The number and location of two particle Schrödinger operators on a lattice

2024/07/19 by Ulashov, Sobir, Khamidov, Shakhobiddin, Lakaev, Shukhrat
#FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2407.14033

Abstract

We study the Schrödinger operators H_λμ(K) with K∈\mathbbT² being the fixed quasimomentum of a pair of particles, associated with a system of two arbitrary particles on a two-dimensional lattice ℤ² with on-site and nearest-neighbor interactions of strengths λ∈ℝ and μ∈ℝ, respectively. We divide the (λ,μ)-plane of parameters λ and μ into connected components, such that in each component, the Schrödinger operator H_λμ(0) has a fixed number of eigenvalues. These eigenvalues are located both below the bottom of the essential spectrum and above its top. Additionally, we establish a sharp lower bound for the number of isolated eigenvalues of H_λμ(K) within each connected component.

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