{
  "id": "1108.3158",
  "version": "v1",
  "published": "2011-08-16T05:51:07.000Z",
  "updated": "2011-08-16T05:51:07.000Z",
  "title": "Some Results on the Scattering Theory for Nonlinear Schrödinger Equations in Weighted $L^{2}$ Space",
  "authors": [
    "Wei Dai"
  ],
  "comment": "28 pages, no figure",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We investigate the scattering theory for the nonlinear Schr\\\"{o}dinger equation $i \\partial_{t}u+ \\Delta u+\\lambda|u|^\\alpha u=0$ in $\\Sigma=H^{1}(\\mathbb{R}^{d})\\cap L^{2}(|x|^{2};dx)$. We show that scattering states $u^{\\pm}$ exist in $\\Sigma$ when $\\alpha_{d}<\\alpha<\\frac{4}{d-2}$, $d\\geq3$, $\\lambda\\in \\mathbb{R}$ with certain smallness assumption on the initial data $u_{0}$, and when $\\alpha(d)\\leq \\alpha< \\frac{4}{d-2}$($\\alpha\\in [\\alpha(d), \\infty)$, if $d=1,2$), $\\lambda>0$ under suitable conditions on $u_{0}$, where $\\alpha_{d}$, $\\alpha(d)$ are the positive root of the polynomial $dx^{2}+dx-4$ and $dx^{2}+(d-2)x-4$ respectively. Specially, when $\\lambda>0$, we obtain the existence of $u^{\\pm}$ in $\\Sigma$ for $u_{0}$ below a mass-energy threshold $M[u_{0}]^{\\sigma}E[u_{0}]<\\lambda^{-2\\tau}M[Q]^{\\sigma}E[Q]$ and satisfying an mass-gradient bound $\\|u_{0}\\|_{L^{2}}^{\\sigma}\\|\\nabla u_{0}\\|_{L^{2}}<\\lambda^{-\\tau}\\|Q\\|_{L^{2}}^{\\sigma}\\|\\nabla Q\\|_{L^{2}}$ with $\\frac{4}{d}<\\alpha<\\frac{4}{d-2}$($\\alpha\\in (\\frac{4}{d}, \\infty)$, if $d=1,2$), and also for oscillating data at critical power $\\alpha=\\alpha(d)$, where $\\sigma=\\frac{4-(d-2)\\alpha}{\\alpha d-4}$, $\\tau=\\frac{2}{\\alpha d-4}$ and $Q$ is the ground state. We also study the convergence of $u(t)$ to the free solution $e^{it\\Delta}u^{\\pm}$ in $\\Sigma$, where $u^{\\pm}$ is the scattering state at $\\pm\\infty$ respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-08-16T05:51:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55"
    ],
    "keywords": [
      "nonlinear schrödinger equations",
      "scattering theory",
      "scattering state",
      "smallness assumption",
      "initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.3158D"
    }
  }
}