{
  "id": "math-ph/0506008",
  "version": "v2",
  "published": "2005-06-02T14:57:08.000Z",
  "updated": "2006-07-12T09:12:55.000Z",
  "title": "On inverse scattering in electromagnetic field in classical relativistic mechanics at high energies",
  "authors": [
    "Alexandre Jollivet"
  ],
  "journal": "Asympotic Analysis 55, 1-2 (2007) 103-123",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the multidimensional Newton-Einstein equation in static electromagnetic field $$\\eqalign{\\dot p = F(x,\\dot x), F(x,\\dot x)=-\\nabla V(x)+{1\\over c}B(x)\\dot x,\\cr p={\\dot x \\over \\sqrt{1-{|\\dot x|^2 \\over c^2}}}, \\dot p={dp\\over dt}, \\dot x={dx\\over dt}, x\\in C^1(\\R,\\R^d),}\\eqno{(*)}$$ where $V \\in C^2(\\R^d,\\R),$ $B(x)$ is the $d\\times d$ real antisymmetric matrix with elements $B\\_{i,k}(x)={\\pa\\over \\pa x\\_i}\\A\\_k(x)-{\\pa\\over \\pa x\\_k}\\A\\_i(x)$, and $|\\pa^j\\_x\\A\\_i(x)|+|\\pa^j\\_x V(x)| \\le \\beta\\_{|j|}(1+|x|)^{-(\\alpha+|j|)}$ for $x\\in \\R^d,$ $|j| \\le 2,$ $i=1..d$ 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 $PB\\_{i,k}$ for $i,k=1..d,$ $i\\neq k.$ 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 $(V,B)$. In addition we show that our high energy asymptotics found for the configuration valued component of the scattering operator doesn't determine uniquely $V$ when $d\\ge 2$ and $B$ when $d=2$ but that it uniquely determines $B$ when $d\\ge 3.$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-07-12T09:12:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34A55",
      "83A05",
      "78A46",
      "44A12"
    ],
    "keywords": [
      "classical relativistic mechanics",
      "electromagnetic field",
      "inverse scattering",
      "scattering operator",
      "velocity valued component"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.ph...6008J"
    }
  }
}