{
  "abstract": "We establish a spectral theorem for compact self-adjoint linear relations on a Hilbert space. The proof relies on the Arens decomposition of a closed linear relation into an operator part and a purely multivalued part, reducing the problem directly to the classical spectral theorem for compact self-adjoint operators. No finite-dimensionality condition is imposed on the multivalued part. We also prove a finiterank approximation theorem for compact bounded linear relations and a Fredholm alternative for compact self-adjoint linear relations. As an application, we employ the theory of boundary triples to show that the inverse of the Neumann Laplacian on a bounded domain is a compact self-adjoint linear relation. The abstract spectral theorem then recovers the classical eigenfunction expansion of solutions to the Neumann problem: an L2-convergent series in terms of the non-constant Neumann eigenfunctions together with an arbitrary additive constant. The framework extends naturally to general elliptic boundary value problems with finite-dimensional kernel.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13385",
  "author_total": 2,
  "authors": [
    {
      "name": "Mathew Thomas"
    },
    {
      "name": "V. B. Kiran Kumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ab4gennwaw59f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13385",
  "fulltext_url": null,
  "id": "ab4gennwaw59f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13385",
  "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.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7221564009",
  "pdf_url": "https://arxiv.org/pdf/2610.13385",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ab4gennwaw59f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13385",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13385"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Spectral theorem for compact self-adjoint linear relations with an application to elliptic boundary value problems",
  "title_text": "Spectral theorem for compact self-adjoint linear relations with an application to elliptic boundary value problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/ab4gennwaw59f",
  "venue": "arXiv",
  "versions": []
}