{
  "abstract": "Building on finite-time convergence results for geometric matrix hydrodynamics, we establish global spectral exactness for Zeitlin's approximation of the two-dimensional Euler equation on the sphere linearized about polynomial zonal vorticities. By separating each azimuthal block into a skew-adjoint transport operator and a compact perturbation, we obtain a dimension-uniform estimate $\\|K_N-K\\|=O(N^{-1})$ and the sharper fixed-block estimate $\\|K_{N,m}-K_m\\|=O(N^{-2})$. These bounds, together with strong-resolvent convergence of transport, yield exact approximation of the isolated spectrum off the imaginary axis: algebraic multiplicities are preserved, Riesz projections converge in norm, and spurious off-axis spectral limits are excluded. Every algebraically simple off-axis eigenvalue has an $O(N^{-2})$ approximation error. For eigenvalues of higher multiplicity we establish second-order convergence of arithmetic means of eigenvalue clusters and $O(N^{-2/α})$ individual eigenvalue bounds, where $α$ is the maximal Jordan-block size. For any prescribed positive growth margin, the numbers of unstable and stable eigenvalues beyond that margin are eventually correct. The argument combines an exact matrix-harmonic coordinate recurrence, uniform bounds for high azimuthal modes, and resolvent factorizations involving compact operators. An explicit degree-three zonal equilibrium has a rigorously enclosed simple unstable eigenvalue, providing a concrete instance of the quantitative theory.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01664",
  "author_total": 1,
  "authors": [
    {
      "name": "Robert Vrabel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5k2v831cm7zge.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01664",
  "fulltext_url": null,
  "id": "5k2v831cm7zge",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01664",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01664",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5k2v831cm7zge.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01664",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01664"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Spectral Exactness and Quantitative Instability Approximation in the Euler-Zeitlin Model for Zonal Equilibria",
  "title_text": "Spectral Exactness and Quantitative Instability Approximation in the Euler-Zeitlin Model for Zonal Equilibria",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/5k2v831cm7zge",
  "venue": "arXiv",
  "versions": []
}