{
  "id": "0710.0085",
  "version": "v1",
  "published": "2007-09-29T19:02:12.000Z",
  "updated": "2007-09-29T19:02:12.000Z",
  "title": "On inverse scattering at high energies for the multidimensional Newton equation in electromagnetic field",
  "authors": [
    "Alexandre Jollivet"
  ],
  "journal": "J. Inverse Ill-posed Probl. 17, 5 (2009) 441--476",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the multidimensional (nonrelativistic) Newton equation in a static electromagnetic field $$\\ddot x = F(x,\\dot x), F(x,\\dot x)=-\\nabla V(x)+B(x)\\dot x, \\dot x={dx\\over dt}, x\\in C^2(\\R,\\R^n),\\eqno{(*)}$$ where $V \\in C^2(\\R^n,\\R),$ $B(x)$ is the $n\\times n$ real antisymmetric matrix with elements $B_{i,k}(x)$, $B_{i,k}\\in C^1(\\R^n,\\R)$ (and $B$ satisfies the closure condition), and $|\\pa^{j_1}_xV(x)| +|\\pa^{j_2}_xB_{i,k}(x)| \\le \\beta_{|j_1|} (1+|x|)^{-(\\alpha+|j_1|)}$ for $x\\in \\R^n,$ $1\\le|j_1|\\le 2,$ $0\\le|j_2|\\le 1$, $|j_2|=|j_1|-1$, $i,k=1... n$ 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 transforms $P\\nabla V$ and $PB_{i,k}$ (on sufficiently rich sets of straight lines). Applying results on inversion of the X-ray transform $P$ we obtain that for $n\\ge 2$ the velocity valued component of the scattering operator at high energies uniquely determines $(\\nabla V,B)$. We also consider the problem of recovering $(\\nabla V,B)$ from our high energies asymptotics found for the configuration valued component of the scattering operator. Results of the present work were obtained by developing the inverse scattering approach of [R. Novikov, 1999] for $(*)$ with $B\\equiv 0$ and of [Jollivet, 2005] for the relativistic version of $(*)$. We emphasize that there is an interesting difference in asymptotics for scattering solutions and scattering data for $(*)$ on the one hand and for its relativistic version on the other.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-09-29T19:02:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "70F99",
      "34A55"
    ],
    "keywords": [
      "multidimensional newton equation",
      "electromagnetic field",
      "inverse scattering",
      "velocity valued component",
      "scattering operator"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.0085J"
    }
  }
}