ray-splitting58.pdf (550.01 kB)
The semiclassical theory of discontinuous systems and ray-splitting billiards
journal contribution
posted on 2016-01-14, 11:32 authored by Dmitry Jakobson, Yuri Safarov, Alexander Strohmaier, Yves C. de VerdiereWe 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.
Funding
Research of the first author supported in part by NSERC, FQRNT, and a Dawson Fellowship
History
School
- Science
Department
- Mathematical Sciences
Published in
American Journal of MathematicsVolume
137Issue
4Pages
859 - 906Citation
JAKOBSON, D. ... et al., 2015. The semiclassical theory of discontinuous systems and ray-splitting billiards. American Journal of Mathematics, 137 (4), pp. 859 - 906.Publisher
© Johns Hopkins University PressVersion
- AM (Accepted Manuscript)
Publisher statement
This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/Publication date
2015-07-10Notes
This article appeared in the American Journal of Mathematics, Volume 137, Issue 4, 2015, pages 859-906, Copyright © 2015, Johns Hopkins University Press.ISSN
0002-9327eISSN
1080-6377Publisher version
Language
- en