{
  "id": "math-ph/0606017",
  "version": "v3",
  "published": "2006-06-05T22:41:59.000Z",
  "updated": "2006-12-10T09:44:01.000Z",
  "title": "Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate",
  "authors": [
    "Laszlo Erdos",
    "Benjamin Schlein",
    "Horng-Tzer Yau"
  ],
  "comment": "Latex file, 66 pages; new version, with an appendix to include a new class of inital states",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where $\\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let $H_N$ denote the Hamiltonian of the system and let $\\psi_{N,t}$ be the solution to the Schr\\\"odinger equation. Suppose that the initial data $\\psi_{N,0}$ satisfies the energy condition \\[ < \\psi_{N,0}, H_N^k \\psi_{N,0} > \\leq C^k N^k \\] for $k=1,2,... $. We also assume that the $k$-particle density matrices of the initial state are asymptotically factorized as $N\\to\\infty$. We prove that the $k$-particle density matrices of $\\psi_{N,t}$ are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schr\\\"odinger equation with the coupling constant given by the scattering length of the potential $V$. We also prove the same conclusion if the energy condition holds only for $k=1$ but the factorization of $\\psi_{N,0}$ is assumed in a stronger sense.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2006-12-10T09:44:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "81Q15",
      "81T18",
      "81V70"
    ],
    "keywords": [
      "gross-pitaevskii equation",
      "bose-einstein condensate",
      "particle density matrices",
      "energy condition",
      "derivation"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 66,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.ph...6017E"
    }
  }
}