Quantum ergodicity for sequences of locally symmetric spaces

Speaker:  A/Prof Simon Marshall 
Affiliation: University of Melbourne

Abstract

If X is a compact Riemannian manifold, the correspondence principle in quantum mechanics predicts that the behaviour of Laplace eigenfunctions on X with large eigenvalue should be related to the dynamics of the geodesic flow.  One way of making this rigorous is the quantum ergodicity theorem of Shnirelman, Zelditch, and Colin de Verdière, which states that if the geodesic flow is ergodic, then the Laplace eigenfunctions are quantum ergodic, meaning that a density 1 subsequence of them distribute their L^2 mass evenly over X.

In my talk, I will first introduce Laplace eigenfunctions and give some background on the quantum ergodicity theorem.  I will then describe a variant of quantum ergodicity, known as quantum ergodicity in the level aspect, in which the Laplace eigenvalue is bounded, but one considers a sequence of manifolds of growing volume.  Finally, I will describe a new result, joint with Brumley, Matz, and Peterson, which establishes quantum ergodicity in the level aspect for many new sequences of manifolds, namely for locally symmetric spaces of arbitrary rank (extending previous results in rank one).