Tuomas Sahlsten: A gentle introduction to quantum ergodicity

We will give an introduction to the topic of “quantum ergodicity” and review the history and current challenges of the problem. The quantum ergodicity theorem states that on Riemannian surfaces with an ergodic geodesic flow, most eigenfunctions of the Laplacian equidistribute spatially in the large eigenvalue limit. In this talk, we will present an alternative equidistribution theorem for eigenfunctions where the eigenvalues stay bounded and we take instead sequences of compact hyperbolic We will give an introduction to the topic of “quantum ergodicity” and review the history and current challenges of the problem. The quantum ergodicity theorem states that on Riemannian surfaces with an ergodic geodesic flow, most eigenfunctions of the Laplacian equidistribute spatially in the large eigenvalue limit. In this talk, we will present an alternative equidistribution theorem for eigenfunctions where the eigenvalues stay bounded and we take instead sequences of compact hyperbolic surfaces that become large in, say, volume. Thus the result combines quantum ergodicity with the theory of limit multiplicities in spectral theory (after DeGeorge and Wallach).
The approach is motivated by the recent works of Anantharaman, Brooks, Le Masson, and Lindenstrauss on eigenvectors of the discrete Laplacian on regular graphs, and the holomorphic form analogues by Nelson, Pitale and Saha. In the dynamics side of the proof we require the exponential mixing structure of the geodesic flow on hyperbolic surfaces, in particular a quantitative mean ergodic theorem by Nevo.
This is a joint work with Etienne Le Masson (Bristol).