The number and location of eigenvalues for the two-particle Schrödinger operators on lattices

2023/04/23 by S. N. Lakaev, Lakaev, Saidakhmat N., M. O. Akhmadova +1 · 1 citation
Materials Science · Mathematics · #FOS: Physical sciences #Mathematical Physics (math-ph) #Quasicrystal Structures and Properties #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2304.11610

openalex publication_date 2023/04/23 · openalex created_date 2023/04/27 · openalex updated_date 2026/07/28

Abstract

We study the Schrödinger operators H_γλμ(K), K∈\T being a fixed (quasi)momentum of the particles pair, associated with a system of two identical bosons on the one-dimensional lattice ℤ, where the real quantities γ, λ and μ describe the interactions between pairs of particles on one site, two nearest neighboring sites and next two neighboring sites, respectively. We found a partition of the three-dimensional space (γ, λ,μ) of interaction parameters into connected components and the exact number of eigenvalues of this operator that lie below and above the essential spectrum, in each component. Moreover, we show that for any K∈\T^d the number of eigenvalues of H_γλμ(K) is not less than the corresponding number of eigenvalues of H_γλμ(0).

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• Threshold Resonances, Critical Couplings, and Eigenvalue Bounds for Two-Particle Operators on ℤ³

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