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arXiv · 2608.28125

Explicit Bound States and Threshold Resonances for Two Identical Fermions on a One-Dimensional Lattice

Sobir S. Ulashov·Shakhobiddin I. Khamidov
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Abstract

We study a two-particle lattice Schr\"odinger operator describing two identical fermions on the one-dimensional lattice $\mathbb Z$ with nearest-neighbor interaction of strength $\lambda\in\mathbb R$. Using the direct-integral decomposition with respect to the total quasi-momentum $k\in\mathbb T:=(-\pi,\pi]$, we reduce the problem to fiber operators $H_\lambda(k)$ acting in the odd relative-coordinate space. For $k\in(-\pi,\pi)$, the essential spectrum is \[ \sigma_{\mathrm{ess}}(H_\lambda(k)) = [4-4\cos(k/2),\,4+4\cos(k/2)]. \] 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 \[ |\lambda|>2\cos(k/2), \] in which case \[ E(k,\lambda) = 4+\lambda+\frac{4\cos^2(k/2)}{\lambda}. \] For $|\lambda|<2\cos(k/2)$ there is no discrete eigenvalue. At the critical coupling $|\lambda|=2\cos(k/2)$, 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 $k=\pi$, where the hopping vanishes and the essential spectrum collapses to $\{4\}$, is treated separately.

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Sobir S. Ulashov, Shakhobiddin I. Khamidov. 2026-08-28. Explicit Bound States and Threshold Resonances for Two Identical Fermions on a One-Dimensional Lattice. https://arxiv.org/abs/2608.28125

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