{
  "id": "0709.2490",
  "version": "v2",
  "published": "2007-09-16T13:10:28.000Z",
  "updated": "2009-12-14T00:01:41.000Z",
  "title": "Semiclassical limit of the scattering cross section as a distribution",
  "authors": [
    "E. Lakshtanov"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider quantum scattering from a compactly supported potential $q$. The semiclassical limit amounts to letting the wavenumber $k \\to \\infty$ while rescaling the potential as $k^2 q$ (alternatively, one can scale Planck's constant $\\hbar \\searrow 0$). It is well-known that, under appropriate conditions, for $\\om \\in \\bbS_{n-1}$ such that there is exactly one outgoing ray with direction $\\om$ (in the sense of geometric optics), the differential scattering cross section $|f(\\om,k)|^{2}$ tends to the classical differential cross section $|f_{cl}(\\om)|^2$ as $k \\uparrow \\infty$. It is also clear that the same can not be true if there is more than one outgoing ray with direction $\\om$ or for \\emph{nonregular} directions (including the forward direction $\\theta_0$). However, based on physical intuition, one could conjecture $|f|^2 \\to |f_{cl}|^2 + \\sigma_{cl} \\delta_{\\theta_0}$ where $|f_{cl}|^2$ is the classical cross section and $\\delta_{\\theta_0}$ is the Dirac measure supported at the forward direction $\\theta_0$. The aim of this paper is to prove this conjecture.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-12-14T00:01:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "semiclassical limit",
      "distribution",
      "scale plancks constant",
      "classical differential cross section",
      "differential scattering cross section"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.2490L"
    }
  }
}