Semiclassical analysis and symmetry reduction II. Equivariant quantum ergodicity for invariant Schr\"odinger operators on compact manifolds

TL;DR

This paper extends quantum ergodicity results to Schr"odinger operators on symmetric compact manifolds, showing that symmetry-reduced flows are ergodic lead to equidistribution of eigenfunctions, generalizing classical theorems.

Contribution

It proves an equivariant quantum ergodicity theorem for invariant Schr"odinger operators on manifolds with symmetry, generalizing classical results to the equivariant setting.

Findings

Establishes an equivariant quantum ergodicity theorem under symmetry assumptions.

Derives an equivariant version of the Shnirelman-Zelditch-Colin-de-Verdière theorem.

Provides a representation theoretic equidistribution result.

Abstract

We study the ergodic properties of Schr\"odinger operators on a compact connected Riemannian manifold $M$ without boundary in case that the underlying Hamiltonian system possesses certain symmetries. More precisely, let $M$ carry an isometric and effective action of a compact connected Lie group $G$. Relying on an equivariant semiclassical Weyl law proved in Part I of this work, we deduce an equivariant quantum ergodicity theorem under the assumption that the symmetry-reduced Hamiltonian flow on the principal stratum of the singular symplectic reduction of $M$ is ergodic. In particular, we obtain an equivariant version of the Shnirelman-Zelditch-Colin-de-Verdi\`ere theorem, as well as a representation theoretic equidistribution theorem. If $M/G$ is an orbifold, similar results were recently obtained by Kordyukov. When $G$ is trivial, one recovers the classical results.