Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II. The compact group case

TL;DR

This paper derives asymptotic formulas for the eigenvalue distribution of certain pseudodifferential operators on bounded domains with symmetry, extending classical results to cases involving singular group actions.

Contribution

It introduces a method to obtain Weyl asymptotics for operators commuting with compact group actions, including singular cases, using partial desingularization and stationary phase techniques.

Findings

Asymptotic formulas for eigenvalue counts in isotypic components

Extension of Weyl law to singular group actions

Estimate for remainder terms in eigenvalue asymptotics

Abstract

Let $G\subset \O(n)$ be a compact group of isometries acting on $n$-dimensional Euclidean space $\R^n$, and ${\bf{X}}$ a bounded domain in $\R^n$ which is transformed into itself under the action of $G$. Consider a symmetric, classical pseudodifferential operator $A_0$ in $\L^2(\R^n)$ that commutes with the regular representation of $G$, and assume that it is elliptic on $\bf{X}$. We show that the spectrum of the Friedrichs extension $A$ of the operator $\mathrm{res} \circ A_0 \circ \mathrm{ext}: \CT({\bf{X}}) \to \L^2({\bf{X}})$ is discrete, and using the method of the stationary phase, we derive asymptotics for the number $N_\chi(\lambda)$ of eigenvalues of $A$ equal or less than $\lambda$ and with eigenfunctions in the $\chi$-isotypic component of $\L^2({\bf{X}})$ as $\lambda \to \infty$, giving also an estimate for the remainder term for singular group actions. Since the considered critical set is a singular variety, we recur to partial desingularization in order to apply the stationary phase theorem.