{
  "abstract": "Let $λ_k(Ω)$ denote the $k$th Dirichlet Laplacian eigenvalue of a bounded planar domain $Ω$, with eigenvalues counted with multiplicity. We disprove a conjectured higher-index extension of the Payne--Pólya--Weinberger inequality that was recorded as an open problem by Ashbaugh. For every integer $m\\geq3$, we construct a bounded planar domain $Ω_m$ with $C^\\infty$ boundary such that $$ \\frac{λ_{2m}(Ω_m)}{λ_m(Ω_m)} >\\frac{13}{5} >\\frac{j_{1,1}^2}{j_{0,1}^2}, $$ where $j_{ν,1}$ is the first positive zero of $J_ν$, the Bessel function of the first kind of order $ν$. For $m=3$, the annulus $\\{x\\in\\mathbb{R}^2:1/10<|x|<1\\}$ already provides such a counterexample.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26198",
  "author_total": 3,
  "authors": [
    {
      "name": "Yixin He"
    },
    {
      "name": "Quanyu Tang"
    },
    {
      "name": "Haiqi Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/708x10a1e7mdh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26198",
  "fulltext_url": null,
  "id": "708x10a1e7mdh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26198",
  "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.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7214088354",
  "pdf_url": "https://arxiv.org/pdf/2609.26198",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/708x10a1e7mdh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26198",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26198"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Counterexamples to a higher-index Dirichlet eigenvalue-ratio conjecture",
  "title_text": "Counterexamples to a higher-index Dirichlet eigenvalue-ratio conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/708x10a1e7mdh",
  "venue": "arXiv",
  "versions": []
}