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

47 pages, typos corrected and presentation restructured

References in corpus (3)

• Ground state for a massive scalar field in BTZ spacetime with Robin boundary conditions
• Initial value problems for wave equations on manifolds
• Resolvent and propagation estimates for Klein-Gordon equations with non-positive energy

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