On the essential and discrete spectrum of a model operator related to three-particle discrete Schrödinger operators

2005/01/11 by Sergio Albeverio, S. N. Lakaev, Albeverio, Sergio +4
Mathematics · Physics and Astronomy · #47N50 #Differential Equations and Boundary Problems #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Primary: 81Q10 #Secondary: 35P20 #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP #msc:35P20 #msc:47N50 #msc:81Q10

paper · pdf · doi:10.48550/arxiv.math-ph/0501024

arxiv created 2005/01/11 · openalex publication_date 2005/01/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A model operator H corresponding to a three-particle discrete Schrödinger operator on a lattice \Z³ is studied. The essential spectrum is described via the spectrum of two Friedrichs models with parameters h_α(p), α=1,2, p ∈ \T³=(-π,π]³. The following results are proven: 1) The operator H has a finite number of eigenvalues lying below the bottom of the essential spectrum in any of the following cases: (i) both operators h_α(0), α=1,2, have a zero eigenvalue; (ii) either h₁(0) or h₂(0) has a zero eigenvalue. 2) The operator H has infinitely many eigenvalues lying below the bottom and accumulating at the bottom of the essential spectrum, if both operators h_α(0),α=1,2, have a zero energy resonance.

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