{
  "abstract": "Let $X=\\mathrm{PSL}_2(\\mathbb Z)\\backslash\\mathbb H$, and let $\\{u_j\\}$ be an orthonormal basis of even Hecke--Maass cusp forms on $X$ with Laplace eigenvalues $1/4+t_j^2$. For a real-valued $ψ\\in C_c^\\infty(0,\\infty)$, define \\[ μ_j(ψ)=\\int_0^\\inftyψ(y)|u_j(iy)|^2\\,\\frac{dy}{y}, \\qquad M(ψ)=\\frac6π\\int_0^\\inftyψ(y)\\,\\frac{dy}{y}. \\] For $T\\ge2$, $1\\le U\\le T$, and every $\\varepsilon>0$, we prove \\[ \\sum_{\\substack{u_j\\ {\\rm even}\\\\T<t_j\\le T+U}} L(1,\\mathrm{sym}^2u_j)|μ_j(ψ)-M(ψ)|^2 \\ll_{ψ,\\varepsilon}T^{1+\\varepsilon}. \\] As a consequence, for fixed $0<θ\\le1$, $0<η<θ/2$, and every $\\varepsilon>0$, if $T^θ\\le U\\le T$, then all but \\[ O_{ψ,θ,η,\\varepsilon} \\!\\left(T^{1+2η+\\varepsilon}\\right) \\] of the even forms with $T<t_j\\le T+U$ satisfy $μ_j(ψ)=M(ψ)+O(T^{-η})$. In particular, the exceptional proportion tends to zero in this short spectral interval.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27454",
  "author_total": 2,
  "authors": [
    {
      "name": "Qingfeng Sun"
    },
    {
      "name": "Qizhi Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5ex3nacy6j98c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27454",
  "fulltext_url": null,
  "id": "5ex3nacy6j98c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27454",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223115427",
  "pdf_url": "https://arxiv.org/pdf/2610.27454",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5ex3nacy6j98c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27454",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27454"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Quantum Variance for Hecke--Maass Cusp Forms on the Vertical Geodesic",
  "title_text": "Quantum Variance for Hecke--Maass Cusp Forms on the Vertical Geodesic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/5ex3nacy6j98c",
  "venue": "arXiv",
  "versions": []
}