---
OA_place: repository
OA_type: green
_id: '22035'
abstract:
- lang: eng
  text: "We prove that solutions of the cubic nonlinear Schr\\\"odinger equation on
    $\\Bbb{R}^2$ can be approximated by a finite-dimensional Hamiltonian system, uniformly
    on bounded sets of initial data. This is despite the wealth of non-compact symmetries:
    scaling, translation, and Galilei boosts.\r\n\r\nComplementing this approximation
    result, we show that all solutions of the finite-dimensional Hamiltonian system
    we use can be approximated by the full PDE.\r\n\r\nA key ingredient in these results
    is the development of a general methodology for transfering uniform global space-time
    bounds to suitable Fourier truncations of dispersive PDE models.\r\n\r\nAs an
    application, we prove symplectic non-squeezing (in the sense of Gromov) for the
    cubic NLS on $\\Bbb{R}^2$. This is the first symplectic non-squeezing result for
    a Hamiltonian PDE in infinite volume. It is also the first unconditional symplectic
    non-squeezing result in a scaling-critical setting.\r\n\r\nFinally, we discuss
    implications of non-squeezing on the nature of scattering."
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. Finite-dimensional approximation and non-squeezing
    for the cubic nonlinear Schrödinger equation on ℝ2. <i>American Journal of Mathematics</i>.
    2021;143(2):613-680. doi:<a href="https://doi.org/10.1353/ajm.2021.0014">10.1353/ajm.2021.0014</a>
  apa: Killip, R., Vişan, M., &#38; Zhang, X. (2021). Finite-dimensional approximation
    and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2. <i>American
    Journal of Mathematics</i>. Johns Hopkins University Press. <a href="https://doi.org/10.1353/ajm.2021.0014">https://doi.org/10.1353/ajm.2021.0014</a>
  chicago: Killip, Rowan, Monica Vişan, and Xiaoyi Zhang. “Finite-Dimensional Approximation
    and Non-Squeezing for the Cubic Nonlinear Schrödinger Equation on ℝ2.” <i>American
    Journal of Mathematics</i>. Johns Hopkins University Press, 2021. <a href="https://doi.org/10.1353/ajm.2021.0014">https://doi.org/10.1353/ajm.2021.0014</a>.
  ieee: R. Killip, M. Vişan, and X. Zhang, “Finite-dimensional approximation and non-squeezing
    for the cubic nonlinear Schrödinger equation on ℝ2,” <i>American Journal of Mathematics</i>,
    vol. 143, no. 2. Johns Hopkins University Press, pp. 613–680, 2021.
  ista: Killip R, Vişan M, Zhang X. 2021. Finite-dimensional approximation and non-squeezing
    for the cubic nonlinear Schrödinger equation on ℝ2. American Journal of Mathematics.
    143(2), 613–680.
  mla: Killip, Rowan, et al. “Finite-Dimensional Approximation and Non-Squeezing for
    the Cubic Nonlinear Schrödinger Equation on ℝ2.” <i>American Journal of Mathematics</i>,
    vol. 143, no. 2, Johns Hopkins University Press, 2021, pp. 613–80, doi:<a href="https://doi.org/10.1353/ajm.2021.0014">10.1353/ajm.2021.0014</a>.
  short: R. Killip, M. Vişan, X. Zhang, American Journal of Mathematics 143 (2021)
    613–680.
das_tickbox: '1'
date_created: 2026-06-19T07:43:41Z
date_published: 2021-04-01T00:00:00Z
date_updated: 2026-06-22T12:56:27Z
day: '01'
doi: 10.1353/ajm.2021.0014
extern: '1'
external_id:
  arxiv:
  - '1606.07738'
fulldoi: https://doi.org/10.1353/ajm.2021.0014
intvolume: '       143'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1606.07738
month: '04'
oa: 1
oa_version: Preprint
page: 613-680
publication: American Journal of Mathematics
publication_identifier:
  eissn:
  - 1080-6377
publication_status: published
publisher: Johns Hopkins University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Finite-dimensional approximation and non-squeezing for the cubic nonlinear
  Schrödinger equation on ℝ2
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 143
year: '2021'
...
