# A Gutzwiller Trace formula for Dirac Operators on a Stationary Spacetime

#### 23.06.2022, 16:00  –  Raum 0.14 Forschungsseminar Differentialgeometrie

Onirban Islam

A Duistermaat-Guillemin-Gutzwiller trace formula for a Dirac-type operator D on a globally hyperbolic spatially compact standard stationary spacetime $$(M,g,Z)$$ is achieved by generalising the recent construction by A. Strohmaier and S. Zelditch [Adv. Math. 376,107434 (2021)] to a vector bundle setting. We have analysed the spectrum of the Lie derivative $$\mathcal{L}_Z$$ with respect to the global timelike Killing vector field $$Z$$ on the kernel $$\mathrm{ker}\ D$$ of $$D$$ and found that it comprises discrete real eigenvalues. The distributional trace $$\mathrm{Tr}\ U_t$$ of the time evolution operator $$U_t$$ is then the trace of $$e^{t\mathcal{L}_Z}$$ on ker D and $$\mathrm{Tr}\ U_t$$ 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. A pivotal technical ingredient to prove these results is the Feynman propagator for $$D$$. In order to obtain a Fourier integral description of this propagator, we have generalised the classic work of J. Duistermaat and L. Hörmander [Acta Math. 128, 183 (1972)] on distinguished parametrices for any normally hyperbolic operator on a globally hyperbolic spacetime by propounding their microlocalisation theorem in a bundle setting. As a by-product of these analyses, another proof on the existence of Hadamard bisolutions for a normally hyperbolic operator and for a Dirac-type operator is reported.

zu den Veranstaltungen