On the spectrum of Schrödinger-type operators on two dimensional lattices

Abstract

We consider a familyHˆa,b(μ)=Hˆ0+μVˆa,bμ>0, of Schrödinger-type operators on the two dimensional lattice Z2, where Hˆ0 is a Laurent-Toeplitz-type convolution operator with a given Hopping matrix eˆ and Vˆa,b is a potential taking into account only the zero-range and one-range interactions, i.e., a multiplication operator by a function vˆ such that vˆ(0)=a, vˆ(x)=b for |x|=1 and vˆ(x)=0 for |x|≥2, where a,b∈R∖{0}. Under certain conditions on the regularity of eˆ we completely describe the discrete spectrum of Hˆa,b(μ) lying above the essential spectrum and study the dependence of eigenvalues on parameters μ, a and b. Moreover, we characterize the threshold eigenfunctions and resonances.