Expansion of eigenvalues of rank-one perturbations of the discrete bilaplacian

2019/10/03 by Khalkhuzhaev, Ahmad, Kholmatov, Shokhrukh Yu., Pardabaev, Mardon
#FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1910.01369

Abstract

We consider the family h_μ:=\varDelta \varDelta - μ v, μ∈ℝ, of discrete Schrödinger-type operators in d-dimensional lattice ℤ^d, where \varDelta is the discrete Laplacian and v is of rank-one. We prove that there exist coupling constant thresholds μ_o,μ^o≥0 such that for any μ∈[-μ^o,μ_o] the discrete spectrum of h_μ is empty and for any μ∈ ℝ∖[-μ^o,μ_o] the discrete spectrum of h_μ is a singleton \e(μ)\, and e(μ)<0 for μ>μ_o and e(μ)>4d² for μ

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