{
  "abstract": "Threshold scattering below the ground-state energy and gradient is established for the three-dimensional focusing energy-critical nonlinear Schrödinger equation. A single solution-dependent Gaussian interaction scale, mass-free angular coercivity, and a frequency-gap source estimate control the projected dynamics. These estimates yield quartic spacetime integrability and a zero-mass contradiction, without radial symmetry or an a priori $L^2$ bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10658",
  "author_total": 2,
  "authors": [
    {
      "name": "Pang-hung Chung"
    },
    {
      "name": "Dan Han"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dma4mg3ynnj2m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10658",
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  "landing_url": "https://arxiv.org/abs/2610.10658",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
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      "2610|Advanced Mathematical Physics Problems"
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  "openalex": "W7221168309",
  "pdf_url": "https://arxiv.org/pdf/2610.10658",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
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  "ris": "https://research.draconitic.com/paper/dma4mg3ynnj2m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.10658",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10658"
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
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  ],
  "title": "Threshold Scattering for the Three-Dimensional Focusing Energy-Critical Nonlinear Schrödinger Equation",
  "title_text": "Threshold Scattering for the Three-Dimensional Focusing Energy-Critical Nonlinear Schrödinger Equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/dma4mg3ynnj2m",
  "venue": "arXiv",
  "versions": []
}