Quantum ergodicity in the level aspect

Abstract: A classical result of Shnirelman and others shows that closed Riemannian manifolds of negative curvature are quantum ergodic, meaning that on average the probability measures $|f|^2 dx$ on $M$, with $f$ running through normalized Laplace eigenfunctions on $M$ with growing eigenvalue, converge towards the Riemannian measure $dx$ on $M$.

Following ideas of Abert, Bergeron, Le Masson, and Sahlsten, we look at a related situation: We want to consider certain sequences of manifolds together with Laplace eigenfunctions of approximately the same eigenvalue instead of high energy eigenfunctions on a fixed manifold. In my talk I want to discuss joint work with F. Brumley in which we study this situation in higher rank for sequences of compact quotients of $SL(n,\mathbb{R})/SO(n)$.

analysis of PDEsnumber theoryrepresentation theory

Audience: researchers in the topic

Harmonic Analysis and Symmetric Spaces 2021