{
  "abstract": "In this paper, we study the Dirichlet eigenvalue problem for the logarithmic Hardy operator, defined as the logarithmic Laplacian with a critical logarithmic potential, in a bounded Lipschitz domain containing the origin. We first establish a logarithmic Hardy inequality and use it to identify the associated energy space, showing that the embedding into \\(L^2\\) is compact when the perturbation parameter exceeds the endpoint value \\(-1\\), but fails to be compact at that endpoint. For parameters above \\(-1\\), we develop a complete variational spectral theory, prove that the problem admits a discrete sequence of eigenvalues tending to infinity, characterized by successive minimizations over orthogonal complements, whose eigenfunctions form a complete orthonormal basis of \\(L^2\\). We further establish a uniform lower bound for the first eigenvalue, a scaling property under domain dilation that determines a critical radius for the sign of the first eigenvalue and leaves the eigenvalue gaps invariant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23593",
  "author_total": 3,
  "authors": [
    {
      "name": "Tangrui Liao"
    },
    {
      "name": "Ying Wang"
    },
    {
      "name": "Minghua Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cesv48ppbcjy6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23593",
  "fulltext_url": null,
  "id": "cesv48ppbcjy6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23593",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7214047751",
  "pdf_url": "https://arxiv.org/pdf/2609.23593",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cesv48ppbcjy6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23593",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23593"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Dirichlet Eigenvalues for the Logarithmic-Hardy operator",
  "title_text": "Dirichlet Eigenvalues for the Logarithmic-Hardy operator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/cesv48ppbcjy6",
  "venue": "arXiv",
  "versions": []
}