Spectral properties of the resolvent difference for singularly perturbed operators

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 Hausdorff dimension $d=\mathbf{N}-1$, the Weyl type asymptotics for eigenvalues is justified.