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)
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.