{
  "id": "math-ph/0410005",
  "version": "v3",
  "published": "2004-10-02T23:00:23.000Z",
  "updated": "2005-10-19T23:54:33.000Z",
  "title": "Derivation of the Gross-Pitaevskii Hierarchy for the Dynamics of Bose-Einstein Condensate",
  "authors": [
    "Laszlo Erdos",
    "Benjamin Schlein",
    "Horng-Tzer Yau"
  ],
  "comment": "Latex file, 66 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider a system of $N$ bosons on the three dimensional unit torus interacting via a pair potential $N^2V(N(x_i-x_j))$, where $\\bx=(x_1, ..., x_N)$ denotes the positions of the particles. Suppose that the initial data $\\psi_{N,0}$ satisfies the condition \\[ < \\psi_{N,0}, H_N^2 \\psi_{N,0} > \\leq C N^2 \\] where $H_N$ is the Hamiltonian of the Bose system. This condition is satisfied if $\\psi_{N,0}= W_N \\phi_{N,0}$ where $W_N$ is an approximate ground state to $H_N$ and $\\phi_{N,0}$ is regular. Let $\\psi_{N,t}$ denote the solution to the Schr\\\"odinger equation with Hamiltonian $H_N$. Gross and Pitaevskii proposed to model the dynamics of such system by a nonlinear Schr\\\"odinger equation, the Gross-Pitaevskii (GP) equation. The GP hierarchy is an infinite BBGKY hierarchy of equations so that if $u_t$ solves the GP equation, then the family of $k$-particle density matrices $\\{\\otimes_k u_t, k\\ge 1 \\}$ solves the GP hierarchy. We prove that as $N\\to \\infty$ the limit points of the $k$-particle density matrices of $\\psi_{N,t}$ are solutions of the GP hierarchy. The uniqueness of the solutions to this hierarchy remains an open question. Our analysis requires that the $N$ boson dynamics is described by a modified Hamiltonian which cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical inter-particle distance. Our proof can be extended to a modified Hamiltonian which only forbids at least $n$ particles from coming close together, for any fixed $n$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2005-10-19T23:54:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "81Q05",
      "81V70"
    ],
    "keywords": [
      "bose-einstein condensate",
      "gross-pitaevskii hierarchy",
      "particle density matrices",
      "gp hierarchy",
      "derivation"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 66,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.ph..10005E"
    }
  }
}