## Analysis and Applied Mathematics Seminar

Joey Zou

Northwestern University

A Gutzwiller trace formula for semiclassical Schrödinger operators with conormal potentials

**Abstract:**We discuss ongoing work, joint with J. Wunsch and M. Yang, which concerns extending the Gutzwiller Trace Formula from the case of smooth potentials to the case of potentials with conormal singularities. In the smooth case, the formula expresses an eigenvalue-counting function of a Schrödinger operator as a sum of certain dynamical quantities over periodic Hamiltonian trajectories. In the conormal case, a consideration of a WKB ansatz for the Schrödinger propagator suggests the sum should incorporate dynamical information about Hamiltonian trajectories which reflect at the site of the singularity. We discuss the variational formulation required to make sense of the dynamics of such trajectories, as well as the further work needed to complete the proof. We also present an explicit example of such a potential whose eigenvalue asymptotics can be computed; such asymptotics show the presence of reflected dynamics when applied to the Gutzwiller Trace Formula.

Monday September 23, 2024 at 4:00 PM in 636 SEO