{
  "abstract": "The Grushin Laplacian $Δ_α= Δ_x +|x|^{2α}Δ_y $ is a degenerate elliptic operator in $\\mathbb{R}^{h+k}$ that degenerates on $\\{0\\} \\times \\mathbb{R}^k$. We explicitly compute the Grushin spherical harmonics on the gauge sphere $\\{(x,y) \\in \\mathbb{R}^{h+k} : |x|^{2(α+1)}+(α+1)^2|y|^2=1\\}$ and the corresponding spectrum when the non-degenerate dimension $h=1$, a case not covered by the existing literature. The eigenfunctions consist of homogeneous Grushin-harmonic polynomials of degrees $n \\equiv 0,1 \\pmod{α+1}$, even and odd in the $x$-variable respectively, which can be described in terms of Jacobi polynomials.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31137",
  "author_total": 2,
  "authors": [
    {
      "name": "Roberto Bramati"
    },
    {
      "name": "Paolo Luzzini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3jqbb5jgjrkye.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31137",
  "fulltext_url": null,
  "id": "3jqbb5jgjrkye",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31137",
  "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.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7223398306",
  "pdf_url": "https://arxiv.org/pdf/2610.31137",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3jqbb5jgjrkye.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31137",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31137"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Grushin spherical harmonics in non-degenerate dimension one",
  "title_text": "Grushin spherical harmonics in non-degenerate dimension one",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/3jqbb5jgjrkye",
  "venue": "arXiv",
  "versions": []
}