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

Abstract

We consider a family H a,b (µ) = H 0 + µ V a,b µ > 0, of Schrödinger-type operators on the two dimensional lattice Z 2 , where H 0 is a Laurent-Toeplitz-type convolution operator with a given Hopping matrix e and V a,b is a potential taking into account only the zero-range and one-range interactions, i.e., a multiplication operator by a function v such that v(0) = a, v(x) = b for |x| = 1 and v(x) = 0 for |x| ≥ 2, where a, b ∈ R \ {0}. Under certain conditions on the regularity of e we completely describe the discrete spectrum of H a,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.