{
  "schema_version": 1,
  "problem_number": "OWR-3389-016",
  "title": "The Harrell–Stubbe Gap Inequality for Dirichlet Eigenvalues Is Never Saturated",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let E_1 ≤ E_2 ≤ ⋯ be the eigenvalues of the Dirichlet Laplacian on a bounded open set Ω ⊂ R^n, and let M_p(J) = ((n+2p)/n · (1/J) Σ_{j≤J} E_j^p)^{1/p}. Harrell and Stubbe proved that M_1(J)^2 − M_2(J)^2 ≥ ¼(E_{J+1} − E_J)^2, and in a problem list of the 2009 Oberwolfach workshop on low eigenvalues of Laplace and Schrödinger operators they asked whether some Ω and J saturate this inequality. We show that the answer is no: the inequality is strict for every bounded open set and every J. Written in terms of the mean and the variance of E_1, …, E_J, the inequality is a gap estimate of Cheng and Yang, so that estimate is strict as well. The proof analyses the case of equality in the Harrell–Stubbe trace identity behind H. C. Yang's inequality. Equality would put each function x_k u_1, where u_1 ≥ 0 is a first eigenfunction, into a finite sum of eigenspaces. Then every partial derivative of u_1 would lie in H^1_0(Ω), so ∫_Ω Δu_1 = 0, which is impossible because Δu_1 = −E_1 u_1. The same argument shows that Yang's first inequality is strict whenever E_{J+1} > E_1, and hence for every J when Ω is connected; in 2002 Ashbaugh left the strictness of this inequality undecided. By contrast, for the harmonic oscillator and on spheres the analogous bounds are equalities at every spectral gap, that is, for every J with E_J < E_{J+1}. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.SP",
    "math.AP"
  ],
  "keywords": [
    "Dirichlet Laplacian",
    "eigenvalue gaps",
    "universal eigenvalue inequalities",
    "Yang's inequality",
    "Harrell–Stubbe inequality",
    "commutator method",
    "sum rules",
    "Oberwolfach Reports",
    "OWR-3389-016",
    "math.SP",
    "math.AP",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-3389-016/",
  "pdf_url": "https://eulersolve.org/papers/owr-3389-016/paper.pdf?v=4c8c05bf41c7",
  "doi": "10.5281/zenodo.23064381",
  "zenodo_record_url": "https://zenodo.org/records/23064381",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers the Harrell–Stubbe question (OWR 6/2009, p. 415) negatively for the Dirichlet Laplacian on every bounded open set in R^n, reading the printed M_2(J) as M_2(J)^2. The identity used is the Harrell–Stubbe trace identity; what is new is the analysis of its equality case. Neumann conditions, Schrödinger operators, manifolds and unbounded domains are not treated.",
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    "source.zip": {
      "sha256": "6751d9ea5525f4bc9741c0c96432b1d0b09a20b7678bd2cdaea18b0e5022bf5d"
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    "verification_report.md": {
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}