Quantum ergodicity of C* dynamical systems

This paper contains a very simple and general proof that eigenfunctions of quantizations of classically ergodic systems become uniformly distributed in phase space. This ergodicity property of eigenfunctions f is shown to follow from a convexity inequality for the invariant states (Af,f). This proof of ergodicity of eigenfunctions simplifies previous proofs (due to A.I. Shnirelman, Colin de Verdiere and the author) and extends the result to the much more general framework of C* dynamical systems.Comment: Only very minor differences with the published versio

Size of nodal domains of the eigenvectors of a G(n,p) graph

Consider an eigenvector of the adjacency matrix of a G(n, p) graph. A nodal domain is a connected component of the set of vertices where this eigenvector has a constant sign. It is known that with high probability, there are exactly two nodal domains for each eigenvector corresponding to a non-leading eigenvalue. We prove that with high probability, the sizes of these nodal domains are approximately equal to each other

Recent developments in mathematical Quantum Chaos

This is a survey of recent results on quantum ergodicity, specifically on the large energy limits of matrix elements relative to eigenfunctions of the Laplacian. It is mainly devoted to QUE (quantum unique ergodicity) results, i.e. results on the possible existence of a sparse subsequence of eigenfunctions with anomalous concentration. We cover the lower bounds on entropies of quantum limit measures due to Anantharaman, Nonnenmacher, and Rivi\`ere on compact Riemannian manifolds with Anosov flow. These lower bounds give new constraints on the possible quantum limits. We also cover the non-QUE result of Hassell in the case of the Bunimovich stadium. We include some discussion of Hecke eigenfunctions and recent results of Soundararajan completing Lindenstrauss' QUE result, in the context of matrix elements for Fourier integral operators. Finally, in answer to the potential question `why study matrix elements' it presents an application of the author to the geometry of nodal sets.Comment: Preliminary version of lecture notes for the 2009 Current Developments in Mathematic