---
OA_place: repository
OA_type: green
_id: '22087'
abstract:
- lang: eng
  text: "We prove that solutions of the cubic nonlinear Schrödinger equation on $\\Bbb{R}^2$
    can be approximated by a finite-dimensional Hamiltonian system, uniformly on bounded
    sets of initial data. This is despite the wealth of non-compact symmetries: scaling,
    translation, and Galilei boosts.\r\n\r\nComplementing this approximation result,
    we show that all solutions of the finite-dimensional Hamiltonian system we use
    can be approximated by the full PDE.\r\n\r\nA key ingredient in these results
    is the development of a general methodology for transfering uniform global space-time
    bounds to suitable Fourier truncations of dispersive PDE models.\r\n\r\nAs an
    application, we prove symplectic non-squeezing (in the sense of Gromov) for the
    cubic NLS on $\\Bbb{R}^2$. This is the first symplectic non-squeezing result for
    a Hamiltonian PDE in infinite volume. It is also the first unconditional symplectic
    non-squeezing result in a scaling-critical setting.\r\n\r\nFinally, we discuss
    implications of non-squeezing on the nature of scattering."
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. Finite-dimensional approximation and non-squeezing
    for the cubic nonlinear Schrödinger equation on ℝ2. <i>American Journal of Mathematics</i>.
    2021;143(2):613-680. doi:<a href="https://doi.org/10.1353/ajm.2021.0014">10.1353/ajm.2021.0014</a>
  apa: Killip, R., Vişan, M., &#38; Zhang, X. (2021). Finite-dimensional approximation
    and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2. <i>American
    Journal of Mathematics</i>. Johns Hopkins University Press. <a href="https://doi.org/10.1353/ajm.2021.0014">https://doi.org/10.1353/ajm.2021.0014</a>
  chicago: Killip, Rowan, Monica Vişan, and Xiaoyi Zhang. “Finite-Dimensional Approximation
    and Non-Squeezing for the Cubic Nonlinear Schrödinger Equation on ℝ2.” <i>American
    Journal of Mathematics</i>. Johns Hopkins University Press, 2021. <a href="https://doi.org/10.1353/ajm.2021.0014">https://doi.org/10.1353/ajm.2021.0014</a>.
  ieee: R. Killip, M. Vişan, and X. Zhang, “Finite-dimensional approximation and non-squeezing
    for the cubic nonlinear Schrödinger equation on ℝ2,” <i>American Journal of Mathematics</i>,
    vol. 143, no. 2. Johns Hopkins University Press, pp. 613–680, 2021.
  ista: Killip R, Vişan M, Zhang X. 2021. Finite-dimensional approximation and non-squeezing
    for the cubic nonlinear Schrödinger equation on ℝ2. American Journal of Mathematics.
    143(2), 613–680.
  mla: Killip, Rowan, et al. “Finite-Dimensional Approximation and Non-Squeezing for
    the Cubic Nonlinear Schrödinger Equation on ℝ2.” <i>American Journal of Mathematics</i>,
    vol. 143, no. 2, Johns Hopkins University Press, 2021, pp. 613–80, doi:<a href="https://doi.org/10.1353/ajm.2021.0014">10.1353/ajm.2021.0014</a>.
  short: R. Killip, M. Vişan, X. Zhang, American Journal of Mathematics 143 (2021)
    613–680.
das_tickbox: '1'
date_created: 2026-06-19T08:46:12Z
date_published: 2021-04-01T00:00:00Z
date_updated: 2026-07-01T12:38:11Z
day: '01'
doi: 10.1353/ajm.2021.0014
extern: '1'
external_id:
  arxiv:
  - '1606.07738'
fulldoi: https://doi.org/10.1353/ajm.2021.0014
intvolume: '       143'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1606.07738
month: '04'
oa: 1
oa_version: Preprint
page: 613-680
publication: American Journal of Mathematics
publication_identifier:
  eissn:
  - 1080-6377
publication_status: published
publisher: Johns Hopkins University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Finite-dimensional approximation and non-squeezing for the cubic nonlinear
  Schrödinger equation on ℝ2
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 143
year: '2021'
...
---
OA_place: repository
OA_type: green
_id: '22041'
abstract:
- lang: eng
  text: We consider the defocusing energy-critical nonlinear Schr\"odinger equation
    in the exterior of a smooth compact strictly convex obstacle in three dimensions.
    For the initial-value problem with Dirichlet boundary condition we prove global
    well-posedness and scattering for all initial data in the energy space.
