Abstract:

This is joint work with Yu. Safarov and A. Strohmaier.

After giving an overview of Quantum Ergodicity results on compact Riemannian manifolds with ergodic geodesic flow (due to Shnirelman, Zelditch, Colin de Verdiere and others), we discuss joint work with Yuri Safarov and Alex Strohmaier, which concerns the semiclassical limit of spectral theory on manifolds whose metrics have jump-like discontinuities. Such systems are quite different from manifolds with smooth Riemannian metrics because the semiclassical limit does not relate to a classical flow but rather to branching (ray-splitting) billiard dynamics. In order to describe this system we introduce a dynamical system on the space of functions on phase space. We prove a quantum ergodicity theorem for discontinuous systems. In order to do this we introduce a new notion of ergodicity for the ray-splitting dynamics. If time permits, we outline an example (provided by Y. Colin de Verdiere) of a system where the ergodicity assumption holds for the discontinuous system.