Ergodic billiards that are not quantum unique ergodic

TL;DR

This paper proves that a broad class of ergodic billiard systems, including the Bunimovich stadium and Sinai billiard, are not quantum unique ergodic for almost all aspect ratios, providing the first such rigorous examples.

Contribution

It establishes the non-quantum unique ergodicity of ergodic billiard systems for a continuous range of parameters, extending previous numerical and theoretical evidence.

Findings

Proves non-QUE for all t in [1,2] except possibly measure zero set.

First rigorous examples of ergodic billiards proven to be non-QUE.

Supports numerical and theoretical evidence with a formal proof.

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