643 research outputs found
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
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