---
OA_place: repository
OA_type: green
_id: '22058'
abstract:
- lang: eng
  text: "We consider the defocusing H˙ 1-critical nonlinear Schr¨odinger equation
    in all dimensions (n ≥ 3) with a quadratic potential V (x) = ± 1/2|x|^2. We show
    global well-posedness for radial initial data obeying ∇u0(x), xu0(x) ∈L^2. In
    view of the potential V , this is the natural energy space. In the repulsive case,
    we also prove scattering. We follow the approach pioneered by Bourgain and Tao
    in the case of no potential; indeed, we include a proof of their results that
    incorporates a\r\ncouple of simplifications discovered while treating the problem
    with quadratic potential."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
- first_name: Xiaoyi
  full_name: Zhang, Xiaoyi
  last_name: Zhang
citation:
  ama: Killip R, Vişan M, Zhang X. Energy-critical NLS with quadratic potentials.
    <i>Communications in Partial Differential Equations</i>. 2009;34(12):1531-1565.
    doi:<a href="https://doi.org/10.1080/03605300903328109">10.1080/03605300903328109</a>
  apa: Killip, R., Vişan, M., &#38; Zhang, X. (2009). Energy-critical NLS with quadratic
    potentials. <i>Communications in Partial Differential Equations</i>. Informa UK
    Limited. <a href="https://doi.org/10.1080/03605300903328109">https://doi.org/10.1080/03605300903328109</a>
  chicago: Killip, Rowan, Monica Vişan, and Xiaoyi Zhang. “Energy-Critical NLS with
    Quadratic Potentials.” <i>Communications in Partial Differential Equations</i>.
    Informa UK Limited, 2009. <a href="https://doi.org/10.1080/03605300903328109">https://doi.org/10.1080/03605300903328109</a>.
  ieee: R. Killip, M. Vişan, and X. Zhang, “Energy-critical NLS with quadratic potentials,”
    <i>Communications in Partial Differential Equations</i>, vol. 34, no. 12. Informa
    UK Limited, pp. 1531–1565, 2009.
  ista: Killip R, Vişan M, Zhang X. 2009. Energy-critical NLS with quadratic potentials.
    Communications in Partial Differential Equations. 34(12), 1531–1565.
  mla: Killip, Rowan, et al. “Energy-Critical NLS with Quadratic Potentials.” <i>Communications
    in Partial Differential Equations</i>, vol. 34, no. 12, Informa UK Limited, 2009,
    pp. 1531–65, doi:<a href="https://doi.org/10.1080/03605300903328109">10.1080/03605300903328109</a>.
  short: R. Killip, M. Vişan, X. Zhang, Communications in Partial Differential Equations
    34 (2009) 1531–1565.
das_tickbox: '1'
date_created: 2026-06-19T07:57:55Z
date_published: 2009-11-01T00:00:00Z
date_updated: 2026-06-29T10:20:21Z
day: '01'
doi: 10.1080/03605300903328109
extern: '1'
external_id:
  arxiv:
  - math/0611394
intvolume: '        34'
issue: '12'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.math/0611394
mathsc:
- 35Q55
- 35B33
month: '11'
oa: 1
oa_version: Preprint
page: 1531-1565
publication: Communications in Partial Differential Equations
publication_identifier:
  issn:
  - 0360-5302
  - 1532-4133
publication_status: published
publisher: Informa UK Limited
quality_controlled: '1'
scopus_import: '1'
status: public
title: Energy-critical NLS with quadratic potentials
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 34
year: '2009'
...