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

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.