Quantum Ergodicity and Mixing
Steve Zelditch
TL;DR
The article surveys how chaotic classical dynamics influence quantum spectra and eigenfunctions in the semiclassical limit, focusing on Laplacians on compact manifolds and quantum maps. It develops the bridge between quantum evolution and geodesic flow via Egorov’s theorem, presents Schnirelman-type quantum ergodicity results, and analyzes rate questions through diagonal/off-diagonal variances and spectral measures. It also discusses strong results like QUE in arithmetic settings, limitations from weak mixing and cat maps, and complementary random-wave perspectives that guide conjectures for eigenfunction statistics. Overall, the work delineates rigorous structures connecting classical chaos to quantum behavior and outlines key open problems and approaches in quantum chaos.
Abstract
This is an expository article for the Encyclopedia of Mathematical Physics on the subject in the title.