Remarks on quantum ergodicity

TL;DR

This paper extends the Quantum Ergodicity Theorem to broader geometric contexts, analyzing eigenfunction distributions on manifolds where classical ergodic assumptions may not hold.

Contribution

It generalizes the Quantum Ergodicity Theorem and explores eigenfunction distribution in non-ergodic geometric settings.

Findings

Proves a generalized Quantum Ergodicity Theorem for smooth compact Riemannian manifolds.

Analyzes eigenfunction distribution where Liouville measure is non-ergodic.

Provides asymptotic properties of Laplacian eigenfunctions in new geometric contexts.

Abstract

We prove a generalized version of the Quantum Ergodicity Theorem on smooth compact Riemannian manifolds without boundary. We apply it to prove some asymptotic properties on the distribution of typical eigenfunctions of the Laplacian in geometric situations where the Liouville measure is not (or not known to be) ergodic.