Presented by: 
Dr. Andrew Hassell (ANU)
Date: 
Mon 12 Apr, 2:00 pm - 3:00 pm
Venue: 
214 in Otto Hirschfeld (81)

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 Δ u = Eu, where u is a function on D vanishing at the boundary, Δ is the Laplacian on D and E is a real number, in the limit as E → ∞. 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 E → ∞. I will talk about Quantum Ergodicity, which is the property that "most of" the eigenfunctions become uniformly distributed in D, asymptotically as E → ∞, 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.

No prior knowledge of any of the topics mentioned is assumed.