Explicit bound states and threshold resonances for two identical fermions on a onedimensional lattice
Year: 2026
Type: article
Abstract: We study a two-particle lattice Schrödinger operator describing two identical fermions on the one-dimensional lattice Z with a nearest-neighbor interaction of strength λ ∈ R. The Hamiltonian acts in the antisymmetric subspace ℓ 2 , a (Z 2 ). Using the direct-integral decomposition with respect to the total quasi-momentum k ∈ T := (−π, π], we reduce the spectral problem to a family of fiber operators H λ (k) acting in the odd relative-coordinate space ℓ 2 , odd (Z). For every k ∈ (−π, π), the fiber Hamiltonian is represented as a one-dimensional discrete Schrödinger operator, equivalently, as a Jacobi operator with a finite-range boundary perturbation, and its essential spectrum is σ ess H λ ( k ) = [4 − 4 cos k/2, 4 + 4 cos k/2]. We give a complete description of the spectral transition at both edges of the essential spectrum. For each k ∈ (−π, π), the operator H λ (k) has a unique simple eigenvalue outside the essential spectrum if and only if |λ| > 2 cos k /2. This eigenvalue is given explicitly by E(k, λ) = 4 + λ + 4 cos 2 k/2 / λ, and the corresponding eigenfunction is obtained in explicit form in the relative coordinate. For |λ| < 2 cos k/2, there is no eigenvalue outside the essential spectrum. The critical threshold transition is also analyzed for each k ∈ (−π, π). At the critical coupling |λ| = 2 cos k/2, the discrete eigenvalue merges with the corresponding edge of the essential spectrum and gives rise to a threshold resonance. The corresponding threshold equation admits a nonzero bounded odd solution belonging to ℓ ∞,odd (Z) \ ℓ 2,odd (Z), showing that the spectral threshold is a resonance rather than an eigenvalue. We further establish the strong-coupling and threshold asymptotics of the discrete spectrum. In particular, in the strong-coupling regime the eigenfunction becomes localized at the interaction sites, whereas at the critical coupling it converges pointwise to the corresponding threshold resonant solution. The exceptional fiber k = π, for which the relative hopping vanishes and the essential spectrum collapses to the point {4}, is treated separately.
Authors: S. S. Ulashov, Sh. I. Khamidov
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W7204775698