}).
$$
%
This proves i).
The behaviour of $\alpha_k (p)$ depends now on how $k$ grows with $p$,
and inserting the asymptotic expansion of $e^{-2\pi ik(\sqrt{p^2-s}-p)}$
leads to the
leading order terms in ii) and iii).\\
For the higher order terms in ii) and iii) one inserts the asymptotic
development of the integrand
in (\ref{alpha}) (notice that it has compact support in $s$),
and uses furthermore that $\int \beta_i dE = 0$ for $i>0$.
\hfill $\Box $\\
%%%%%%%%%%%%%%%
{\bf Remark 1}. In the asymptotic expansion ii) of $\alpha_k (p)$ there appear
all moments of the measures $\{ \beta_i\}$. So knowing these moments and
the coefficients of the polynomials $R(t)$, one can recover the
asymptotics of $\alpha_k (p)$ and vice versa. This is a relation between
the spectrum and geometric properties of $(M,g)$.
{\bf Remark 2}. The asymptotics ii) have a similar structure, especially
the dependence on $k$, to
the one derived by Guillemin in \cite{G2} for an $\alpha_k (p)$
coming from a nondegenerate elliptic orbit,
$$
\alpha_k (p) = c_1 + c_2 p^{-1} +O(p^{-2})
$$
with
$$
c_2 = 2\pi ik c_1 (\cdots)\,\, .
$$
Here $k$ denotes the number of iterates of the primitive orbit, and
the expression in brackets depends on the symbol of the Hamilton operator
in a neighborhood of the orbit.
\medskip
Now we come to the proof of theorem 1.1. $N(E)$ is given by
$$
N(E)=\int_0^{\sqrt{E}}d(p) dp= \int_0^{\sqrt{E}}\alpha_0(p)dp
+\int_0^{\sqrt{E}}\sum_{k\not= 0}\alpha_k (p)e^{i\frac{\pi}{2}\nu k}
e^{-2\pi ikp}dp .
$$
The first integral gives
$$
\int_0^{\sqrt{E}}\alpha_0(p) dp=\frac{b_1}{d}E^{d/2}+O(E^{d/2-1})\quad ,
$$
%and $(k \not =0)$
%$$
%\int_0^{\sqrt{E}}\alpha_k (p)e^{i\frac{\pi}{2}\nu k}
%e^{-2\pi ikp}dp=\frac{ib_1}{2\pi k}e^{i\frac{\pi}{2}\nu k}E^{(d-1)/2}
%e^{-2\pi ik\sqrt{E}}+O(\frac{E^{d/2-1}}{k})\quad .
%$$
so we get
$$
N(E)=\frac{b_1}{d}E^{d/2}+E^{(d-1)/2}\Theta(\sqrt{E}),
$$
with $\Theta(x)$ given by
$$
\Theta(x)=\frac{1}{x^{d-1}}\sum_{k\not= 0} e^{i\frac{\pi}{2}\nu k}
\int_0^x \alpha_k(p)
e^{-2\pi ikp} dp\,+O(1/x)\quad .
$$
The sum converges in the $B^2$ norm to the function
$$
\sum_{k\not= 0}\frac{ib_1}{2\pi k}e^{i\frac{\pi}{2}\nu k}
e^{-2\pi ikx}\quad .
$$
To see this we compute $||\Theta_N(x)-\sum_{k\not= 0}
\frac{ib_1}{2\pi k}e^{i\frac{\pi}{2}\nu k}e^{-2\pi ikx}||_{B^2}$ with
$$
\Theta_N(x)=
\sum_{\tiny\begin{array}{c}k\not= 0\\|k|\le N\end{array}}
\frac{e^{i\frac{\pi}{2}\nu k}e^{-2\pi ikx}}{x^{d-1}}
\int_0^x \alpha_k(p)
e^{-2\pi ik(p-x)}dp\, +O(1/x) \quad .
$$
The triangle inequality gives
\begin{eqnarray*}
||\Theta_N(x)-\sum_{k\not= 0}
\frac{ib_1}{2\pi k}e^{i\frac{\pi}{2}\nu k}e^{-2\pi ikx}||_{B^2}
&\le &
||\Theta_N(x)-
\sum_{\tiny\begin{array}{c}k\not= 0\\|k|\le N\end{array}}
\frac{ib_1}{2\pi k}e^{i\frac{\pi}{2}\nu k}e^{-2\pi ikx}||_{B^2}\\
& &+||\sum_{|k|>N}
\frac{ib_1}{2\pi k}e^{i\frac{\pi}{2}\nu k}e^{-2\pi ikx}||_{B^2}\quad ,
\end{eqnarray*}
but
$$\frac{1}{x^{d-1}}\int_0^x \alpha_k(p)e^{-2\pi ik(p-x)}dp=
\frac{ib_1}{2\pi k}+O(\frac{1}{x})$$
by lemma 3.1, so
the first term on the right hand side vanishes.
The remaining term can be estimated by
$$\sum_{|k|>N}\frac{b_1^2}{(2\pi k)^2}
\sim \frac{b_1^2}{(2\pi)^2}\frac{1}{N}\quad ,$$
so
$$
\lim_{N\to\infty}||\Theta_N(x)-\sum_{k\not= 0}
\frac{ib_1}{2\pi k}e^{i\frac{\pi}{2}\nu k}e^{-2\pi ikx}||_{B^2}=0
$$
and we have proven (\ref{N}) and (\ref{Fser}).
To prove (7) we notice that for a periodic function $f(x)$, with
period $1$ and for every continuous $g$,
\begin{equation}\label{erg}
\lim_{T\to\infty}\frac{1}{T}\int_0^T g(f(x))dx =\int_0^1 g(f(\theta))d\theta
\quad ,
\end{equation}
because
\begin{eqnarray*}
\frac{1}{T}\int_0^T g(f(x))dx &=&\frac{1}{T}\int_0^{[T]} g(f(x))dx +
\frac{1}{T}\int_{[T]}^T g(f(x))dx \\&=&\frac{[T]}{T}\int_0^1 g(f(x))dx +
\frac{1}{T}\int_{[T]}^T g(f(x))dx \quad .
\end{eqnarray*}
Inserting (\ref{Fsum}) in (\ref{erg}) leads to
$$
\lim_{T\to\infty}\frac{1}{T}\int_0^T g(\Theta(x))dx = \int_0^1
g(b_1f(x-\nu/4))dx =\frac{1}{b_1}\int_{-\frac{b_1}{2}}^{\frac{b_1}{2}}
g(x) dx \quad ,
$$
so we have proven (\ref{step}) for $\rho$ in (\ref{lim}) a step function
which is $1$ on $[0,1]$ and $0$ elsewhere. The proof for general
$\rho$ works with an approximation of $\rho$ by step functions and can
be found in \cite{B1,KMS}.
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\end{document}