{
  "abstract": "We determine the sharp observation-time threshold for the autonomous, pure-potential wave equation with an inverse-square boundary singularity on the unit ball. For every fixed $κ<0$, full-boundary observability holds on $(-T,T)$ whenever $T>1$ and fails when $T<1$. This resolves, for this model, the optimal-time question raised by Enciso, Shao, and Vergara [J. Eur. Math. Soc. 23 (2021), 3459--3495] and extends their parameter range $-\\frac12<κ<0$ to all $κ<0$, including the limit-point regime $κ\\leq-\\frac12$. Our approach is based on the Ingham--Beurling theory of nonharmonic Fourier series. Spherical harmonic decomposition reduces the problem to one-dimensional singular operators, for which we establish frequency separation, local counting estimates, and lower bounds for the boundary norming constants, uniformly in the angular parameter. A quantitative Ingham--Beurling inequality then yields the sharp-time observability estimate. As a by-product, this spectral approach provides an independent proof of boundary observability and an explicit functional characterization of its maximal $L^2$ observation domain. The latter is described by a weighted $\\ell^2$ condition on the spectral coefficients and consists precisely of energy solutions whose distributional Neumann trace belongs to $L^2$. It contains the boundary-admissible energy solutions considered in Enciso--Shao--Vergara's original work.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39475",
  "author_total": 1,
  "authors": [
    {
      "name": "Longben Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e2x03mht4kahn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39475",
  "fulltext_url": null,
  "id": "e2x03mht4kahn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39475",
  "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/2610.39475",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e2x03mht4kahn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39475",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39475"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp observation time for wave equations with an inverse-square boundary potential",
  "title_text": "Sharp observation time for wave equations with an inverse-square boundary potential",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/e2x03mht4kahn",
  "venue": "arXiv",
  "versions": []
}