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Expansion of eigenvalues of the perturbed discrete bilaplacian

We consider the family [Formula: see text] of discrete Schrödinger-type operators in d-dimensional lattice [Formula: see text] , where [Formula: see text] is the discrete Laplacian and [Formula: see text] is of rank-one. We prove that there exist coupling constant thresholds [Formula: see text] such...

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Detalles Bibliográficos
Autores principales: Kholmatov, Shokhrukh Yu., Khalkhuzhaev, Ahmad, Pardabaev, Mardon
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer Vienna 2022
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8983584/
https://www.ncbi.nlm.nih.gov/pubmed/35464818
http://dx.doi.org/10.1007/s00605-022-01678-1
Descripción
Sumario:We consider the family [Formula: see text] of discrete Schrödinger-type operators in d-dimensional lattice [Formula: see text] , where [Formula: see text] is the discrete Laplacian and [Formula: see text] is of rank-one. We prove that there exist coupling constant thresholds [Formula: see text] such that for any [Formula: see text] the discrete spectrum of [Formula: see text] is empty and for any [Formula: see text] the discrete spectrum of [Formula: see text] is a singleton [Formula: see text] and [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text] Moreover, we study the asymptotics of [Formula: see text] as [Formula: see text] and [Formula: see text] as well as [Formula: see text] The asymptotics highly depends on d and [Formula: see text]