Bound states of Schrödinger-type operators on one and two dimensional lattices

Abstract

We study the spectral properties of the Schrödinger-type operator

$${\hat{\mathit{H}}}_{\mu }:={\hat{\mathit{H}}}_0+\mu \hat{\mathit{V}},\mu \ge 0,$$

associated to a one-particle system in d-dimensional lattice ${\mathbb{Z}}^d$, $d=1,2$, where the non-perturbed operator ${\hat{\mathit{H}}}_0$ is a self-adjoint convolution-type operator generated by a Hopping matrix $\hat{\mathfrak{e}}:{\mathbb{Z}}^d\to \mathbb{C}$ and the potential $\hat{\mathit{V}}$ is the multiplication operator by $\hat{v}:{\mathbb{Z}}^d\to \mathbb{R}$. Under certain regularity assumption on $\hat{\mathfrak{e}}$ and a decay assumption on $\hat{v}$, we establish the existence or non-existence and also the finiteness of eigenvalues of ${\hat{\mathit{H}}}_{\mu }$. Moreover, in the case of existence we study the asymptotics of eigenvalues of ${\hat{\mathit{H}}}_{\mu }$ as $\mu \searrow 0$.