## 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 e^{tZ} of the timelike Killing vector field Z on □. The spectrum of Z on ker□ is discrete and the singularities of Tre^{tZ}|_{ker□} occur at periods of periodic orbits of exptZ on the symplectic manifold of null geodesics. The trace formula gives a Weyl law for the eigenvalues of Z on ker□.

Original language | English (US) |
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Article number | 107434 |

Journal | Advances in Mathematics |

Volume | 376 |

DOIs | |

State | Published - Jan 6 2021 |

## Keywords

- Spectral theory
- Stationary spacetimes
- Trace formula

## ASJC Scopus subject areas

- General Mathematics