Spectral properties of Schr\"odinger operator with translations and Neumann boundary conditions

TL;DR

This paper analyzes the spectral properties of a nonlocal Schr"odinger operator with Neumann boundary conditions and translations, providing explicit eigenvalue asymptotics and basis properties of eigenfunctions.

Contribution

It introduces a novel spectral asymptotic expansion for the eigenvalues of a nonlocal Schr"odinger operator with translations, including higher-order terms and error estimates.

Findings

Eigenvalues are represented as convergent series in negative powers of n.

The series converge uniformly in parameters and provide spectral asymptotics.

Eigenfunctions form a Bari basis in L^2 space.

Abstract

We consider a nonlocal differential--difference Schr\"odinger operator on a segment with the Neumann conditions and two translations in the free term. The values of the translations are denoted by $\alpha$ and $\beta$ and are treated as parameters. The spectrum of this operator consists of countably many discrete eigenvalues, which are taken in the ascending order of their absolute values and are indexed by the natural parameter $n.$ Our main result is the representation of the eigenvalues as convergent series in negative powers of $n$ with the coefficients depending on $n,$ $\alpha,$ and $\beta.$ We show that these series converge absolutely and uniformly in $n,$ $\alpha,$ and $\beta$ and they can be also treated as spectral asymptotics for the considered operator with uniform in $\alpha$ and $\beta$ estimates for the error terms. As an example, we find the four--term spectral asymptotics for the eigenvalues with the error term of order $O(n^{-3}).$ This asymptotics involves additional nonstandard terms and 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 in the space of functions square integrable on the unit segment.