The number and location of eigenvalues of the one particle discrete Schr¨odinger operators

Abstract

We consider a quantum particle moving in the one dimensional lattice Z and interacting with a indefinite sign external fieldv. We prove that the associated discrete Schrödinger 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.Date: July 16, 2018.