{
  "id": "1111.6222",
  "version": "v3",
  "published": "2011-11-27T03:20:33.000Z",
  "updated": "2013-02-22T05:50:01.000Z",
  "title": "Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in $d=3$ based on spacetime norms",
  "authors": [
    "Thomas Chen",
    "Nataša Pavlović"
  ],
  "comment": "44 pages, AMS Latex",
  "categories": [
    "math-ph",
    "math.AP",
    "math.MP"
  ],
  "abstract": "We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in dimension $d=3$, from an $N$-body Schr\\\"{o}dinger equation describing a gas of interacting bosons in the GP scaling, in the limit $N\\rightarrow\\infty$. The main result of this paper is the proof of convergence of the corresponding BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work on the well-posedness of the Cauchy problem for GP hierarchies, \\cite{chpa2,chpa3,chpa4}, which are inspired by the solutions spaces based on space-time norms introduced by Klainerman and Machedon in \\cite{klma}. We note that in $d=3$, this has been a well-known open problem in the field. While our results do not assume factorization of the solutions, consideration of factorized solutions yields a new derivation of the cubic, defocusing nonlinear Schr\\\"odinger equation (NLS) in $d=3$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-02-22T05:50:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "81V70"
    ],
    "keywords": [
      "gross-pitaevskii hierarchy",
      "manybody dynamics",
      "spacetime norms",
      "cubic nls",
      "derivation"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014AnHP...15..543C"
    }
  }
}