Spectral Properties of Schrödinger Operator With Translations and Neumann Boundary Conditions

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 (Formula presented.) We represent the eigenvalues as series in negative powers of (Formula presented.) with the coefficients depending on (Formula presented.) (Formula presented.) and (Formula presented.) which converge absolutely and uniformly in (Formula presented.) (Formula presented.) and (Formula presented.) and serve as spectral asymptotics for the considered operator with uniform in (Formula presented.) and (Formula presented.) estimates for the error terms. As an example, we find the four-term spectral asymptotics for the eigenvalues with the error term of order (Formula presented.) 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. © 2025 John Wiley & Sons Ltd.