{
  "id": "1709.00748",
  "version": "v1",
  "published": "2017-09-03T17:40:12.000Z",
  "updated": "2017-09-03T17:40:12.000Z",
  "title": "Recovering the singularities of a potential from Backscattering data",
  "authors": [
    "Cristóbal J. Meroño"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove that, up to dimension 7, the main singularities of a complex potential $q$ having some a priori regularity are contained in the Born approximation $q_B$ constructed from backscattering data. In particular in dimension 2,3,4 we show that under certain assumptions $q-q_B$ is one derivative more regular than $q$ and that this result is optimal. These results are based on new Sobolev estimates of the double, triple and quadruple dispersion operators in general dimension $n$. We construct low regularity counterexamples which among other things show that when $n>4$ the double dispersion operator can have worse Sobolev regularity than the potential, even if $q$ is compactly supported, radial and real. We also give H\\\"older estimates of the double dispersion operator for $n\\ge3$, known up to now only in dimension 2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-03T17:40:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "backscattering data",
      "singularities",
      "double dispersion operator",
      "construct low regularity counterexamples",
      "worse sobolev regularity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}