# Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons

Elgart A, Erdös L, Schlein B, Yau H. 2006. Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons. Archive for Rational Mechanics and Analysis. 179(2), 265–283.

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Author

Elgart, Alexander;
Erdös, László

^{IST Austria}^{}; Schlein, Benjamin; Yau, Horng-TzerAbstract

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 [InlineMediaObject not available: see fulltext.] solves the GP hierarchy. Under the assumption that a = Nε for 0 < ε < 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.

Publishing Year

Date Published

2006-02-01

Journal Title

Archive for Rational Mechanics and Analysis

Volume

179

Issue

2

Page

265 - 283

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### Cite this

Elgart A, Erdös L, Schlein B, Yau H. Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons.

*Archive for Rational Mechanics and Analysis*. 2006;179(2):265-283. doi:10.1007/s00205-005-0388-zElgart, A., Erdös, L., Schlein, B., & Yau, H. (2006). Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons.

*Archive for Rational Mechanics and Analysis*. Springer. https://doi.org/10.1007/s00205-005-0388-zElgart, Alexander, László Erdös, Benjamin Schlein, and Horng Yau. “Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons.”

*Archive for Rational Mechanics and Analysis*. Springer, 2006. https://doi.org/10.1007/s00205-005-0388-z.A. Elgart, L. Erdös, B. Schlein, and H. Yau, “Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons,”

*Archive for Rational Mechanics and Analysis*, vol. 179, no. 2. Springer, pp. 265–283, 2006.Elgart, Alexander, et al. “Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons.”

*Archive for Rational Mechanics and Analysis*, vol. 179, no. 2, Springer, 2006, pp. 265–83, doi:10.1007/s00205-005-0388-z.