Spectral Asymptotics for Nonlocal One-dimensional Schrödinger Operator with Neumann Condition and Translation in Free Term

We consider a nonlocal differential-difference Schrödinger operator on a segment with the Neumann conditions and a translation in the free term. This translation is regarded as a nonlocal perturbation and the value of the translation is regarded as a parameter. We show that the considered operator is $m$-sectorial and its spectrum consists of discrete eigenvalues accumulating at infinity only. Our first result is the uniform in this parameter spectral asymptotics for such operator, that is, the asymptotics for the eigenvalues in their index uniformly in the translation parameter. The asymptotics exhibits a non-trivial high-frequency phenomenon generated by the translation. Our second main result says that the eigenfunctions and corresponding generalized eigenfunctions of the considered operator form a Bari basis.