{
  "abstract": "By using spherical harmonicas decomposition and Gaussian-type Poincaré inequalities, we establish several sharp stability estimates for the following second-order weighted Heisenberg Uncertainty Principle \\begin{equation*} \\int_{\\mathbb{R}^N} \\!\\frac{|Δu|^2} {|x|^{2α}} \\mathrm{d}x \\int_{\\mathbb{R}^N} |x|^{2α+2}|\\nabla u|^2\\mathrm{d}x \\geq \\frac{(N+4α+2)^2}{4} \\left(\\int_{\\mathbb{R}^N} |\\nabla u|^2 \\mathrm{d}x\\right)^2, \\end{equation*} and \\begin{equation*} \\int_{\\mathbb{R}^{N}} \\frac{|Δu|^{2}} {|x|^{2α}} \\mathrm{d}x +\\int_{\\mathbb{R}^{N}} \\left|x\\right|^{2α+2} |\\nabla u|^{2}\\mathrm{d}x \\ge\\left(N+4α+2\\right) \\int_{\\mathbb{R}^{N}} |\\nabla u|^{2}\\mathrm{d}x. \\end{equation*} We also provide the explicit value and the necessary and sufficient condition for attainability of the sharp stability constants. Moreover, when $α=0$, our results reduce into those of [\\emph{Calc. Var. Partial Differential Equations} \\textbf{64} (2025), Paper No. 129], [\\emph{J. Funct. Anal.} \\textbf{290} (2026), Paper No. 111321] and [arXiv:2510.00453].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29170",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiao-Ping Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/86h841rqkgav7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29170",
  "fulltext_url": null,
  "id": "86h841rqkgav7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29170",
  "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|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7214321978",
  "pdf_url": "https://arxiv.org/pdf/2609.29170",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/86h841rqkgav7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29170",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29170"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp stability for the second-order weighted Heisenberg Uncertainty Principle with explicit stability constants and optimizers",
  "title_text": "Sharp stability for the second-order weighted Heisenberg Uncertainty Principle with explicit stability constants and optimizers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/86h841rqkgav7",
  "venue": "arXiv",
  "versions": []
}