# Quantum Billiards

#### by Dr Andrew Hassell

Institution: Australian National University
Date: Tue 12th October 2010
Time: 12:00 PM
Location: Russell Love Theatre, Richard Berry Bldg, The Uni of Melb

Abstract: A mathematical "billiard" is a bounded plane domain D, with smooth (enough) boundary. Quantum billiards is the study of properties of eigenfunctions of the Laplacian on D, i.e. solutions of Î”u = Eu, where u is a function on D vanishing at the boundary, Î” is the Laplacian on D and E is a real number, in the limit as Eâ ’âˆž. This large-E limit is the "classical limit" --- it turns out that eigenfunctions with large E exhibit behaviour related to the classical billiard system (a billiard ball moving around inside D, bouncing elastically off the boundary). That is, dynamical properties of the classical billiard system such as ergodicity, chaos, etc, are reflected in properties of the eigenfunctions in the limit Eâ ’âˆž.

I will talk about Quantum Ergodicity, which is the property that "most of" the eigenfunctions become uniformly distributed in D, asymptotically as Eâ� ’âˆž, i.e. they are the same size, on average, in all parts of the domain D; and the stronger property of Quantum Unique Ergodicity, which is the same property with the words "most of" deleted. In particular, I will discuss a recent result of mine showing that the "stadium billiard" is not Quantum Unique Ergodic, establishing a conjecture first made in the 1980s.