The semiclassical theory of discontinuous systems and ray-splitting billiards

TL;DR

This paper develops a semiclassical framework for manifolds with jump discontinuities, revealing ray-splitting billiard dynamics and establishing a quantum ergodicity theorem for such systems.

Contribution

It introduces a novel semiclassical approach to discontinuous manifolds, linking quantum and ray-splitting classical dynamics, and proves a quantum ergodicity theorem in this context.

Findings

Established a relation between quantum and ray-splitting classical dynamics.

Proved a quantum ergodicity theorem for discontinuous systems.

Constructed examples of systems with ray-splitting billiard dynamics.

Abstract

We analyze 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. To identify the quantum dynamics in the semiclassical limit we compute the principal symbols of the Fourier integral operators associated to reflected and refracted geodesic rays and identify the relation between classical and quantum dynamics. In particular 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. The paper contains an Appendix written by Yves Colin de Verdiere in which a non-trivial class of examples is constructed.