{
  "abstract": "We prove a three-term asymptotic formula for the eigenvalues of the fractional Laplacian on the bounded interval $(-1,1)$. This improves the eigenvalue asymptotics of Kulczycki--Kwaśnicki--Małecki--Stós and Kwaśnicki, and confirms the conjectural $O_α(n^{-2})$ remainder suggested by the numerical simulations of Kaleta--Kwaśnicki--Małecki. Moreover, we prove that the normalized eigenfunctions are bounded uniformly in the eigenvalue index $n$ and the fractional order $α$. This settles the conjecture proposed by Kwaśnicki through numerical experiments. Furthermore, we prove that the $n$-th eigenfunction has exactly $n-1$ zeros in the interval $(-1,1)$ and every zero is simple, and hence there are exactly $n$ nodal domains. A key ingredient in the proof is an explicit representation of the eigenfunction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.23457",
  "author_total": 1,
  "authors": [
    {
      "name": "Cheng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3v4bc2kzsdtkw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.23457",
  "fulltext_url": null,
  "id": "3v4bc2kzsdtkw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.23457",
  "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",
      "cs.NA"
    ],
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      "arxiv",
      "math-ph"
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      "arxiv",
      "math.CA"
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      "arxiv",
      "math.MP"
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      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7204166814",
  "pdf_url": "https://arxiv.org/pdf/2608.23457",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3v4bc2kzsdtkw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:52Z",
      "native_id": "2608.23457",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.23457"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Eigenvalues and eigenfunctions of the fractional Laplacian on the interval",
  "title_text": "Eigenvalues and eigenfunctions of the fractional Laplacian on the interval",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/3v4bc2kzsdtkw",
  "venue": "arXiv",
  "versions": []
}