{
  "id": "1310.6302",
  "version": "v1",
  "published": "2013-10-23T17:30:21.000Z",
  "updated": "2013-10-23T17:30:21.000Z",
  "title": "Dispersive estimates for four dimensional Schrödinger and wave equations with obstructions at zero energy",
  "authors": [
    "M. Burak Erdogan",
    "Michael Goldberg",
    "William R. Green"
  ],
  "comment": "32 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We investigate $L^1(\\mathbb R^4)\\to L^\\infty(\\mathbb R^4)$ dispersive estimates for the Schr\\\"odinger operator $H=-\\Delta+V$ when there are obstructions, a resonance or an eigenvalue, at zero energy. In particular, we show that if there is a resonance or an eigenvalue at zero energy then there is a time dependent, finite rank operator $F_t$ satisfying $\\|F_t\\|_{L^1\\to L^\\infty} \\lesssim 1/\\log t$ for $t>2$ such that $$\\|e^{itH}P_{ac}-F_t\\|_{L^1\\to L^\\infty} \\lesssim t^{-1},\\,\\,\\,\\,\\,\\text{for} t>2.$$ We also show that the operator $F_t=0$ if there is an eigenvalue but no resonance at zero energy. We then develop analogous dispersive estimates for the solution operator to the four dimensional wave equation with potential.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-23T17:30:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "zero energy",
      "dimensional schrödinger",
      "obstructions",
      "eigenvalue",
      "dimensional wave equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.6302B"
    }
  }
}