article
Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons
published
Alexander
Elgart
author
László
Erdös
author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603
Benjamin
Schlein
author
Horng
Yau
author
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.
Springer2006
Archive for Rational Mechanics and Analysis10.1007/s00205-005-0388-z
1792265 - 283
yes
A. Elgart, L. Erdös, B. Schlein, and H. Yau, “Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons,” <i>Archive for Rational Mechanics and Analysis</i>, vol. 179, no. 2. Springer, 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. <i>Archive for Rational Mechanics and Analysis</i>. Springer. <a href="https://doi.org/10.1007/s00205-005-0388-z">https://doi.org/10.1007/s00205-005-0388-z</a>
Elgart A, Erdös L, Schlein B, Yau H. Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons. <i>Archive for Rational Mechanics and Analysis</i>. 2006;179(2):265-283. doi:<a href="https://doi.org/10.1007/s00205-005-0388-z">10.1007/s00205-005-0388-z</a>
Elgart, Alexander, et al. “Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons.” <i>Archive for Rational Mechanics and Analysis</i>, vol. 179, no. 2, Springer, 2006, pp. 265–83, doi:<a href="https://doi.org/10.1007/s00205-005-0388-z">10.1007/s00205-005-0388-z</a>.
Elgart, Alexander, László Erdös, Benjamin Schlein, and Horng Yau. “Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons.” <i>Archive for Rational Mechanics and Analysis</i>. Springer, 2006. <a href="https://doi.org/10.1007/s00205-005-0388-z">https://doi.org/10.1007/s00205-005-0388-z</a>.
A. Elgart, L. Erdös, B. Schlein, H. Yau, Archive for Rational Mechanics and Analysis 179 (2006) 265–283.
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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