{
  "id": "math/0508206",
  "version": "v1",
  "published": "2005-08-11T16:17:37.000Z",
  "updated": "2005-08-11T16:17:37.000Z",
  "title": "A counterexample to dispersive estimates for Schrödinger operators in higher dimensions",
  "authors": [
    "M. Goldberg",
    "M. Visan"
  ],
  "journal": "Comm. Math. Phys. 266 no. 1 (2006), 211-238.",
  "doi": "10.1007/s00220-006-0013-5",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In dimension $n>3$ we show the existence of a compactly supported potential in the differentiability class $C^\\alpha$, $\\alpha < \\frac{n-3}2$, for which the solutions to the linear Schr\\\"odinger equation in $\\R^n$, $$ -i\\partial_t u = - \\Delta u + Vu, \\quad u(0)=f, $$ do not obey the usual $L^1\\to L^{\\infty}$ dispersive estimate. This contrasts with known results in dimensions $n \\leq 3$, where a pointwise decay condition on $V$ is generally sufficient to imply dispersive bounds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-08-11T16:17:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dispersive estimate",
      "schrödinger operators",
      "higher dimensions",
      "counterexample",
      "differentiability class"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer",
      "journal": "Commun. Math. Phys."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}