arXiv 0807.06662008-07math.APmath-phmath.MP

Ergodic billiards that are not quantum unique ergodic

Andrew Hassell, Luc Hillairet

备注：: 11 pages, 1 figure. The paper, authored by Andrew Hassell, now includes an appendix by Andrew Hassell and Luc Hillairet, extending the result to all partially rectangular billiards

英文摘要：

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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