{
  "abstract": "This paper constructs explicit closed-form Green's functions for the Helmholtz equation on annular and spherical-shell domains with two independent Robin impedances on the inner and outer boundaries. Graf's and the hyperspherical addition theorems reduce each angular mode to an explicit $2\\times2$ linear system, and the resonance spectrum is governed by a characteristic determinant bilinear in the two impedances. This bilinearity has a direct inverse-problem consequence: two resonant frequencies of one non-radial angular mode generate at most two candidate impedance pairs via an explicit quadratic equation, a third resonance selects the physical pair, and an exact reflection-symmetry obstruction identifies where the radial spherical mode cannot recover the impedances. Spectrally, we prove the branches positive, simple and strictly increasing in both impedances; derive first-order asymptotics at the four corners of the impedance plane, with coefficients given by boundary masses and normal derivatives of the limiting eigenfunctions, and an explicit mixed second-order coefficient at the Dirichlet--Dirichlet corner, which for the radial mode evaluates in closed form to $2π/(R_2-R_1)^3$; establish a low-frequency resonance-free band, with a second-order threshold expansion explicit in dimension three and a rational approximation accurate over the whole impedance range; and prove the universal high-frequency spacing law with a shell-curvature correction. Jacobian-based sensitivity and conditioning criteria are included. The kernels and spectra are computable to machine precision, all asymptotic regimes are confirmed numerically, and the kernels provide reference solutions for finite-element and boundary-element validation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39643",
  "author_total": 1,
  "authors": [
    {
      "name": "Ming Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/93w096hx6z4kt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39643",
  "fulltext_url": null,
  "id": "93w096hx6z4kt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39643",
  "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.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7215069304",
  "pdf_url": "https://arxiv.org/pdf/2609.39643",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/93w096hx6z4kt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:54Z",
      "native_id": "2609.39643",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39643"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Explicit Robin Green's Functions, Resonance Spectra, and Impedance Recovery on Annuli and Spherical Shells",
  "title_text": "Explicit Robin Green's Functions, Resonance Spectra, and Impedance Recovery on Annuli and Spherical Shells",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/93w096hx6z4kt",
  "venue": "arXiv",
  "versions": []
}