Spectral asymptotics and estimates for matrix Birman-Schwinger operators with singular measures

TL;DR

This paper develops spectral asymptotics and estimates for matrix-valued operators involving singular measures, extending Weyl asymptotics and sharp bounds to operators with complex measure supports.

Contribution

It introduces Weyl type asymptotics for matrix operators with singular measures and proves sharp spectral estimates for Ahlfors-regular measures.

Findings

Established Weyl asymptotics for singular numbers and eigenvalues.

Proved order sharpness of spectral estimates for Ahlfors-regular measures.

Extended spectral analysis to matrix-valued operators with singular measures.

Abstract

We consider operators of the form $\mathbf{T}=\mathbf{A^*}(V\mu)\mathbf{A}$ in $\mathbb{R}^\mathbf{N}$, where $\mathbf{A}$ is a pseudodifferential operator of order $-l$, $\mu$ is a compactly supported singular measure, order $s>0$ Ahlfors-regular, and $V$ is a weight function on the support of $\mu$. The scalar type operator $\mathbf{A}$ and the weight function $V$ are supposed to be $m\times m$ matrix valued. We establish Weyl type asymptotic formulas for singular numbers and eigenvalues of $\mathbf{T}$ for $\mu$ being the natural measure on a compact Lipschitz surface. For a general Ahlfors-regular measure $\mu$, we prove that the previously found upper spectral estimates are order sharp.