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


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.


Neural Network Complex System Nonlinear Dynamics Electromagnetism Nonlinear Term 
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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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