A Gutzwiller trace formula for stationary space-times

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 $\ker \mathrm{\square }$ of the wave operator and define $U(t)$ to be translation by the flow $e^{tZ}$ of the timelike Killing vector field $Z$ on $\mathrm{\square }$. The spectrum of $Z$ on $\ker \mathrm{\square }$ is discrete and the singularities of $\mathrm{T}\mathrm{r}{e}^{tZ}{\mathrm{\mid }}_{\ker \mathrm{\square }}$ 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 $\ker \mathrm{\square }$.