{
  "abstract": "We construct quasiperiodic Schrödinger operators with Gevrey potentials for which the Lyapunov exponent is discontinuous in the spectral energy. More precisely, let \\(α\\in\\R\\setminus\\mathbb Q\\) satisfy the Diophantine condition \\(\\|kα\\|_{\\T}\\geqγ|k|^{-τ}\\) for all \\(k\\in\\Z\\setminus\\{0\\}\\), with \\(γ>0\\) and \\(τ\\geq1\\). For every Gevrey index \\(\\mathfrak s>1+τ\\) and every sufficiently large coupling \\(λ\\), we construct a real-valued potential \\(v_λ\\in G^{\\mathfrak s}(\\T,\\R)\\) and a sequence of energies \\(E_j\\downarrow0\\) such that \\[ L_{v_λ}(0)\\geq(1-\\varepsilon_λ)\\logλ, \\qquad \\sup_{j\\geq1} L_{v_λ}(E_j)\\leq(1-c_*)\\logλ, \\] where \\(L_{v_λ}(E)\\) denotes the Lyapunov exponent of the Schrödinger operator with potential \\(λv_λ\\), \\(c_*>0\\) is an absolute constant, and \\(\\varepsilon_λ\\to0\\) as \\(λ\\to\\infty\\). For bounded-type frequencies (\\(τ=1\\)), the regularity range \\(\\mathfrak s>2\\) is sharp up to the endpoint \\(\\mathfrak s=2\\), since the Lyapunov exponent is known to be continuous in energy for \\(1<\\mathfrak s<2\\).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32623",
  "author_total": 2,
  "authors": [
    {
      "name": "Jiahao Xu"
    },
    {
      "name": "Shuo Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/18qtq1adg9vtq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32623",
  "fulltext_url": null,
  "id": "18qtq1adg9vtq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32623",
  "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.DS"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7223656450",
  "pdf_url": "https://arxiv.org/pdf/2610.32623",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/18qtq1adg9vtq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32623",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32623"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Energy Discontinuity of the Lyapunov Exponent for Quasiperiodic Schrödinger Operators with Gevrey Potentials",
  "title_text": "Energy Discontinuity of the Lyapunov Exponent for Quasiperiodic Schrödinger Operators with Gevrey Potentials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/18qtq1adg9vtq",
  "venue": "arXiv",
  "versions": []
}