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  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  with respect to the global timelike Killing vector field  on the kernel of and found that it comprises discrete real eigenvalues. The distributional trace  of the time evolution operator  is then the trace of  on ker D and  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 . 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.

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