On spectrum of a periodic operator with a small localized perturbation

We study the spectrum of a periodic self-adjoint operator on the axis perturbed by a small localized nonself-adjoint operator. It is shown that the continuous spectrum is independent of the perturbation, the residual spectrum is empty, and the point spectrum has no finite accumulation points. We address the existence of the embedded eigenvalues. We establish the necessary and sufficient conditions of the existence of the eigenvalues and construct their asymptotics expansions. The asymptotics expansions for the associated eigenfunctions are also obtained. The examples are given.