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. Quintic NLS in the exterior of a strictly convex
    obstacle. <i>American Journal of Mathematics</i>. 2016;138(5):1193-1346. doi:<a
    href="https://doi.org/10.1353/ajm.2016.0039">10.1353/ajm.2016.0039</a>
  apa: Killip, R., Vişan, M., &#38; Zhang, X. (2016). Quintic NLS in the exterior
    of a strictly convex obstacle. <i>American Journal of Mathematics</i>. Johns Hopkins
    University Press. <a href="https://doi.org/10.1353/ajm.2016.0039">https://doi.org/10.1353/ajm.2016.0039</a>
  chicago: Killip, Rowan, Monica Vişan, and Xiaoyi Zhang. “Quintic NLS in the Exterior
    of a Strictly Convex Obstacle.” <i>American Journal of Mathematics</i>. Johns
    Hopkins University Press, 2016. <a href="https://doi.org/10.1353/ajm.2016.0039">https://doi.org/10.1353/ajm.2016.0039</a>.
  ieee: R. Killip, M. Vişan, and X. Zhang, “Quintic NLS in the exterior of a strictly
    convex obstacle,” <i>American Journal of Mathematics</i>, vol. 138, no. 5. Johns
    Hopkins University Press, pp. 1193–1346, 2016.
  ista: Killip R, Vişan M, Zhang X. 2016. Quintic NLS in the exterior of a strictly
    convex obstacle. American Journal of Mathematics. 138(5), 1193–1346.
  mla: Killip, Rowan, et al. “Quintic NLS in the Exterior of a Strictly Convex Obstacle.”
    <i>American Journal of Mathematics</i>, vol. 138, no. 5, Johns Hopkins University
    Press, 2016, pp. 1193–346, doi:<a href="https://doi.org/10.1353/ajm.2016.0039">10.1353/ajm.2016.0039</a>.
  short: R. Killip, M. Vişan, X. Zhang, American Journal of Mathematics 138 (2016)
    1193–1346.
das_tickbox: '1'
date_created: 2026-06-19T07:45:53Z
date_published: 2016-10-01T00:00:00Z
date_updated: 2026-06-22T13:20:45Z
day: '01'
doi: 10.1353/ajm.2016.0039
extern: '1'
external_id:
  arxiv:
  - '1208.4904'
fulldoi: https://doi.org/10.1353/ajm.2016.0039
intvolume: '       138'
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1208.4904
month: '10'
oa: 1
oa_version: Preprint
page: 1193-1346
publication: American Journal of Mathematics
publication_identifier:
  eissn:
  - 1080-6377
publication_status: published
publisher: Johns Hopkins University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Quintic NLS in the exterior of a strictly convex obstacle
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 138
year: '2016'
...
---
OA_place: repository
OA_type: green
_id: '22023'
abstract:
- lang: eng
  text: "We consider the focusing energy-critical nonlinear Schrödinger equation iut
    + ∆u = −|u|\r\n4 d−2 u in dimensions d ≥ 5. We prove that if a maximal-lifespan
    solution u : I × Rd → C obeys supt∈I k∇u(t)k2 < k∇Wk2, then it is global and scatters
    both forward and backward in time. Here W denotes the ground state, which is a
    stationary solution of the equation. In\r\nparticular, if a solution has both
    energy and kinetic energy less than those\r\nof the ground state W at some point
    in time, then the solution is global and\r\nscatters. We also show that any solution
    that blows up with bounded kinetic\r\nenergy must concentrate at least the kinetic
    energy of the ground state. Similar\r\nresults were obtained by Kenig and Merle
    in [17, 18] for spherically symmetric\r\ninitial data and dimensions d = 3, 4,
    5."
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
citation:
  ama: Killip R, Vişan M. The focusing energy-critical nonlinear Schrödinger equation
    in dimensions five and higher. <i>American Journal of Mathematics</i>. 2010;132(2):361-424.
    doi:<a href="https://doi.org/10.1353/ajm.0.0107">10.1353/ajm.0.0107</a>
  apa: Killip, R., &#38; Vişan, M. (2010). The focusing energy-critical nonlinear
    Schrödinger equation in dimensions five and higher. <i>American Journal of Mathematics</i>.
    Johns Hopkins University Press. <a href="https://doi.org/10.1353/ajm.0.0107">https://doi.org/10.1353/ajm.0.0107</a>
  chicago: Killip, Rowan, and Monica Vişan. “The Focusing Energy-Critical Nonlinear
    Schrödinger Equation in Dimensions Five and Higher.” <i>American Journal of Mathematics</i>.
