Spectral properties of the resolvent difference for singularly perturbed operators

Journal article, 2025

We obtain order sharp spectral estimates for the difference of resolvents of singularly perturbed elliptic operators A+V-1 and A+V-2 in a domain Omega subset of R-N with perturbations V-1,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 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=N-1, the Weyl type asymptotics for eigenvalues is justified.