Igor Wigman: Ergodic billiards that are not quantum unique ergodic

Igor Wigman will speak about Hassel's construction of bouncing ball modes — namely eigenfunctions of the Laplacian that concentrate/localize on the rectangular part of a “stadium billiard”.

It is not strictly required, but we recommend people to have at least glanced at Terry Tao’s blog entry “Hassell’s proof of scarring for the Bunimovich stadium”.

Abstract. 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 ∈ [1,2] excluding, possibly, a set of Lebesgue measure zero. This yields the first examples of ergodic billiard systems proven to be non-QUE.