Ergodic billiards that are not quantum unique ergodic

Details Ergodic billiards that are not quantum unique ergodic Ergodic billiards that are not quantum unique ergodic

2008-07-04

arxiv.org/abs/0807.0666v3

Mathematics

Analysis of PDEs

11 pages, 1 figure. The paper, authored by Andrew Hassell, now includes an appendix by Andrew Hassell and Luc Hillairet, exten

Scientific paper

Partially rectangular domains are compact two-dimensional Riemannian manifolds $X$, either closed or with boundary, that contain a flat rectangle or cylinder. In this paper we are interested in partially rectangular domains with ergodic billiard flow; examples are the Bunimovich stadium, the Sinai billiard or Donnelly surfaces. We consider a one-parameter family $X_t$ of such domains parametrized by the aspect ratio $t$ of their rectangular part. There is convincing theoretical and numerical evidence that the Laplacian on $X_t$ with Dirichlet or Neumann boundary conditions is not quantum unique ergodic (QUE). We prove that this is true for all $t \in [1,2]$ excluding, possibly, a set of Lebesgue measure zero. This yields the first examples of ergodic billiard systems proven to be non-QUE.

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