{
  "abstract": "For area-$4π$ metrics on the two-sphere, we obtain quantitative refinements of classical eigenvalue inequalities and study the joint behavior of the first two Laplace eigenvalues. We prove a sharp quadratic improvement of Hersch's reciprocal-sum inequality in terms of the first eigenvalue, together with a simultaneous quadratic estimate for the first three eigenvalues and a geometric stability result for admissible measures. We also establish a quantitative refinement of Nadirashvili's second-eigenvalue inequality with sharp exponential scale. Finally, using explicit spectral families due to Petrides, a degree argument, and a refined finite-dimensional matrix method, we obtain nontrivial inner and outer bounds for the joint range of $(λ_1,λ_2)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32443",
  "author_total": 2,
  "authors": [
    {
      "name": "Zuoqin Wang"
    },
    {
      "name": "Qingwei Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e5wme5yv3tmt3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32443",
  "fulltext_url": null,
  "id": "e5wme5yv3tmt3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32443",
  "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.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7214755907",
  "pdf_url": "https://arxiv.org/pdf/2609.32443",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e5wme5yv3tmt3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32443",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32443"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Improvements of Classical Eigenvalue Inequalities I: Quantitative Hersch-deficit and the Joint Spectral Range on the Two-Sphere",
  "title_text": "Improvements of Classical Eigenvalue Inequalities I: Quantitative Hersch-deficit and the Joint Spectral Range on the Two-Sphere",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/e5wme5yv3tmt3",
  "venue": "arXiv",
  "versions": []
}