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 \mathbf{A}+\mathbf{V}_{1} and \mathbf{A}+\mathbf{V}_{2} in a domain \Omega\subseteq\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.