{
  "id": "math-ph/0604030",
  "version": "v2",
  "published": "2006-04-14T03:51:52.000Z",
  "updated": "2006-09-26T21:05:48.000Z",
  "title": "Completeness of the set of scattering amplitudes",
  "authors": [
    "A. G. Ramm"
  ],
  "doi": "10.1016/j.physleta.2006.07.054",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $f\\in L^2(S^2)$ be an arbitrary fixed function with small norm on the unit sphere $S^2$, and $D\\subset \\R^3$ be an arbitrary fixed bounded domain. Let $k>0$ and $\\alpha\\in S^2$ be fixed. It is proved that there exists a potential $q\\in L^2(D)$ such that the corresponding scattering amplitude $A(\\alpha')=A_q(\\alpha')=A_q(\\alpha',\\alpha,k)$ approximates $f(\\alpha')$ with arbitrary high accuracy: $\\|f(\\alpha')-A_q(\\alpha')_{L^2(S^2)}\\|\\leq\\ve$ where $\\ve>0$ is an arbitrarily small fixed number. This means that the set $\\{A_q(\\alpha')\\}_{\\forall q\\in L^2(D)}$ is complete in $L^2(S^2)$. The results can be used for constructing nanotechnologically \"smart materials\".",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-09-26T21:05:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J10",
      "35P25",
      "35R30",
      "74J25",
      "81V05"
    ],
    "keywords": [
      "scattering amplitude",
      "completeness",
      "arbitrary high accuracy",
      "small norm",
      "unit sphere"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}