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

2020/07/08 by Shokhrukh Yu. Kholmatov, Kholmatov, Shokhrukh, S. N. Lakaev +3
Computer Science · Mathematics · #41A25 #47A55 #47A75 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Operator Algebras (math.OA) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2007.04035

openalex publication_date 2020/07/08 · openalex created_date 2020/07/16 · openalex updated_date 2026/07/28

Abstract

We study the spectral properties of discrete Schrödinger operator \widehat H_μ=\widehat H₀ + μ\widehatV, μ≥0, associated to a one-particle system in d-dimensional lattice ℤ^d, d=1,2, where the non-perturbed operator H₀ is a self-adjoint Laurent-Toeplitz-type operator generated by e:ℤ^d→ℂ and the potential V is the multiplication operator by v:ℤ^d→ℝ. Under certain regularity assumption on e and a decay assumption on v, we establish the existence or non-existence and also the finiteness of eigenvalues of H_μ. Moreover, in the case of existence we study the asymptotics of eigenvalues of H_μ as μ\searrow 0.

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