{
  "id": "1508.05738",
  "version": "v1",
  "published": "2015-08-24T10:04:49.000Z",
  "updated": "2015-08-24T10:04:49.000Z",
  "title": "Wave Operators for Schrödinger Operators with Threshold Singuralities, Revisited",
  "authors": [
    "Kenji Yajima"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "The continuity property in the Sobolev space $W^{k,p}({\\bf R}^m)$ of wave operators of scattering theory for $m$-dimensional single-body Schr\\\"odinger operator is considered when the resolvent of the operator has singularities at the bottom of the continuous spectrum. It is shown that they are continuous in $W^{k,p}({\\bf R}^m)$, $0\\leq k \\leq 2$, for $1<p<3$ but not for $p>3$ if $m=3$ and, for $1<p<m/2$ but not for $p>m/2$ if $m\\geq 5$. This extends the previously known interval of $p$ for the continuity, $3/2<p<3$ for $m=3$ and $m/(m-2)<p<m/2$ for $m\\geq 5$. The formula which represents the integral kernel of the resolvent of the even dimensional free Sch\\\"odinger operator as the superposition of exponential-polynomial like functions substantially simplifies the proof of the previous paper when $m \\geq 6$ is even.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-24T10:04:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81U05"
    ],
    "keywords": [
      "wave operators",
      "schrödinger operators",
      "threshold singuralities",
      "functions substantially simplifies",
      "dimensional single-body"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150805738Y"
    }
  }
}