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

TL;DR: In this paper, the authors considered compact, Birman-Schwinger type pseudodifferential operators with singular Borel measures and proved a proper version of H.Weyl's asymptotic law for eigenvalues with order depending on dimensional characteristics of the measure.

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 $-l e -\mathbf{N}/2$ 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.