{
  "abstract": "We disprove Kenig's 1994 spectral-radius conjecture for the double-layer operator in every dimension $d\\ge2$. For every $t>1/2$ sufficiently close to $1/2$, we construct a bounded simply connected Lipschitz domain with connected boundary such that the double-layer operator on $L^2$ with respect to the surface measure has $\\pm \\mathrm{i} t$ in its Fredholm essential spectrum. In particular, the essential spectral radius is greater than $1/2$. In the plane, the boundary may be chosen differentiable everywhere and smooth away from a single point. Our planar construction develops a refinement mechanism that produces unbounded growth of adjoint resolvent solutions with fixed forcing while keeping the graph slopes uniformly bounded. A closed recursion for the density and its image under an auxiliary singular integral reduces the growth estimate to a certified inequality for Hermitian $2\\times2$ matrices. The certificate uses continuous estimates, exact rational arithmetic and rigorous interval enclosures. Localisation and rescaled graph insertions then produce a weakly null sequence of normalised approximate eigenvectors for the adjoint double-layer operator on one bounded boundary. A compact intertwining relation on Cartesian products of the planar domain with cubes gives the higher-dimensional examples without changing the spectral parameter.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07947",
  "author_total": 2,
  "authors": [
    {
      "name": "Matthew J. Colbrook"
    },
    {
      "name": "Siavash Sadeghi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5g6d9rv46sfdd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07947",
  "fulltext_url": null,
  "id": "5g6d9rv46sfdd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07947",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
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      "arxiv",
      "math-ph"
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      "arxiv",
      "math.AP"
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      "arxiv",
      "math.MP"
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      "arxiv",
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      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
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  "openalex": "W7212023307",
  "pdf_url": "https://arxiv.org/pdf/2609.07947",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5g6d9rv46sfdd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:52Z",
      "native_id": "2609.07947",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07947"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A counterexample to Kenig's conjecture for the Laplace double-layer operator",
  "title_text": "A counterexample to Kenig's conjecture for the Laplace double-layer operator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/5g6d9rv46sfdd",
  "venue": "arXiv",
  "versions": []
}