    Johns Hopkins University Press, 2010. <a href="https://doi.org/10.1353/ajm.0.0107">https://doi.org/10.1353/ajm.0.0107</a>.
  ieee: R. Killip and M. Vişan, “The focusing energy-critical nonlinear Schrödinger
    equation in dimensions five and higher,” <i>American Journal of Mathematics</i>,
    vol. 132, no. 2. Johns Hopkins University Press, pp. 361–424, 2010.
  ista: Killip R, Vişan M. 2010. The focusing energy-critical nonlinear Schrödinger
    equation in dimensions five and higher. American Journal of Mathematics. 132(2),
    361–424.
  mla: Killip, Rowan, and Monica Vişan. “The Focusing Energy-Critical Nonlinear Schrödinger
    Equation in Dimensions Five and Higher.” <i>American Journal of Mathematics</i>,
    vol. 132, no. 2, Johns Hopkins University Press, 2010, pp. 361–424, doi:<a href="https://doi.org/10.1353/ajm.0.0107">10.1353/ajm.0.0107</a>.
  short: R. Killip, M. Vişan, American Journal of Mathematics 132 (2010) 361–424.
date_created: 2026-06-19T07:31:39Z
date_published: 2010-03-31T00:00:00Z
date_updated: 2026-06-19T10:24:15Z
day: '31'
doi: 10.1353/ajm.0.0107
extern: '1'
external_id:
  arxiv:
  - '0804.1018'
fulldoi: https://doi.org/10.1353/ajm.0.0107
intvolume: '       132'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.0804.1018
month: '03'
oa: 1
oa_version: Preprint
page: 361-424
publication: American Journal of Mathematics
publication_identifier:
  eissn:
  - 1080-6377
  issn:
  - 0002-9327
publication_status: published
publisher: Johns Hopkins University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: The focusing energy-critical nonlinear Schrödinger equation in dimensions five
  and higher
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 132
year: '2010'
...
---
OA_place: repository
OA_type: green
_id: '22089'
abstract:
- lang: eng
  text: "We consider the focusing energy-critical nonlinear Schr\\\"odinger equation
    \r\n$iu_t+\\Delta u = - |u|^{4\\over{d-2}}u$ in dimensions $d\\geq 5$. We prove
    \r\nthat if a maximal-lifespan solution $u\\colon \\ I\\times {\\Bbb R}^d\\to
    {\\Bbb C}$ obeys $\\sup_{t\\in I}\\|\\nabla u(t)\\|_2&lt;\\|\\nabla W\\|_2$, then
    it is \r\nglobal and scatters both forward and backward in time. Here $W$ denotes
    \r\nthe ground state, which is a stationary solution of the equation. In \r\nparticular,
    if a solution has both energy and kinetic energy less than \r\nthose of the ground
    state $W$ at some point in time, then the solution is \r\nglobal and scatters.
    We also show that any solution that blows up with \r\nbounded kinetic energy must
    concentrate at least the kinetic energy of the \r\nground state. Similar results
    were obtained by Kenig and Merle for \r\nspherically symmetric initial data and
    dimensions $d=3,4,5$.\r\n</jats:p>"
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
citation:
  ama: Killip R, Vişan M. The focusing energy-critical nonlinear Schrödinger equation
    in dimensions five and higher. <i>American Journal of Mathematics</i>. 2010;132(2):361-424.
    doi:<a href="https://doi.org/10.1353/ajm.0.0107">10.1353/ajm.0.0107</a>
  apa: Killip, R., &#38; Vişan, M. (2010). The focusing energy-critical nonlinear
    Schrödinger equation in dimensions five and higher. <i>American Journal of Mathematics</i>.
    Johns Hopkins University Press. <a href="https://doi.org/10.1353/ajm.0.0107">https://doi.org/10.1353/ajm.0.0107</a>
  chicago: Killip, Rowan, and Monica Vişan. “The Focusing Energy-Critical Nonlinear
    Schrödinger Equation in Dimensions Five and Higher.” <i>American Journal of Mathematics</i>.
    Johns Hopkins University Press, 2010. <a href="https://doi.org/10.1353/ajm.0.0107">https://doi.org/10.1353/ajm.0.0107</a>.
  ieee: R. Killip and M. Vişan, “The focusing energy-critical nonlinear Schrödinger
    equation in dimensions five and higher,” <i>American Journal of Mathematics</i>,
    vol. 132, no. 2. Johns Hopkins University Press, pp. 361–424, 2010.
  ista: Killip R, Vişan M. 2010. The focusing energy-critical nonlinear Schrödinger
    equation in dimensions five and higher. American Journal of Mathematics. 132(2),
    361–424.
  mla: Killip, Rowan, and Monica Vişan. “The Focusing Energy-Critical Nonlinear Schrödinger
    Equation in Dimensions Five and Higher.” <i>American Journal of Mathematics</i>,
    vol. 132, no. 2, Johns Hopkins University Press, 2010, pp. 361–424, doi:<a href="https://doi.org/10.1353/ajm.0.0107">10.1353/ajm.0.0107</a>.
