{
  "abstract": "We study the inverse problem of determining a connected, closed Riemannian manifold of dimension at least two from wave observations. The sources are supported in an open set $S$, and the resulting waves are observed on an open receiver set $R$, which may be at a positive distance from $S$. We prove uniqueness up to isometry within the class of manifolds that satisfy a subcritical spectral lower bound: $\\|ϕ|_R\\|_{L^2(R)} \\ge ce^{-Cλ^θ}\\|ϕ\\|_{L^2(M)}$ for every eigenfunction $ϕ$, where $λ$ is the corresponding eigenvalue, $θ\\in[0,1/2)$, and the constants $c,C>0$ do not depend on $ϕ$. If a bound of this type holds also on $S$, then uniqueness holds within the class of all connected, closed manifolds. By the spectral inequality, the bound always holds with $θ=1/2$. Uniqueness nevertheless fails at this endpoint: in every dimension $n\\ge 2$, we construct a pair of nonisometric connected, closed Riemannian $n$-manifolds that produce identical wave observations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00871",
  "author_total": 1,
  "authors": [
    {
      "name": "Lauri Ylinen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1gg1wh6bfxdcf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00871",
  "fulltext_url": null,
  "id": "1gg1wh6bfxdcf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00871",
  "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.DG"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7215995483",
  "pdf_url": "https://arxiv.org/pdf/2610.00871",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1gg1wh6bfxdcf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00871",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00871"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Uniqueness and nonuniqueness in the disjoint data inverse problem: a sharp spectral threshold",
  "title_text": "Uniqueness and nonuniqueness in the disjoint data inverse problem: a sharp spectral threshold",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/1gg1wh6bfxdcf",
  "venue": "arXiv",
  "versions": []
}