Ergodic billiards that are not quantum unique ergodic

• 成果类型: Article
• 署名作者: Hassell, Andrew; Hillairet, Luc
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2010.171.605
• 发表日期: 2010
• 页码: 605-618
• 关键词: boundary-values; eigenfunctions; localization

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, Neumann or Robin boundary conditions is not quantum unique ergodic (QUE). We prove that this is true for all t is an element of [1, 2] excluding, possibly, a set of Lebesgue measure zero. This yields the first examples of ergodic billiard systems proven to be non-QUE.