{
  "abstract": "The generalized Zernike family $H_{(N)} = p_1^2 + p_2^2 + \\sum_{n=1}^N γ_n\\,(q_1 p_1 + q_2 p_2)^n$ is a parametric family of two-dimensional superintegrable Hamiltonians, admitting $N$ integrals of motion of degree $N$ in the momenta. A theorem of Nozaleda, Tempesta, and Tondo guarantees that canonical separation coordinates (Darboux--Haantjes coordinates) exist for any such system; the challenge is to construct them explicitly. This paper solves the problem for $N = 2$ -- the classical Zernike system, which is canonically equivalent to the isotropic harmonic oscillator on flat space or on a space of constant curvature -- covering all four known separation types: polar, two Cartesian-type, and elliptic. The key structural fact is that the Haantjes operators associated with all integrals of $H_{(2)}$ have no momentum-dependent off-diagonal block (lift form). We prove that this implies the separation coordinates are reachable by an extended point transformation: the new positions depend only on the old positions, with no momentum entering the coordinate change. In the polar and Cartesian-type cases the new position coordinates involve at most a square root of a single-variable rational function; in the elliptic case they are given by the two roots of a quadratic polynomial in the original coordinates, and the resulting branch structure introduces a fourth regular singular point in the quantum separated ODE, placing it in the Heun class, in agreement with results of Atakishiyev, Pogosyan, Vicent, Wolf, and Yakhno. For $N \\geq 3$ we prove an obstruction: no lift-form Haantjes operator can generate an integral independent of the angular momentum. The separation coordinates for higher Zernike Hamiltonians therefore require momentum-dependent canonical transformations, whose explicit construction is the subject of future work.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: J. Phys. A: Math. Theor. 59 (2026) 425203",
  "arxiv": "2605.23748",
  "author_total": 2,
  "authors": [
    {
      "name": "Ondřej Kubů"
    },
    {
      "name": "Danilo Latini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1x7rm4wjpx96t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.23748",
  "fulltext_url": null,
  "id": "1x7rm4wjpx96t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.23748",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2605.23748",
  "pmcid": null,
  "pmid": null,
  "published": "2026-05-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1x7rm4wjpx96t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:15Z",
      "native_id": "2605.23748",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.23748"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Haantjes algebras, Zernike system and separation of variables",
  "title_text": "Haantjes algebras, Zernike system and separation of variables",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/1x7rm4wjpx96t",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1088/1751-8121/aeaf25",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}