Quantum ergodicity of C* dynamical systems

TL;DR

This paper presents a simplified and general proof that eigenfunctions of quantized classically ergodic systems become uniformly distributed in phase space, extending previous results to C* dynamical systems.

Contribution

It introduces a new, simpler proof of quantum ergodicity that applies to a broader class of C* dynamical systems, improving upon earlier methods.

Findings

Eigenfunctions become uniformly distributed in phase space

The proof extends to general C* dynamical systems

Simplifies previous proofs of quantum ergodicity

Abstract

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.