{
  "id": "0802.3877",
  "version": "v3",
  "published": "2008-02-26T19:11:50.000Z",
  "updated": "2009-03-16T10:03:52.000Z",
  "title": "Rigorous Derivation of the Gross-Pitaevskii Equation with a Large Interaction Potential",
  "authors": [
    "Laszlo Erdos",
    "Benjamin Schlein",
    "Horng-Tzer Yau"
  ],
  "comment": "LateX file; 53 pages. Final version",
  "categories": [
    "math-ph",
    "math.AP",
    "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 \\psi_{N,0} > \\leq C N >. \\] and that the one-particle density matrix converges to a projection as $N \\to \\infty$. Then, we prove that the $k$-particle density matrices of $\\psi_{N,t}$ factorize in the limit $N \\to \\infty$. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schr\\\"odinger equation with the coupling constant proportional to the scattering length of the potential $V$. In \\cite{ESY}, we proved the same statement under the condition that the interaction potential $V$ is sufficiently small; in the present work we develop a new approach that requires no restriction on the size of the potential.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-03-16T10:03:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "81Q15",
      "81T18",
      "81V70"
    ],
    "keywords": [
      "large interaction potential",
      "rigorous derivation",
      "one-particle density matrix converges",
      "particle orbital wave function",
      "repulsive short range pair potential"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1090/S0894-0347-09-00635-3",
      "journal": "Journal of the American Mathematical Society",
      "year": 2009,
      "month": "Oct",
      "volume": 22,
      "number": 4,
      "pages": 1099
    },
    "note": {
      "typesetting": "LaTeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009JAMS...22.1099E"
    }
  }
}