On the spectrum of discrete Schr\"odinger equation with one-dimensional perturbation

TL;DR

This paper analyzes the spectrum of a discrete Schrödinger equation with a one-dimensional perturbation, providing explicit formulas for the scattering matrix, conditions for spectrum singularity absence, and eigenvalue calculations, with brief remarks on higher-dimensional cases.

Contribution

It derives explicit scattering matrix formulas and spectrum conditions for the discrete Schrödinger equation with one-dimensional perturbation, extending understanding of spectral properties.

Findings

Explicit scattering matrix form obtained

Condition for absence of spectrum singularity established

Eigenvalues identified when the condition fails

Abstract

We consider the spectrum of the discrete Schr\"odinger equation with one-dimensional perturbation. We obtain the explicit form of scattering matrix and find the exact condition of absence of singular part of the spectrum. We calculated also the eigenvalue that appears if this condition is not true. In the last part of our paper we give few remarks on the case of two-dimensional perturbations.