On the spectrum of Schr\"odinger-type operators on two dimensional lattices

TL;DR

This paper analyzes the spectral properties of a family of Schr"odinger-type operators on two-dimensional lattices, focusing on the discrete spectrum, eigenvalue dependence on parameters, and threshold phenomena.

Contribution

It provides a complete description of the discrete spectrum and characterizes eigenfunctions and resonances for these lattice operators.

Findings

Complete spectral characterization above the essential spectrum.

Eigenvalue dependence on parameters μ, a, and b.

Identification of threshold eigenfunctions and resonances.

Abstract

We consider a family $$ \widehat H_{a,b}(\mu)=\widehat H_0 +\mu \widehat V_{a,b}\quad \mu>0, $$ of Schr\"odinger-type operators on the two dimensional lattice $\mathbb{Z}^2,$ where $\widehat H_0$ is a Laurent-Toeplitz-type convolution operator with a given Hopping matrix $\hat{e}$ and $\widehat V_{a,b}$ 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 $|x|\ge2,$ 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.