{
  "id": "1606.03575",
  "version": "v1",
  "published": "2016-06-11T10:52:56.000Z",
  "updated": "2016-06-11T10:52:56.000Z",
  "title": "On wave operators for Schrödinger operators with threshold singuralities in three dimensions",
  "authors": [
    "Kenji Yajima"
  ],
  "comment": "21 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We show that wave operators for three dimensional Schr\\\"odinger operators $H=-\\Delta + V$ with threshold singularities are bounded in $L^1({\\mathbb R}^3)$ if and only if zero energy resonances are absent from $H$ and the existence of zero energy eigenfunctions does not destroy the $L^1$-boundedness of wave operators for $H$ with the regular threshold behavior. We also show in this case that they are bounded in $L^p({\\mathbb R}^3)$ for all $1\\leq p \\leq \\infty$ if all zero energy eigenfunctions $\\phi(x)$ have vanishing first three moments: $\\int_{{\\mathbb R}^3} x^\\alpha V(x)\\phi(x)dx=0$, $|\\alpha|=0,1,2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-11T10:52:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P25",
      "81U05",
      "47A40"
    ],
    "keywords": [
      "wave operators",
      "schrödinger operators",
      "threshold singuralities",
      "zero energy eigenfunctions",
      "dimensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}