{
  "id": "math/0611394",
  "version": "v2",
  "published": "2006-11-13T16:41:59.000Z",
  "updated": "2010-10-20T18:09:23.000Z",
  "title": "Energy-critical NLS with quadratic potentials",
  "authors": [
    "Rowan Killip",
    "Monica Visan",
    "Xiaoyi Zhang"
  ],
  "comment": "Incorporates corrections to Lemma 6.5",
  "journal": "Comm. PDE. 34 (2009), 1531--1565",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the defocusing $\\dot H^1$-critical nonlinear Schr\\\"odinger equation in all dimensions ($n\\geq 3$) with a quadratic potential $V(x)=\\pm \\tfrac12 |x|^2$. We show global well-posedness for radial initial data obeying $\\nabla u_0(x), xu_0(x) \\in L^2$. 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 couple of simplifications discovered while treating the problem with quadratic potential.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-10-20T18:09:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quadratic potential",
      "energy-critical nls",
      "natural energy space",
      "radial initial data obeying",
      "global well-posedness"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....11394K"
    }
  }
}