Estimates on the number of eigenvalues of two-particle discrete Schrödinger operators

2005/01/12 by Sergio Albeverio, Albeverio, Sergio, S. N. Lakaev +5
Computer Science · Mathematics · Physics and Astronomy · #47N50 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Primary: 81Q10 #Secondary: 35P20 #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #advanced mathematical theories #math-ph #math.MP #math.SP #msc:35P20 #msc:47N50 #msc:81Q10

paper · pdf · doi:10.48550/arxiv.math-ph/0501036

12 pages

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

Abstract

Two-particle discrete Schrödinger operators H(k)=H₀(k)-V on the three-dimensional lattice \Z³, k being the two-particle quasi-momentum, are considered. An estimate for the number of the eigenvalues lying outside of the band of H₀(k) via the number of eigenvalues of the potential operator V bigger than the width of the band of H₀(k) is obtained. The existence of non negative eigenvalues below the band of H₀(k) is proven for nontrivial values of the quasi-momentum k∈ \T³≡ (-π,π]³, provided that the operator H(0) has either a zero energy resonance or a zero eigenvalue. It is shown that the operator H(k), k∈ \T³, has infinitely many eigenvalues accumulating at the bottom of the band from below if one of the coordinates k^(j),j=1,2,3, of k∈ \T³ is π.

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