{
  "abstract": "Consider the Mathieu operator $L=-d^2/dx^2+2a\\cos(2x)$ for fixed $a\\in\\mathbb{C}\\setminus\\{0\\}$. For large enough $n$, the periodic ($n$ even) or anti-periodic ($n$ odd) problem on $[0,π]$ has exactly two algebraically simple eigenvalues $λ_n^-,λ_n^+$ in the disc $|λ-n^2|<1$. We prove that, as $n\\to\\infty$, $λ_n^+-λ_n^-=\\pm\\frac{8(a/4)^n}{[(n-1)!]^2}\\exp\\left[-\\frac{a^2(n+2)}{4(n^2-1)^2}-\\frac{15a^4}{128n^7}+O(n^{-8})\\right]$, where the sign depends on the labelling. For real $a$ with $λ_n^-<λ_n^+$, this difference is the length of the $n$th spectral gap. Expanding the exponential determines the relative corrections of orders $n^{-4}$ to $n^{-7}$, beyond the correction $-a^2/(4n^3)$ obtained by Anahtarci and Djakov (2012) with a relative $O(n^{-4})$ remainder. By isospectrality, the same formula holds for the potential $ce^{-2ix}+de^{2ix}$ with $a^2=cd\\ne0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12751",
  "author_total": 1,
  "authors": [
    {
      "name": "Berkay Anahtarci"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4dev7ax4deazk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12751",
  "fulltext_url": null,
  "id": "4dev7ax4deazk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12751",
  "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.CA"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7221561983",
  "pdf_url": "https://arxiv.org/pdf/2610.12751",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4dev7ax4deazk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12751",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12751"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "High-Index Asymptotics of Mathieu Spectral Gaps and Splittings",
  "title_text": "High-Index Asymptotics of Mathieu Spectral Gaps and Splittings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/4dev7ax4deazk",
  "venue": "arXiv",
  "versions": []
}