Eigenvalues of the Birman-Schwinger operator for singular measures: the noncritical case

TL;DR

This paper studies the eigenvalues of Birman-Schwinger type operators involving singular measures and pseudodifferential operators, providing asymptotic estimates, Weyl law extensions, and singular number bounds in the noncritical case.

Contribution

It extends eigenvalue and singular number estimates to operators with singular measures and noncritical order pseudodifferential operators, including Weyl law and CLR estimates.

Findings

Derived eigenvalue asymptotics depending on measure dimension

Established Weyl law for a class of singular measure operators

Proved CLR estimate for operators with singular measures

Abstract

In a domain $\Omega\subseteq \mathbb{R}^\mathbf{N}$ we consider compact, Birman-Schwinger type, operators of the form $\mathbf{T}_{P,\mathfrak{A}}=\mathfrak{A}^*P\mathfrak{A}$; here $P$ is a singular Borel measure in $\Omega$ and $\mathfrak{A}$ is a noncritical order $-l\ne -\mathbf{N}/2$ pseudodifferential operator. For a class of such operators, we obtain estimates and a proper version of H.Weyl's asymptotic law for eigenvalues, with order depending on dimensional characteristics of the measure. A version of the CLR estimate for singular measures is proved. For non-selfadjoint operators of the form $P_2 \mathfrak{A} P_1$ and $\mathfrak{A}_2 P \mathfrak{A}_1$ with singular measures $P,P_1,P_2$ and negative order pseudodifferential operators $\mathfrak{A},\mathfrak{A}_1,\mathfrak{A}_2$ we obtain estimates for singular numbers.