{
  "id": "math-ph/0502040",
  "version": "v2",
  "published": "2005-02-16T09:10:08.000Z",
  "updated": "2006-07-10T17:07:59.000Z",
  "title": "On inverse scattering for the multidimensional relativistic Newton equation at high energies",
  "authors": [
    "Alexandre Jollivet"
  ],
  "journal": "Journal of Mathematical Physics 47, 6 (2006) 062902",
  "doi": "10.1063/1.2206690",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider the Newton equation in the relativistic case (that is the Newton-Einstein equation) $$\\eqalign{\\dot p = F(x),& F(x)=-\\nabla V(x),\\cr p={\\dot x \\over \\sqrt{1-{|\\dot x|^2 \\over c^2}}},& \\dot p={dp\\over dt}, \\dot x={dx\\over dt}, x\\in C^1(\\R,\\R^d),}\\eqno{(*)}$$ $${\\rm where\\}V \\in C^2(\\R^d,\\R), |\\pa^j\\_x V(x)| \\le \\beta\\_{|j|}(1+|x|)^{-(\\alpha+|j|)}$$ for $|j| \\le 2$ and some $\\alpha > 1$. We give estimates and asymptotics for scattering solutions and scattering data for the equation $(*)$ for the case of small angle scattering. We show that at high energies the velocity valued component of the scattering operator uniquely determines the X-ray transform $PF.$ Applying results on inversion of the X-ray transform $P$ we obtain that for $d\\ge 2$ the velocity valued component of the scattering operator at high energies uniquely determines $F$. In addition we show that our high energy asymptotics found for the configuration valued component of the scattering operator doesn't determine uniquely $F$. The results of the present work were obtained in the process of generalizing some results of Novikov [R.G. Novikov, Small angle scattering and X-ray transform in classical mechanics, Ark. Mat. 37, pp. 141-169 (1999)] to the relativistic case.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-07-10T17:07:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "78A46",
      "65R32",
      "45Q05",
      "34A55"
    ],
    "keywords": [
      "multidimensional relativistic newton equation",
      "high energy",
      "x-ray transform",
      "inverse scattering",
      "scattering operator"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AIP",
      "journal": "J. Math. Phys."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}