On the spectrum of the Schrödinger operator for a three-particle system on a lattice

• Published in: Учёные записки Казанского университета: Серия Физико-математические науки
• Main Authors: A. M. Khalkhuzhaev, Kh. G. Khayitova, I. A. Khujamiyorov
• Format: Article
• Language: English
• Published: Kazan Federal University 2025-10-01
• Subjects: lattice; hamiltonian; schr¨odinger operator; contact potential; fermion; eigenvalue; quasi-momentum; invariant subspace
• Online Access: https://uzakufismat.elpub.ru/jour/article/view/228
• ISSN: 2541-7746; 2500-2198

A three-particle discrete Schrödinger operator Hµ,γ(K) :≡ Hµ,γ(K), K = (K, K, K) ∈ 𝕋3 , which is associated with a system of three particles (two fermions of mass 1 and one other particle of mass m = 1/γ ,) interacting via pairwise repulsive contact potentials µ > 0 on a three-dimensional lattice ℤ3 , was analyzed. Critical values of mass ratios γs(K) and γas(K) were determined such that the operator Hµ,γ(K) has no eigenvalues if γ ∈ (0, γs(K)), the operator Hµ,γ(K) has a single eigenvalue if γ ∈ (γs(K), γas(K)), and the operator Hµ,γ(K) has three eigenvalues lying to the right of the essential spectrum for sufficiently large µ > 0 if γ ∈ (γas(K), +∞).