Asymptotics of eigenvalues of the zero-range perturbation of the discrete bilaplacian

2019/09/25 by Shokhrukh Yu. Kholmatov, Kholmatov, Shokhrukh Yu., Mardon Pardabaev +1
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Cold Atom Physics and Bose-Einstein Condensates #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.FA #math.MP #math.SP

paper · pdf · doi:10.48550/arxiv.1909.11789

8 pages

openalex publication_date 2019/09/25 · arxiv created 2019/11/19 · arxiv updated 2019/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the family \bf h_μ:=\varDelta \varDelta - μ \bf v, μ∈ℝ, of discrete Schrödinger-type operators in one-dimensional lattice ℤ, where \varDelta is the discrete Laplacian and \bf v is of zero-range. We prove that for any μ≠0 the discrete spectrum of \bf h_μ is a singleton \e(μ)\, and e(μ)<0 for μ>0 and e(μ)>4 for μ<0. Moreover, we study the properties of e(μ) as a function of μ, in particular, we find the asymptotics of e(μ) as μ\searrow0 and μ\nearrow0.

Related

• Expansion of eigenvalues of rank-one perturbations of the discrete bilaplacian
• Bound states of discrete Schrödinger operators on one and two dimensional lattices
• Eigenvalues of the discrete Schrödinger operator in the large coupling constant limit
• Note on the spectrum of discrete Schrödinger operators
• On the discrete spectrum of a family of differential operators