Quantum Ergodicity for a Class of Mixed Systems

TL;DR

This paper investigates the behavior of high energy eigenfunctions in quantum systems with mixed classical dynamics, proving that defect measures on ergodic components are proportional to the Liouville measure, confirming part of Percival's conjecture.

Contribution

It establishes that defect measures on ergodic components in mixed systems are proportional to the Liouville measure, advancing understanding of quantum ergodicity in such systems.

Findings

Defect measures on ergodic components are proportional to Liouville measure.

Proves a part of Percival's conjecture for mixed dynamical systems.

Shows eigenfunctions concentrate according to classical invariant measures.

Abstract

We examine high energy eigenfunctions for the Dirichlet Laplacian on domains where the billiard flow exhibits mixed dynamical behavior. (More generally, we consider semiclassical Schrodinger operators with mixed assumptions on the Hamiltonian flow.) Specificially, we assume that the billiard flow has an invariant ergodic component, U, and study defect measures, mu, of positivie density subsequences of eigenfunctions (and, more generally, of almost orthogonal quasimodes). We show that any defect measure associated to such a subsequence, when restricted to U, satisfies mu = c mu_L where mu_L is the Liouville measure. This proves part of a conjecture of Percival.