Quantum ergodicity and symmetry reduction

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.