The existence and location of eigenvalues of the one particle discrete Schroedinger operators

TL;DR

This paper investigates the eigenvalues of one-dimensional discrete Schrödinger operators with indefinite external fields, establishing conditions for the existence and placement of one or two eigenvalues relative to the essential spectrum.

Contribution

It provides new results on the possible number and locations of eigenvalues for discrete Schrödinger operators with indefinite sign potentials.

Findings

Operators can have one or two eigenvalues below the essential spectrum.

Operators can have one or two eigenvalues above the essential spectrum.

Operators can have two eigenvalues, one below and one above the essential spectrum.

Abstract

We consider a quantum particle moving in the one dimensional lattice Z and interacting with a indefinite sign external field v. We prove that the associated discrete Schroedinger operator H can have one or two eigenvalues, situated as below the bottom of the essential spectrum, as well as above its top. Moreover, we show that the operator H can have two eigenvalues outside of the essential spectrum such that one of them is situated below the bottom of the essential spectrum, and other one above its top.