95-220 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 $$ \# \{n|E_n\le E\}= \frac{\mbox{Vol}(M)}{(4\pi )^{d/2}\, \Gamma(d/2+1)}\, E^{d/2}+R(E) $$ 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.

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