Spectral Properties of Schrödinger Operator With Translations and Neumann Boundary Conditions

ABSTRACT We consider a nonlocal differential–difference Schrödinger operator on a segment with the Neumann conditions and two translations in the lower‐order term. The spectrum of this operator consists of countably many discrete eigenvalues taken in the ascending order of their absolute values and indexed by the natural parameter We represent the eigenvalues as series in negative powers of with the coefficients depending on and which converge absolutely and uniformly in and and serve as spectral asymptotics for the considered operator with uniform in and estimates for the error terms. As an example, we find the four‐term spectral asymptotics for the eigenvalues with the error term of order and show that this asymptotics exhibits a non‐trivial high‐frequency phenomenon generated by the translations. We also establish that the system of eigenfunctions and generalized eigenfunctions of the considered operator forms the Bari basis.