The semiclassical theory of discontinuous systems and ray-splitting billiards
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.
Keywords
Cite
@article{arxiv.1301.6783,
title = {The semiclassical theory of discontinuous systems and ray-splitting billiards},
author = {Dmitry Jakobson and Yuri Safarov and Alexander Strohmaier and Yves Colin de Verdiere},
journal= {arXiv preprint arXiv:1301.6783},
year = {2015}
}
Comments
51 pages, 6 figures, proof in Appendix A corrected and some more typos fixed, error in the formula for the transmission and reflection coefficient due to different choice of boundary conditions corrected, Appendix by Yves Colin de Verdiere