{
  "abstract": "We introduce a bi-oriented interlaced matrix product on 2 × 2 matrices over a unital, possibly non-associative algebra, extending a matrix mnemonic for Cayley-Dickson doubling. The opposite orientations ⋄ R and ⋄ L coincide precisely when the underlying algebra is commutative. For the canonical position and momentum operators Q, P on L 2 (R), a matrix D and its companion D satisfy, on the common Schwartz domain,\n\nWe study the associated interlaced eigenvalue equations for operator-valued columns. The combinations Φ = A + iB and Ψ = A -iB decouple the equations into two-sided Sylvester equations. We construct explicit rank-one generalized eigenoperators from position and momentum eigenstates. Their distributional kernels belong to the doubled rigged Hilbert space S(R 2 ) ⊂ L 2 (R 2 ) ⊂ S ′ (R 2 ); the corresponding position and momentum operators act on different tensor factors and therefore commute. The rank-one families do not exhaust the distributional solutions: higher-order generalized solutions are also exhibited. We distinguish the algebraic identities on the Schwartz core from questions concerning closed-operator realizations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28511",
  "author_total": 1,
  "authors": [
    {
      "name": "Jean-Pierre Gazeau"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a4m3s6jyt9qm9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28511",
  "fulltext_url": null,
  "id": "a4m3s6jyt9qm9",
  "issns": [],
  "landing_url": "https://hal.science/hal-05772840v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "0.phys"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7214312466",
  "pdf_url": "https://hal.science/hal-05772840/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a4m3s6jyt9qm9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:45Z",
      "native_id": "hal-05772840",
      "source": "hal",
      "url": "https://hal.science/hal-05772840v1"
    },
    {
      "first_seen": "2026-10-30T12:24:11Z",
      "native_id": "2609.28511",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28511"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Bi-Oriented Interlaced Matrix Multiplication and an Operator Factorisation of the Quantum Harmonic Oscillator",
  "title_text": "Bi-Oriented Interlaced Matrix Multiplication and an Operator Factorisation of the Quantum Harmonic Oscillator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/a4m3s6jyt9qm9",
  "venue": "HAL",
  "versions": []
}