{
  "abstract": "In this work, we study the sharp constant \\(\\mathcal H_d\\) in the second-order Hydrogen Uncertainty Principle in every dimension \\(d\\ge2\\). Using a spectral basis built from generalized Laguerre polynomials, together with a linearization procedure and spherical harmonic decomposition, we establish a spectral characterization of \\(\\mathcal H_d\\). More precisely, we show that \\[ \\mathcal H_d = \\frac12\\min\\{d+1,c_d^*\\}, \\] where \\(c_d^*\\in(0,d+3)\\) is the unique parameter such that \\[ \\|S_{c_d^*}^{(d)}\\|_{\\ell^2\\to\\ell^2}=1. \\] Here, for \\(0<c<d+3\\), \\(S_c^{(d)}\\) is an explicit positive Hilbert--Schmidt operator on \\(\\ell^2(\\mathbb N_0)\\). This characterization reveals a competition between the radial sector, which yields the value \\(d+1\\), and the first non-radial sector, encoded by the spectral parameter \\(c_d^*\\). For \\(d\\ge4\\), we recover the known identity \\[ \\mathcal H_d=\\frac{d+1}{2} \\] through a spectral argument, giving an alternative proof of the results of Cazacu, Flynn and Lam \\cite{CazacuFlynnLam} for \\(d\\ge5\\) and Huang and Ye \\cite{HuangYe} for \\(d=4\\). For \\(d\\in\\{2,3\\}\\), this characterization yields \\[ \\mathcal H_d=\\frac{c_d^*}{2}. \\] We then exploit this spectral formulation to derive an explicit finite recursive scheme yielding effective two-sided bounds for \\(\\mathcal H_2\\) and \\(\\mathcal H_3\\), substantially improving the estimates of Huang and Ye \\cite{HuangYe}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34916",
  "author_total": 2,
  "authors": [
    {
      "name": "Mohamed Jleli"
    },
    {
      "name": "Bessem Samet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1s8wf0tdyk3v7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34916",
  "fulltext_url": null,
  "id": "1s8wf0tdyk3v7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34916",
  "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.AP"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7223870213",
  "pdf_url": "https://arxiv.org/pdf/2610.34916",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1s8wf0tdyk3v7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34916",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34916"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Sharp Constant in the Second-Order Hydrogen Uncertainty Principle: Spectral Characterization and Quantitative Approximation",
  "title_text": "The Sharp Constant in the Second-Order Hydrogen Uncertainty Principle: Spectral Characterization and Quantitative Approximation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/1s8wf0tdyk3v7",
  "venue": "arXiv",
  "versions": []
}