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

2022/01/08 by Shokhrukh Yu. Kholmatov, Kholmatov, Shokhrukh Yu., S. N. Lakaev +3
Computer Science · Mathematics · #41A25 #47A55 #47A75 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical functions and polynomials #Matrix Theory and Algorithms #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2201.02800

openalex publication_date 2022/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a family \widehat H_a,b(μ)=\widehat H₀ +μ \widehat V_a,b \mugt;0, of Schr"odinger-type operators on the two dimensional lattice ℤ², where \widehat H₀ is a Laurent-Toeplitz-type convolution operator with a given Hopping matrix 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 v such that v(0)=a, v(x)=b for |x|=1 and v(x)=0 for |x|≥2, where a,b∈ℝ∖\0\. Under certain conditions on the regularity of e we completely describe the discrete spectrum of H_a,b(μ) lying above the essential spectrum and study the dependence of eigenvalues on parameters μ, a and b. Moreover, we characterize the threshold eigenfunctions and resonances.

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