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arXiv · 2007.04035

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

Shokhrukh Kholmatov·Saidakhmat Lakaev·Firdavs Almuratov
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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 $\mu\searrow 0.$

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Shokhrukh Kholmatov, Saidakhmat Lakaev, Firdavs Almuratov. 2020-07-08. Bound states of discrete Schr\"odinger operators on one and two dimensional lattices. https://arxiv.org/abs/2007.04035

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