On the resolvents of periodic operators with distant perturbations

We consider distant perturbations for an abstract periodic operator. The unperturbed operator is introduced as a closed operator on the Sobolev space defined on a periodic domain in a multidimensional space. We impose certain condition for the unperturbed operator being a natural generalization of the ellipticity and periodicity conditions for the differential operators. The perturbations are described by abstract relatively bounded operators being localized in a certain sense. We study the case when the distance between the domains, where the perturbations are localized, increases unboundedly. The main obtained result is the explicit representation for the resolvent of the perturbed operator.