{
  "id": "0906.3211",
  "version": "v2",
  "published": "2009-06-17T15:23:11.000Z",
  "updated": "2009-06-21T10:40:32.000Z",
  "title": "Inverse scattering with non-overdetermined data",
  "authors": [
    "A. G. Ramm"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $A(\\beta,\\alpha,k)$ be the scattering amplitude corresponding to a real-valued potential which vanishes outside of a bounded domain $D\\subset \\R^3$. The unit vector $\\alpha$ is the direction of the incident plane wave, the unit vector $\\beta$ is the direction of the scattered wave, $k>0$ is the wave number. The governing equation for the waves is $[\\nabla^2+k^2-q(x)]u=0$ in $\\R^3$. For a suitable class of potentials it is proved that if $A_{q_1}(-\\beta,\\beta,k)=A_{q_2}(-\\beta,\\beta,k)$ $\\forall \\beta\\in S^2,$ $\\forall k\\in (k_0,k_1),$ and $q_1,$ $q_2\\in M$, then $q_1=q_2$. This is a uniqueness theorem for the solution to the inverse scattering problem with backscattering data. It is also proved for this class of potentials that if $A_{q_1}(\\beta,\\alpha_0,k)=A_{q_2}(\\beta,\\alpha_0,k)$ $\\forall \\beta\\in S^2_1,$ $\\forall k\\in (k_0,k_1),$ and $q_1,$ $q_2\\in M$,then $q_1=q_2$. Here $S^2_1$ is an arbitrarily small open subset of $S^2$, and $|k_0-k_1|>0$ is arbitrarily small.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-06-21T10:40:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R30",
      "81U40"
    ],
    "keywords": [
      "non-overdetermined data",
      "unit vector",
      "incident plane wave",
      "arbitrarily small open subset",
      "wave number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.3211R"
    }
  }
}