A Gutzwiller trace formula for stationary space-times

Abstract

We give a relativistic generalization of the Gutzwiller-Duistermaat-Guillemin trace formula for the wave group of a compact Riemannian manifold to globally hyperbolic stationary space-times with compact Cauchy hypersurfaces. We introduce several (essentially equivalent) notions of trace of self-adjoint operators on the null-space ker⁡□ of the wave operator and define U(t) to be translation by the flow etZ of the timelike Killing vector field Z on □. The spectrum of Z on ker⁡□ is discrete and the singularities of TretZ|ker⁡□ occur at periods of periodic orbits of exp⁡tZ on the symplectic manifold of null geodesics. The trace formula gives a Weyl law for the eigenvalues of Z on ker⁡□.