Ergodic properties of the quantum geodesic flow on tori

TL;DR

This paper investigates the ergodic behavior of quantum geodesic flows on tori, showing that ergodic averages of certain operators become translationally invariant, up to negligible corrections, under Schrödinger evolution.

Contribution

It establishes the translational invariance of ergodic averages for pseudodifferential operators on flat tori, linking quantum dynamics with classical geodesic flow properties.

Findings

Ergodic averages are translationally invariant operators.

Results hold up to semi-classically negligible corrections.

Connects quantum ergodic properties with classical geodesic flow on tori.

Abstract

We study ergodic averages for a class of pseudodifferential operators on the flat N-dimensional torus with respect to the Schr\"odinger evolution. The later can be consider a quantization of the geodesic flow on $\bT^N$. We prove that, up to semi-classically negligible corrections, such ergodic averages are translationally invariant operators.