Asymptotic Behavior of the Eigenvalues of a Periodic Operator with Two Distant Perturbations on the Axis

Abstract

We consider a second-order periodic operator with two distant perturbations on the real axis. Perturbations are real finite continuous potentials. The objective is to investigate a behavior of the eigenvalues of the perturbed operator when the distance between the potentials tends to infinity.  The study issue is an existence of perturbed eigenvalues in the case of a double limiting eigenvalue (the simple and isolated eigenvalue of a periodic operator with first potential + the simple and isolated eigenvalue of a periodic operator with а second potential).The paper aim is to construct the first terms of the asymptotic expansions of the perturbed eigenvalues and the corresponding eigenfunctions in the case of a double limiting eigenvalue.The technique to obtain the results can also find application when constructing complete asymptotic expansions of perturbed eigenvalues and their corresponding eigenfunctions. The finiteness of the distant potentials allowed us to reveal the complex exponential structure of the asymptotics obtained.The main results include the following:the first terms of the asymptotic expansions of the perturbed eigenvalues and their corresponding eigenfunctions;symmetry of the first corrections of the asymptotics of the perturbed eigenvalues with respect to zero;exponential structure of the asymptotics of perturbed eigenvalues and their corresponding eigenfunctions.