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

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...

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Bibliographic Details
Published in:arXiv.org 2022-01
Main Authors: Shokhrukh Yu Kholmatov, Lakaev, Saidakhmat N, Almuratov, Firdavsjon M
Format: Article
Language:English
Subjects:
Eigenvalues
Eigenvectors
Lattices
Multiplication
Operators (mathematics)
Online Access:Get full text
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Summary: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.
ISSN:2331-8422