## 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 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.

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Communicated by F. Otto

On leave from GeorgiaTech, Atlanta

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Elgart, A., Erdős, L., Schlein, B. *et al.* Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons.
*Arch. Rational Mech. Anal.* **179**, 265–283 (2006). https://doi.org/10.1007/s00205-005-0388-z

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DOI: https://doi.org/10.1007/s00205-005-0388-z