Spectral properties of the resolvent difference for singularly perturbed operators

We obtain order sharp spectral estimates for the difference of resolvents of singularly perturbed elliptic operators A+V₁ and A+V₂ in a domain Ω⊆N with perturbations V₁,V₂ generated by V₁μ,V₂μ , where μ is a measure singular with respect to the Lebesgue measure and satisfying two-sided or one-sided conditions of Ahlfors type, while V₁,V₂ 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 Ω or, more generally, a rectifiable set of Hausdorff dimension d=N-1 , the Weyl type asymptotics for eigenvalues is justified.