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

A. Elgart, L. Erdös, B. Schlein, H. Yau, Archive for Rational Mechanics and Analysis 179 (2006) 265–283.

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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 [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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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-z
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. https://doi.org/10.1007/s00205-005-0388-z
Elgart, 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 179, no. 2 (2006): 265–83. 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, pp. 265–283, 2006.
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.
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.

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