{
  "abstract": "We investigate the eigenvalues of Hermitian Toeplitz matrices generated by the symmetric Fisher-Hartwig symbol $\\hat a(z) = (1-z)^{α/2}(1-z^{-1})^{α/2}$ for $α\\in (0,2)$. Using Dirichlet-Neumann bracketing of the discrete Laplacian, we establish explicit, non-asymptotic bounds for the individual eigenvalues. As a consequence, we prove that all eigenvalues are simple. We also obtain a two-term approximation for every eigenvalue, with explicit bounds on the remainder, as the size of the matrix tends to infinity. While known results are limited to $α> 1$, we bridge this gap by covering the full range of $α\\in (0,2)$. Our approach uses a construction of approximate eigenvectors that has not previously been applied in this setting.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01498",
  "author_total": 1,
  "authors": [
    {
      "name": "Jacek Wszoła"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1mt5287sxyg3v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01498",
  "fulltext_url": null,
  "id": "1mt5287sxyg3v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01498",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.SP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.01498",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1mt5287sxyg3v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01498",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01498"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Eigenvalues of Hermitian Toeplitz matrices with Fisher--Hartwig symbols",
  "title_text": "Eigenvalues of Hermitian Toeplitz matrices with Fisher--Hartwig symbols",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/1mt5287sxyg3v",
  "venue": "arXiv",
  "versions": []
}