On the Spectrum of Periodic Operators with Distant Perturbations

Abstract

We consider an arbitrary periodic operator with a finite number of distant perturbations in an arbitrary domain of a multidimensional space. The perturbations are arbitrary localized operators. We introduce distant perturbations through shift operators, perturbing operators, and certain weight functions that satisfy a set of the certain conditions. The main aim of the paper is to study the spectrum of a perturbed operator when the distances between domains in which perturbations are located tend to infinity. The formulation of theses problem is quite general and has not been investigated before. Inprevious papers only the spectral properties of differential operators with distant perturbations were considered. The main results are as follows: invariance of the essential spectrum of the perturbed operator with regard to distant perturbations; the existence of a simple and isolated eigenvalue of a perturbed operator that converges to a simple and isolated eigenvalue of the limit operator. complete asymptotic series for the perturbed eigenvalue and the perturbed eigenfunction. proof of uniform convergence of these series and derivation of explicit formulas for their coefficients. The technique used to obtain the results is to reduce the eigenvalue equation to a regularly perturbed operator equation in a special Hilbert space. The perturbation smallness was described by two small characteristic parameters. The adapted version of the Birman-Schwinger method then applied allows us to reduce the problem to the analysis of an operator equation and to search of some holomorphic function zeros.