{
  "abstract": "We prove dispersive estimates for the three-dimensional defocusing energy-critical nonlinear Schrödinger equation associated with $\\mathcal L_a=-Δ+a|x|^{-2}$. For nonnegative potentials, we extend the known finite-$p$ theory to the endpoint $L^1\\to L^\\infty$. For negative potentials in the global well-posedness range, set $σ=\\frac12-\\sqrt{\\frac14+a}$. We obtain the free strong decay rate for $2<p<3/σ$ and the limiting Lorentz estimate from $L^{(3/σ)',1}$ to $L^{3/σ,\\infty}$. The restriction $p<3/σ$ is sharp for strong Lebesgue decay. The proof combines a finite-interval bootstrap, nonlinear real interpolation on bounded energy sets, and endpoint Sobolev--Lorentz estimates adapted to $\\mathcal L_a$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07585",
  "author_total": 2,
  "authors": [
    {
      "name": "Tiesong Jiang"
    },
    {
      "name": "Kexue Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aq720jp1byksh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07585",
  "fulltext_url": null,
  "id": "aq720jp1byksh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07585",
  "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",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7212032129",
  "pdf_url": "https://arxiv.org/pdf/2609.07585",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aq720jp1byksh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07585",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07585"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Strong and Weak-Type Dispersive Estimates for the Energy-Critical Nonlinear Schrödinger Equation with an Inverse-Square Potential",
  "title_text": "Strong and Weak-Type Dispersive Estimates for the Energy-Critical Nonlinear Schrödinger Equation with an Inverse-Square Potential",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/aq720jp1byksh",
  "venue": "arXiv",
  "versions": []
}