{
  "abstract": "We intend to prove that for defocusing nonlinear Schrödinger equations, solutions with finite pseudo conformal energy must be global and scatter. This fact is first proved by Bourgain when $0<s_c<1$: the equation is locally well-posed in $H_x^{s_c}$, and if one further assumes that the initial data satisfies $xu_0\\in L_x^2$, then the solution is global and scatters. The $s_c<0$ case remains not studied, and the best result is given by Beceanu, Deng, Soffer, and Wu: If the initial data is in $H_x^{s_c}$, radial, and compactly supported, then the solution is globally well-posed. In this paper, we extend Bourgain's result to the $s_c<0$ case. We prove that if the initial data satisfies $|x|^{-s_c}u_0\\in L_x^2$ and $xu_0\\in L_x^2$, or if the radial initial data satisfies $u_0\\in \\dot{H}_x^{s_c}$ and $xu_0\\in L_x^2$, then the solution is global and scatters. Bourgain's result is based on the subcritical a priori control on $L_x^{p+2}$, while our argument relies on the supercritical bound on $\\mathcal{F}\\dot H_x^1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Journal of Differential Equations 487 (2027), 114801",
  "arxiv": "2609.37714",
  "author_total": 3,
  "authors": [
    {
      "name": "Yujin Guo"
    },
    {
      "name": "Jia Shen"
    },
    {
      "name": "Changping Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1vgy4dzkm9zpr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37714",
  "fulltext_url": null,
  "id": "1vgy4dzkm9zpr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37714",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.37714",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1vgy4dzkm9zpr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37714",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37714"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Global theory for NLS in the weighted spaces I: finite pseudo conformal energy",
  "title_text": "Global theory for NLS in the weighted spaces I: finite pseudo conformal energy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/1vgy4dzkm9zpr",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1016/j.jde.2026.114801",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}