The trace formula and the distribution of eigenvalues of Schroedinger operators on manifolds all of whose geodesics are closed

We investigate the behaviour of the remainder term R(E) in the Weyl formula {nvertical stroke En≤E}=Vol(M).Ed/2/[(4π)d/2Γ(d/2+1)]+R(E) for the eigenvalues En of a Schroedinger 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)/2Θ(√E), where Θ(x) is an almost periodic function of Besicovitch class B2 which has a limit distribution whose density is a box-shaped function. 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. (orig.)