A Gutzwiller Trace formula for Dirac Operators on a Stationary Spacetime

TL;DR

This paper extends the Gutzwiller trace formula to Dirac operators on stationary spacetimes, analyzing spectral properties and singularities related to Killing flows and lightlike geodesics.

Contribution

It generalizes the trace formula to a vector bundle setting on stationary spacetimes, analyzing the spectrum of the Lie derivative and trace singularities.

Findings

Spectrum of Lie derivative is discrete and real.

Trace has singularities at periods of Killing flow.

Weyl law asymptotics at zero period.

Abstract

A Duistermaat-Guillemin-Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact stationary spacetime is achieved by generalising the recent construction by A. Strohmaier and S. Zelditch [Adv. Math. \textbf{376}, 107434 (2021)] to a vector bundle setting. We have analysed the spectrum of the Lie derivative with respect to a global timelike Killing vector field on the solution space of the Dirac equation and found that it consists of discrete real eigenvalues. The distributional trace of the time evolution operator has singularities at the periods of induced Killing flow on the space of lightlike geodesics. This gives rise to the Weyl law asymptotic at the vanishing period.