Bound states of discrete Schrödinger operators on one and two dimensional lattices

Abstract

We study the spectral properties of discrete Schrödinger operator $${\hat{H}}_{\mu }={\hat{H}}_0+\mu \hat{V},\mu \ge 0,$$ associated to a one-particle system in $d$-dimensional lattice ${\mathbb{Z}}^d,$ $d=1,2,$ where the non-perturbed operator ${\hat{H}}_0$ 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 $\mu \searrow 0.$

23 pages

References in corpus (1)

• On spectral estimates for the Schrödinger operators in global dimension 2