Explicit Bound States and Threshold Resonances for Two Identical Fermions on a One-Dimensional Lattice
Abstract
We study a two-particle lattice Schrödinger operator describing two identical fermions on the one-dimensional lattice ℤ with nearest-neighbor interaction of strength λ∈ℝ. Using the direct-integral decomposition with respect to the total quasi-momentum k∈𝕋:=(−π,π], we reduce the problem to fiber operators Hλ(k) acting in the odd relative-coordinate space. For k∈(−π,π), the essential spectrum is σess(Hλ(k)) = [4−4cos(k/2), 4+4cos(k/2)]. We give a complete description of the spectral transition at both edges. The operator has a unique simple eigenvalue outside the essential spectrum if and only if |λ|>2cos(k/2), in which case E(k,λ) = 4+λ+4cos2(k/2)/λ. For |λ|<2cos(k/2) there is no discrete eigenvalue. At the critical coupling |λ|=2cos(k/2), the eigenvalue merges with the corresponding spectral edge and becomes a threshold resonance, with a bounded non-square-integrable odd solution. We also derive the threshold and strong-coupling asymptotics of the eigenvalue and eigenfunction. In the strong-coupling regime the eigenfunction localizes at the interaction sites, whereas at critical coupling it converges pointwise to the corresponding resonant solution. The exceptional fiber k=π, where the hopping vanishes and the essential spectrum collapses to {4}, is treated separately.
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BibTeX
@misc{ulashov2026explicit,
title = {{Explicit Bound States and Threshold Resonances for Two Identical Fermions on a One-Dimensional Lattice}},
author = {Sobir S. Ulashov and Shakhobiddin I. Khamidov},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28125},
eprint = {2608.28125},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28125},
}
RIS
TY - UNPB
TI - Explicit Bound States and Threshold Resonances for Two Identical Fermions on a One-Dimensional Lattice
AU - Sobir S. Ulashov
AU - Shakhobiddin I. Khamidov
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28125
UR - https://arxiv.org/abs/2608.28125
AB - We study a two-particle lattice Schrödinger operator describing two identical fermions on the one-dimensional lattice ℤ with nearest-neighbor interaction of strength λ∈ℝ. Using the direct-integral decomposition with respect to the total quasi-momentum k∈𝕋:=(−π,π], we reduce the problem to fiber operators H_λ(k) acting in the odd relative-coordinate space. For k∈(−π,π), the essential spectrum is σ_ess(H_λ(k)) = [4−4cos(k/2), 4+4cos(k/2)]. We give a complete description of the spectral transition at both edges. The operator has a unique simple eigenvalue outside the essential spectrum if and only if |λ|>2cos(k/2), in which case E(k,λ) = 4+λ+4cos^2(k/2)/λ. For |λ|<2cos(k/2) there is no discrete eigenvalue. At the critical coupling |λ|=2cos(k/2), the eigenvalue merges with the corresponding spectral edge and becomes a threshold resonance, with a bounded non-square-integrable odd solution. We also derive the threshold and strong-coupling asymptotics of the eigenvalue and eigenfunction. In the strong-coupling regime the eigenfunction localizes at the interaction sites, whereas at critical coupling it converges pointwise to the corresponding resonant solution. The exceptional fiber k=π, where the hopping vanishes and the essential spectrum collapses to {4}, is treated separately.
ER -