Perturbation of Hill operator by narrow potentials

We consider a perturbation of periodic operator of the second order on the axis, which is a special case of the Hill operator. Perturbation is realized by a complex sum of two complex-valued potentials with compact supports depending on two small parameters, one of which describes the length of the carriers of potentials and the reciprocal of the second one corresponds to the maximum values of potential modules. We obtain the sufficient condition, under which from the edges of non-degenerate lacunas of continuous spectrum their eigenvalues arise, and construct asymptotics. We adduce a sufficient condition under which the eigenvalues do not arise.