{
  "abstract": "Let $λ_k$ be the $k$th eigenvalue of the Dirichlet Laplacian on an open set $Ω\\subset\\mathbb R^n$ of finite positive measure. The direct individual consequence of the Berezin--Li--Yau sum inequality has leading coefficient $n/(n+2)$ relative to the Weyl term. The main contribution of this paper is a strict improvement of this coefficient. Energy orthogonality gives a frequency-dependent cap on the Fourier density of the first $k$ eigenfunctions. Combining this cap with the standard Bessel bound and a bathtub principle for the radial capacity yields \\[ λ_k\\geq c_n(2π)^2ω_n^{-2/n} |Ω|^{-2/n}k^{2/n}, \\qquad \\frac{n}{n+2}<c_n<1, \\] for every $k\\geq1$ and every $n\\geq2$, without boundary regularity. The constants are characterized by explicit scalar equations; in dimension two, $c_2=0.5383068077\\ldots$, giving a $7.66\\%$ improvement over the individual Li--Yau coefficient. We emphasize that this improves the leading coefficient in the individual eigenvalue bound and the new constant $c_n$ is independent of the geometry and index $k$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23923",
  "author_total": 2,
  "authors": [
    {
      "name": "Yifan Wang"
    },
    {
      "name": "Hehu Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/14ffkdds3m9en.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23923",
  "fulltext_url": null,
  "id": "14ffkdds3m9en",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23923",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7213995874",
  "pdf_url": "https://arxiv.org/pdf/2609.23923",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/14ffkdds3m9en.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23923",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23923"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Improved leading coefficient in the individual Berezin-Li-Yau bound via energy orthogonality",
  "title_text": "Improved leading coefficient in the individual Berezin-Li-Yau bound via energy orthogonality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/14ffkdds3m9en",
  "venue": "arXiv",
  "versions": []
}