A Gutzwiller trace formula for Dirac operators on a stationary spacetime

Mr Onirban Islam (University of Leeds) will present his research in Analysis and Geometry.

The Duistermaat-Guillemin-Gutzwiller trace formula connects spectral asymptotics of a differential operator with the geometry of the manifold where the operator acts. In this talk, we will start with a few rather elementary identities as predecessors of the trace formula to motivate Gutzwiller's idea. Then we will explain the Lorentzian framework due to A. Strohmaier and S. Zelditch [Adv. Math. 376, 107434 (2021)] and its bundle generalisation for Dirac-type operators on a spatially compact stationary spacetime. On the spectral side, we have studied the spectrum of the Lie derivative with respect to the Killing vector field on the solution space of the Dirac equation and found that it consists of discrete real eigenvalues. The geometric information is obtained by investigating the Dirac-wave-trace which has singularities at the periods of induced Killing flow on the space of lightlike geodesics. The Weyl law is achieved at the vanishing period as a corollary of the singularity analysis. (Based on arXiv:2109.09219 [math.AP]).


The talks are held using Zoom. Anyone interested in attending (outside the School of Mathematics) should email Ben Sharp at b.g.sharp@leeds.ac.uk to request zoom coordinates.