Semiclassical analysis and symmetry reduction I. Equivariant Weyl law for invariant Schr\"odinger operators on compact manifolds

TL;DR

This paper extends the semiclassical Weyl law to Schr"odinger operators on compact manifolds with symmetries, providing an equivariant spectral asymptotics framework that generalizes classical results.

Contribution

It introduces a generalized equivariant Weyl law for invariant Schr"odinger operators on manifolds with symmetry, including estimates for the remainder term.

Findings

Proves an equivariant Weyl law with remainder estimates.

Utilizes semiclassical functional calculus and singular equivariant asymptotics.

Lays groundwork for an equivariant quantum ergodicity theorem in Part II.

Abstract

We study the spectral 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, if $M$ carries an isometric and effective action of a compact connected Lie group $G$, we prove a generalized equivariant version of the semiclassical Weyl law with an estimate for the remainder, using a semiclassical functional calculus for $h$-dependent functions and relying on recent results on singular equivariant asymptotics. These results will be used to derive an equivariant quantum ergodicity theorem in Part II of this work. When $G$ is trivial, one recovers the classical results.