{
  "id": "1508.06300",
  "version": "v1",
  "published": "2015-08-25T20:37:02.000Z",
  "updated": "2015-08-25T20:37:02.000Z",
  "title": "The $L^p$ boundedness of wave operators for Schrödinger Operators with threshold singularities : Odd dimensions",
  "authors": [
    "Michael Goldberg",
    "William R. Green"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $H=-\\Delta+V$ be a Schr\\\"odinger operator on $L^2(\\mathbb R^n)$ with real-valued potential $V$ for $n>3$ odd, and let $H_0=-\\Delta$. If $V$ decays sufficiently, the wave operators $W_{\\pm}=s-\\lim_{t\\to \\pm\\infty} e^{itH}e^{-itH_0}$ are known to be bounded on $L^p(\\mathbb R^n)$ for all $1\\leq p\\leq \\infty$ if zero is not an eigenvalue, and on $\\frac{n} {n-2}<p<\\frac{n}{2}$ if zero is an eigenvalue. We show, that if $\\int_{\\mathbb R^n} V(x) \\phi(x) \\, dx=0$ for all eigenfunctions $\\phi$, then the wave operators are bounded for $\\frac{n}{n-1}<p<n$. If, in addition $\\int_{\\mathbb R^n} xV(x) \\phi(x) \\, dx=0$, then the wave operators are bounded for $1<p<\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-25T20:37:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J10",
      "35Q41",
      "35P25"
    ],
    "keywords": [
      "wave operators",
      "odd dimensions",
      "schrödinger operators",
      "threshold singularities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150806300G"
    }
  }
}