Bound states of discrete Schr\"odinger operators on one and two dimensional lattices

TL;DR

This paper investigates the spectral properties and bound states of discrete Schrödinger operators on one- and two-dimensional lattices, focusing on eigenvalue existence, non-existence, and asymptotics as potential strength varies.

Contribution

It provides new results on eigenvalue existence, non-existence, and asymptotic behavior for discrete Schrödinger operators with specific regularity and decay conditions.

Findings

Eigenvalues can be finite or infinite depending on potential and regularity conditions.

Existence of eigenvalues is established under certain decay assumptions on the potential.

Asymptotic behavior of eigenvalues as potential strength approaches zero is characterized.

Abstract

We study the spectral properties of discrete Schr\"odinger operator $$ \widehat H_\mu=\widehat H_0 + \mu \widehat{V},\qquad \mu\ge0, $$ associated to a one-particle system in $d$-dimensional lattice $\mathbb{Z}^d, $ $d=1,2,$ where the non-perturbed operator $\hat H_0$ is a self-adjoint Laurent-Toeplitz-type operator generated by $\hat e:\mathbb{Z}^d\to\mathbb{C}$ and the potential $\hat V$ is the multiplication operator by $\hat v:\mathbb{Z}^d\to\mathbb{R}.$ Under certain regularity assumption on $\hat e$ and a decay assumption on $\hat v$, we establish the existence or non-existence and also the finiteness of eigenvalues of $\hat H_\mu.$ Moreover, in the case of existence we study the asymptotics of eigenvalues of $\hat H_\mu$ as $\mu\searrow 0.$