{
  "id": "1602.07037",
  "version": "v1",
  "published": "2016-02-23T04:45:20.000Z",
  "updated": "2016-02-23T04:45:20.000Z",
  "title": "Remarks on $L^p$-boundedness of wave operators for Schrödinger operators with threshold singularities",
  "authors": [
    "Kenji Yajima"
  ],
  "comment": "58 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the continuity property in Lebesgue spaces $L^p(\\R^m)$ of wave operators $W_\\pm$ of scattering theory for Schr\\\"odinger operator $H=-\\lap + V$ on $\\R^m$, $|V(x)|\\leq C\\ax^{-\\delta}$ for some $\\delta>2$ when $H$ is of exceptional type, i.e. $\\Ng=\\{u \\in \\ax^{-s} L^2(\\R^m) \\colon (1+ (-\\lap)^{-1}V)u=0 \\}\\not=\\{0\\}$ for some $1/2<s<\\delta-1/2$. It has recently been proved by Goldberg and Green for $m\\geq 5$ that $W_\\pm$ are bounded in $L^p(\\R^m)$ for $1\\leq p<m/2$, the same holds for $1\\leq p<m$ if all $\\f\\in \\Ng$ satisfy $\\int_{\\R^m} V\\f dx=0$ and, for $1\\leq p<\\infty$ if in addition $\\int_{\\R^m} x_i V\\f dx=0$, $i=1, \\dots, m$. We make the results for $p>m/2$ more precise and prove in particular that these conditions are also necessary for the stated properties of $W_\\pm$. We also prove that, for $m=3$, $W_\\pm$ are bounded in $L^p(\\R^3)$ for $1<p<3$ and that the same holds for $1<p<\\infty$ if and only if all $\\f\\in \\Ng$ satisfy $\\int_{\\R^3}V\\f dx=0$ and $\\int_{\\R^3} x_i V\\f dx=0$, $i=1, 2, 3$, simultaneously.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-23T04:45:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P25",
      "81U50"
    ],
    "keywords": [
      "wave operators",
      "schrödinger operators",
      "threshold singularities",
      "boundedness",
      "lebesgue spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}