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

Abstract

In a domain $\Omega \subseteq {\mathbb{R}}^{\mathbf{N}}$ we consider compact, Birman-Schwinger type, operators of the form ${\mathbf{T}}_{P,\mathfrak{A}}={\mathfrak{A}}^{\ast }P\mathfrak{A}$; 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.

40 P