A Gutzwiller trace formula for stationary space-times

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