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Title: The semiclassical theory of discontinuous systems and ray-splitting billiards
Authors: Jakobson, Dmitry
Safarov, Yuri
Strohmaier, Alexander
de Verdiere, Yves C.
Issue Date: 2015
Publisher: © Johns Hopkins University Press
Citation: JAKOBSON, D. ... et al., 2015. The semiclassical theory of discontinuous systems and ray-splitting billiards. American Journal of Mathematics, 137 (4), pp. 859 - 906.
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 (raysplitting) 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.
Description: This article appeared in the American Journal of Mathematics, Volume 137, Issue 4, 2015, pages 859-906, Copyright © 2015, Johns Hopkins University Press.
Sponsor: Research of the first author supported in part by NSERC, FQRNT, and a Dawson Fellowship
Version: Accepted for publication
DOI: 10.1353/ajm.2015.0027
URI: https://dspace.lboro.ac.uk/2134/20023
Publisher Link: http://dx.doi.org/10.1353/ajm.2015.0027
ISSN: 0002-9327
Appears in Collections:Published Articles (Maths)

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