The Trace Formula and the Distribution of Eigenvalues of Schrr Odinger Operators on Manifolds All of Whose Geodesics Are Closed

We investigate the behaviour of the remainder term R(E) in the Weyl formula #fnjE n Eg = Vol(M) (4) d=2 ?(d=2 + 1) E d=2 + R(E) for the eigenvalues E n of a Schrr odinger operator on a d-dimensional compact Rieman-nian manifold all of whose geodesics are closed. We show that R(E) is of the form E (d?1)=2 (p E), where (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 19, 1]. Furthermore we derive a trace formula and study higher order terms in the asymptotics of the coeecients related to the periodic orbits. The periodicity of the geodesic ow leads to a very simple structure of the trace formula which is the reason why the limit distribution can be computed explicitly.