  short: R. Killip, M. Vişan, American Journal of Mathematics 132 (2010) 361–424.
das_tickbox: '1'
date_created: 2026-06-19T08:48:46Z
date_published: 2010-04-01T00:00:00Z
date_updated: 2026-07-01T12:52:05Z
day: '01'
doi: 10.1353/ajm.0.0107
extern: '1'
external_id:
  arxiv:
  - '0804.1018'
fulldoi: https://doi.org/10.1353/ajm.0.0107
intvolume: '       132'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.0804.1018
month: '04'
oa: 1
oa_version: Preprint
page: 361-424
publication: American Journal of Mathematics
publication_identifier:
  eissn:
  - 1080-6377
publication_status: published
publisher: Johns Hopkins University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: The focusing energy-critical nonlinear Schrödinger equation in dimensions five
  and higher
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 132
year: '2010'
...
---
OA_place: repository
OA_type: green
_id: '22057'
abstract:
- lang: eng
  text: " We obtain global well-posedness, scattering, uniform regularity, and global
    L6t,x spacetime bounds for energy-space solutions to the defocusing energy-critical
    nonlinear Schrödinger equation in â\x84\x9DÃ\x97â\x84\x9D4. Our arguments closely
    follow those of Colliander, Keel, et al., though our derivation of the frequency-localized
    interaction Morawetz estimate is somewhat simpler. As a consequence, our method
    yields a better bound on the L6t,x-norm. "
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: E
  full_name: Ryckman, E
  last_name: Ryckman
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
citation:
  ama: Ryckman E, Vişan M. Global well-posedness and scattering for the defocusing
    energy-critical nonlinear Schrödinger equation in R 1+4. <i>American Journal of
    Mathematics</i>. 2007;129(1):1-60. doi:<a href="https://doi.org/10.1353/ajm.2007.0004">10.1353/ajm.2007.0004</a>
  apa: Ryckman, E., &#38; Vişan, M. (2007). Global well-posedness and scattering for
    the defocusing energy-critical nonlinear Schrödinger equation in R 1+4. <i>American
    Journal of Mathematics</i>. Johns Hopkins University Press. <a href="https://doi.org/10.1353/ajm.2007.0004">https://doi.org/10.1353/ajm.2007.0004</a>
  chicago: Ryckman, E, and Monica Vişan. “Global Well-Posedness and Scattering for
    the Defocusing Energy-Critical Nonlinear Schrödinger Equation in R 1+4.” <i>American
    Journal of Mathematics</i>. Johns Hopkins University Press, 2007. <a href="https://doi.org/10.1353/ajm.2007.0004">https://doi.org/10.1353/ajm.2007.0004</a>.
  ieee: E. Ryckman and M. Vişan, “Global well-posedness and scattering for the defocusing
    energy-critical nonlinear Schrödinger equation in R 1+4,” <i>American Journal
    of Mathematics</i>, vol. 129, no. 1. Johns Hopkins University Press, pp. 1–60,
    2007.
  ista: Ryckman E, Vişan M. 2007. Global well-posedness and scattering for the defocusing
    energy-critical nonlinear Schrödinger equation in R 1+4. American Journal of Mathematics.
    129(1), 1–60.
  mla: Ryckman, E., and Monica Vişan. “Global Well-Posedness and Scattering for the
    Defocusing Energy-Critical Nonlinear Schrödinger Equation in R 1+4.” <i>American
    Journal of Mathematics</i>, vol. 129, no. 1, Johns Hopkins University Press, 2007,
    pp. 1–60, doi:<a href="https://doi.org/10.1353/ajm.2007.0004">10.1353/ajm.2007.0004</a>.
  short: E. Ryckman, M. Vişan, American Journal of Mathematics 129 (2007) 1–60.
das_tickbox: '1'
date_created: 2026-06-19T07:57:32Z
date_published: 2007-02-01T00:00:00Z
date_updated: 2026-06-29T10:30:01Z
day: '01'
doi: 10.1353/ajm.2007.0004
extern: '1'
external_id:
  arxiv:
  - math/0501462
fulldoi: https://doi.org/10.1353/ajm.2007.0004
intvolume: '       129'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.math/0501462
month: '02'
oa: 1
oa_version: Preprint
page: 1-60
publication: American Journal of Mathematics
publication_identifier:
  eissn:
  - 1080-6377
publication_status: published
publisher: Johns Hopkins University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Global well-posedness and scattering for the defocusing energy-critical nonlinear
  Schrödinger equation in R 1+4
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 129
year: '2007'
...