{
  "id": "2311.14184",
  "version": "v1",
  "published": "2023-11-23T20:05:03.000Z",
  "updated": "2023-11-23T20:05:03.000Z",
  "title": "Mean Values and Quantum Variance for Degenerate Eisenstein Series of Higher Rank",
  "authors": [
    "Dimitrios Chatzakos",
    "Corentin Darreye",
    "Ikuya Kaneko"
  ],
  "comment": "20 pages. LaTeX2e",
  "categories": [
    "math.NT",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the mean value and quantum variance for ${\\mathrm{SL}_{n}({\\mathbb Z})}$ degenerate maximal parabolic Eisenstein series. We first establish a mean value result for the inner products \\begin{equation*} \\mu_{t, n}(f) = \\int_{\\mathrm{SL}_{n}(\\mathbb{Z}) \\backslash \\mathfrak{h}^{n}} f(z) \\left|E_{(n-1, 1)} \\left(z, \\frac{1}{2}+it, \\mathbf{1} \\right) \\right|^{2} d\\mu(z) \\end{equation*} with $f$ in a restricted space of incomplete parabolic Eisenstein series. Our estimate breaks the barrier with a polynomial power saving beyond the prediction of the Generalised Lindel\\\"{o}f Hypothesis. Moreover, we prove a quantum variance formula for $\\mu_{t, n}(f)$. This paper builds on the work of Zhang (2019) on quantum unique ergodicity for $\\mathrm{SL}_{n}({\\mathbb{Z}})$ degenerate Eisenstein series as well as the work of Huang (2021) on quantum variance for $\\mathrm{SL}_{2}({\\mathbb{Z}})$ Eisenstein series.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-23T20:05:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F12",
      "11F72",
      "58J51",
      "81Q50"
    ],
    "keywords": [
      "degenerate eisenstein series",
      "quantum variance",
      "higher rank",
      "degenerate maximal parabolic eisenstein series",
      "incomplete parabolic eisenstein series"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}