Spectral properties of the resolvent difference for singularly perturbed operators

TL;DR

This paper derives sharp spectral estimates for the difference of resolvents of singularly perturbed elliptic operators with measure-based perturbations, including special cases like Robin boundary conditions, and establishes Weyl asymptotics for eigenvalues.

Contribution

It provides new order sharp spectral estimates for resolvent differences involving measure-based singular perturbations and extends results to Robin realizations and eigenvalue asymptotics.

Findings

Spectral estimates for resolvent differences are established.

Weyl asymptotics for eigenvalues are justified in specific geometric cases.

Results apply to operators with measure-supported perturbations and Robin boundary conditions.

Abstract

We obtain order sharp spectral estimates for the difference of resolvents of singularly perturbed elliptic operators $\mathbf{A}+\mathbf{V}_1$ and $\mathbf{A}+\mathbf{V}_2$ in a domain $\Omega\subseteq \mathbb{R}^\mathbf{N}$ with perturbations $\mathbf{V}_1, \mathbf{V}_2$ generated by $V_1\mu,V_2\mu,$ where $\mu$ is a measure singular with respect to the Lebesgue measure and satisfying two-sided or one-sided conditions of Ahlfors type, while $V_1,V_2$ are weight functions subject to some integral conditions. As an important special case, spectral estimates for the difference of resolvents of two Robin realizations of the operator $\mathbf{A}$ with different weight functions are obtained. For the case when the support of the measure is a compact Lipschitz hypersurface in $\Omega$ or, more generally, a rectifiable set of Hau{\ss}dorff dimension $d=\mathbf{N}-1$, the Weyl type asymptotics for eigenvalues is justified.