{
  "abstract": "We construct representations of the symmetry algebra and derive the exact spectrum of a three-dimensional oscillator with two spin-$\\tfrac12$ degrees of freedom. The Hamiltonian and its first-order scalar and pseudo-scalar integrals were obtained in earlier classifications. Here we determine their action on the complete spectral basis. At fixed total angular momentum, a quadratic algebra acts on the three triplet orbital channels and admits a three-dimensional deformed-oscillator representation, while an overlapping two-state exchange algebra joins the middle triplet to the singlet. These finite modules determine angular--spin multiplicities but do not quantize the radial energy. A separate positive-discrete-series $su(1,1)$ representation of the common radial Hamiltonian generates the energy sequence. We give the finite matrices, Casimir and structure functions, normalized eigenfunctions, radial ladder actions, the exceptional $j=0$ sector, complete energy-shell multiplicities, and the effect of particle interchange.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16317",
  "author_total": 3,
  "authors": [
    {
      "name": "Ian Marquette"
    },
    {
      "name": "O. Ogulcan Tuncer"
    },
    {
      "name": "I. Yurdusen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/18ekfqtr3vhcp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16317",
  "fulltext_url": null,
  "id": "18ekfqtr3vhcp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16317",
  "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"
    ],
    [
      "arxiv",
      "quant-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.16317",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/18ekfqtr3vhcp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:06Z",
      "native_id": "2610.16317",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16317"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Nested polynomial algebras and the exact spectrum of a three-dimensional two-spin oscillator",
  "title_text": "Nested polynomial algebras and the exact spectrum of a three-dimensional two-spin oscillator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/18ekfqtr3vhcp",
  "venue": "arXiv",
  "versions": []
}