The existence and location of eigenvalues of the one particle Hamiltonians on lattices

Abstract

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