A Gutzwiller Trace formula for Dirac Operators on a Stationary Spacetime

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 Strohmaier and 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 manifold of lightlike geodesics. This gives rise to the Weyl law asymptotic at the vanishing period. A pivotal technical ingredient to prove these results is the Feynman propagator. In order to obtain a Fourier integral description of this propagator, we have generalised the classic work of Duistermaat and H\"{o}rmander [Acta Math. \textbf{128}, 183 (1972)] on distinguished parametrices for normally hyperbolic operators on a globally hyperbolic spacetime by propounding their microlocalisation theorem to a bundle setting. As a by-product of these analyses, another proof of the existence of Hadamard bisolutions for a normally hyperbolic operator (resp. Dirac-type operator) is reported.