{
  "abstract": "For $A$ positive definite, $B$ positive semidefinite and $p\\ge1$ we show that $\\mathrm{tr}[(A+tB)^{-p}]=\\int(1+t\\ell)^{-p}\\,dρ_p(\\ell)$ for a positive measure $ρ_p$ on the spectral interval of $L=A^{-1/2}BA^{-1/2}$, and we identify the measure: it has atoms $\\mathrm{tr}[(P_sA^{-1}P_s)^p]$ at the eigenvalues of $L$ and a continuous density, an integral of Heinävaara's tracial joint spectral measure along rays. For integer $p$ the positivity of $ρ_p$ follows from the positivity of Heinävaara's measure alone, and the density is a sum of Curry-Schoenberg B-splines; for real $p$ the representation is obtained by inverting Heinävaara's transform. For $0<p<1$ the function is a Stieltjes function of order $p$ only if $[A,B]=0$, but a positive representation of order one holds, with an explicit density. Along $A_p=pI+A_0$, $p\\to\\infty$, the measures converge to the measure of Stahl's theorem (the Bessis-Moussa-Villani conjecture) for $\\mathrm{tr}\\,e^{-(A_0+tB)}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03208",
  "author_total": 1,
  "authors": [
    {
      "name": "Luc Rey-Bellet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fs5qbkr9h6cqq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03208",
  "fulltext_url": null,
  "id": "fs5qbkr9h6cqq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03208",
  "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-ph"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03208",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fs5qbkr9h6cqq.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:04Z",
      "native_id": "2611.03208",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03208"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "A Stieltjes representation of order $p$ for $ \\mathrm{tr}[(A+tB)^{-p}]$: Heinävaara's measure, B-splines, and Stahl's theorem",
  "title_text": "A Stieltjes representation of order p for $ \\mathrm{tr}[(A+tB)^{-p}]$: Heinävaara's measure, B-splines, and Stahl's theorem",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/fs5qbkr9h6cqq",
  "venue": "arXiv",
  "versions": []
}