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

Abstract

We consider a family $${\hat{H}}_{a,b}(\mu )={\hat{H}}_0+\mu {\hat{V}}_{a,b}\mu >0,$$ of Schrödinger-type operators on the two dimensional lattice ${\mathbb{Z}}^2,$ where ${\hat{H}}_0$ is a Laurent-Toeplitz-type convolution operator with a given Hopping matrix $\hat{e}$ and ${\hat{V}}_{a,b}$ is a potential taking into account only the zero-range and one-range interactions, i.e., a multiplication operator by a function $\hat{v}$ such that $\hat{v}(0)=a,$ $\hat{v}(x)=b$ for $|x|=1$ and $\hat{v}(x)=0$ for $|x|\ge 2,$ where $a,b\in \mathbb{R}\setminus \{0\}.$ Under certain conditions on the regularity of $\hat{e}$ we completely describe the discrete spectrum of ${\hat{H}}_{a,b}(\mu )$ lying above the essential spectrum and study the dependence of eigenvalues on parameters $\mu ,$ $a$ and $b.$ Moreover, we characterize the threshold eigenfunctions and resonances.

28 pages, 3 figures