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Archive for Rational Mechanics and Analysis

, Volume 179, Issue 2, pp 265–283 | Cite as

Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons

  • Alexander ElgartEmail author
  • László Erdős
  • Benjamin Schlein
  • Horng-Tzer Yau
Article

Abstract

We consider the dynamics of N boson systems interacting through a pair potential N −1 V a (x i x j ) where V a (x)=a −3 V(x/a). We denote the solution to the N-particle Schrödinger equation by Ψ N, t . Recall that the Gross-Pitaevskii (GP) equation is a nonlinear Schrödinger equation and 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 Open image in new window solves the GP hierarchy. Under the assumption that a=N −ɛ for 0<ɛ<3/5, we prove that as N→∞ the limit points of the k-particle density matrices of Ψ N, t are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by ∫V(x)dx. The uniqueness of the solutions of this hierarchy remains an open question.

Keywords

Neural Network Complex System Nonlinear Dynamics Electromagnetism Nonlinear Term 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Alexander Elgart
    • 1
    Email author
  • László Erdős
    • 2
  • Benjamin Schlein
    • 1
  • Horng-Tzer Yau
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Institute of MathematicsUniversity of MunichMunichGermany

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