 95220 Roman Schubert
 The Trace Formula and the Distribution of Eigenvalues of
Schroedinger Operators on Manifolds all of whose
Geodesics are closed. (LaTeX, 37 K)
(37K, LaTeX)
May 14, 95

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Abstract. We investigate the behaviour of the remainder term
$R(E)$ in the Weyl formula
$$
\# \{nE_n\le E\}=
\frac{\mbox{Vol}(M)}{(4\pi )^{d/2}\, \Gamma(d/2+1)}\, E^{d/2}+R(E)
$$
for the eigenvalues $E_n$ of a Schr\"odinger operator on
a ddimensional compact Riemannian manifold all of whose
geodesics are closed.
We show that $R(E)$ is of the form $E^{(d1)/2}\,\Theta(\sqrt{E})$,
where $\Theta(x)$ is an almost periodic function of Besicovitch class $B^2$
which has
a limit distribution whose density is a boxshaped 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.
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