The seminars will take place on Wednesdays sometimes using zoom, and (hopefully) sometimes in person.
If you would like further information on how to attend, please contact Ben Sharp (organiser).
A Gutzwiller trace formula for Dirac operators on a stationary spacetime
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]).
Does quantum dynamics emerge from classical chaotic dynamics?
We first explain the meaning of that question. Then we show that it is suggested from the study of the geodesic flow on a surface with constant negative curvature. Finally we show a positive answer in the case of a general geodesic flow that is Anosov or contact Anosov flow. Work done with Masato Tsujii, based on arxiv:1706.09307 and arxiv:2102.11196.
Nodal counts of the Dirichlet-to Neumann operators
The zero set of an eigenfunction is called the nodal set and the connected components of its complement are called the Nodal domains. The well-known Courant Nodal domain theorem gives an upper bound for the nodal count of Laplace eigenfunctions on a compact manifold. However, almost nothing is known when we consider Dirichlet-to-Neumann eigenfunctions. We discuss how we can obtain an asymptotic Courant Nodal domain theorem for the Dirichlet to Neumann operators.
This is joint work with David Sher.