Bound states of Schrödinger-type operators on one and two dimensional lattices

Abstract

We study the spectral properties of the Schrödinger-type operatorHˆμ:=Hˆ0+μVˆ,μ≥0, associated to a one-particle system in d-dimensional lattice Zd, d=1,2, where the non-perturbed operator Hˆ0 is a self-adjoint convolution-type operator generated by a Hopping matrix eˆ:Zd→C and the potential Vˆ is the multiplication operator by vˆ:Zd→R. Under certain regularity assumption on eˆ and a decay assumption on vˆ, we establish the existence or non-existence and also the finiteness of eigenvalues of Hˆμ. Moreover, in the case of existence we study the asymptotics of eigenvalues of Hˆμ as μ↘0.