On spectral properties of a certain class of perturbed operators

Using the method of similar operators, we examine spectral properties of the perturbed operator ${A-B}:D(A)\subset H \to H$, where $A$ is a self-adjoint operator with a compact resolvent and $B$ is an unbounded perturbation. Under certain conditions on the spectrum of the unperturbed operator $A$ and the perturbation $B$, we obtain estimates of the spectral sets of the perturbed operator. The results obtained are applied to the study of the spectrum of differential operators with periodic boundary conditions and a nonsmooth potential.