Quantum ergodicity and symmetry reduction

TL;DR

This paper establishes an equivariant quantum ergodicity theorem for eigenfunctions of Schrödinger operators on manifolds with symmetries, linking ergodic properties of symmetry-reduced flows to eigenfunction distribution.

Contribution

It introduces an equivariant quantum ergodicity theorem for symmetric manifolds, extending classical results to the setting with group actions and singular symplectic reduction.

Findings

Proves an equivariant quantum ergodicity theorem under symmetry conditions.

Derives an equivariant semiclassical Weyl law using singular equivariant asymptotics.

Generalizes classical quantum ergodicity results to symmetric settings.

Abstract

We study the ergodic properties of eigenfunctions of Schr\"odinger operators on a closed connected Riemannian manifold $M$ in case that the underlying Hamiltonian system possesses certain symmetries. More precisely, let $M$ carry an isometric effective action of a compact connected Lie group $G$. We prove an equivariant quantum ergodicity theorem assuming that the symmetry-reduced Hamiltonian flow on the principal stratum of the singular symplectic reduction of $M$ is ergodic. We deduce the theorem by proving an equivariant version of the semiclassical Weyl law, relying on recent results on singular equivariant asymptotics. It implies an equivariant version of the Shnirelman-Zelditch-Colin-de-Verdi\`{e}re theorem, as well as a representation theoretic equidistribution theorem. In case that $G$ is trivial, one recovers the classical results.