{
  "abstract": "We study a two-particle lattice Schrödinger operator describing two identical fermions on the one-dimensional lattice $\\mathbb Z$ with nearest-neighbor interaction of strength $λ\\in\\mathbb R$. Using the direct-integral decomposition with respect to the total quasi-momentum $k\\in\\mathbb T:=(-π,π]$, we reduce the problem to fiber operators $H_λ(k)$ acting in the odd relative-coordinate space. For $k\\in(-π,π)$, the essential spectrum is \\[ σ_{\\mathrm{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. The operator has a unique simple eigenvalue outside the essential spectrum if and only if \\[ |λ|>2\\cos(k/2), \\] in which case \\[ E(k,λ) = 4+λ+\\frac{4\\cos^2(k/2)}λ. \\] For $|λ|<2\\cos(k/2)$ there is no discrete eigenvalue. At the critical coupling $|λ|=2\\cos(k/2)$, 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 $k=π$, where the hopping vanishes and the essential spectrum collapses to $\\{4\\}$, is treated separately.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28125",
  "author_total": 2,
  "authors": [
    {
      "name": "Sobir S. Ulashov"
    },
    {
      "name": "Shakhobiddin I. Khamidov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cph67bk4fq743.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28125",
  "fulltext_url": null,
  "id": "cph67bk4fq743",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28125",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7204775698",
  "pdf_url": "https://arxiv.org/pdf/2608.28125",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cph67bk4fq743.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:22Z",
      "native_id": "2608.28125",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28125"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Explicit Bound States and Threshold Resonances for Two Identical Fermions on a One-Dimensional Lattice",
  "title_text": "Explicit Bound States and Threshold Resonances for Two Identical Fermions on a One-Dimensional Lattice",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/cph67bk4fq743",
  "venue": "arXiv",
  "versions": []
}