{
  "abstract": "Does the Neumann--Poincaré spectrum determine a planar domain up to an extended Möbius transformation? We construct a pair showing that the answer is no, even for analytic, strictly convex, algebraic boundaries. Their operators are similar. We then prove a positive result for prescribed finite-dimensional families of normalised exterior conformal maps. For every $m\\ge1$, the first $m$ positive eigenvalues determine $m$ real Laurent coefficients on a small truncated convex cone, provided the higher coefficients are a prescribed analytic remainder of quadratic order. Recovery is Lipschitz stable, uniformly down to the disk. The proof combines the Grunsky representation with a nondegeneracy argument for weighted Hankel matrices and differentiated perturbation estimates. Explicit two- and three-parameter examples, single-mode families with repeated leading eigenvalues, and Möbius equivalence classes illustrate the result. Direct proofs for complete polarization data show what spectral weights add to the eigenvalues.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29661",
  "author_total": 2,
  "authors": [
    {
      "name": "Habib Ammari"
    },
    {
      "name": "Matthew J. Colbrook"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4y6rdpdvkq6sy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29661",
  "fulltext_url": null,
  "id": "4y6rdpdvkq6sy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29661",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7223237589",
  "pdf_url": "https://arxiv.org/pdf/2610.29661",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4y6rdpdvkq6sy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:19Z",
      "native_id": "2610.29661",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29661"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "On the planar Neumann--Poincaré inverse spectral problem",
  "title_text": "On the planar Neumann--Poincaré inverse spectral problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/4y6rdpdvkq6sy",
  "venue": "arXiv",
  "versions": []
}