Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate

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Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate

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Title: Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate
Author: Erdos, Laszlo; Schlein, Benjamin; Yau, Horng-Tzer

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Citation: Erdős, László, Benjamin Schlein, and Horng-Tzer Yau. 2010. “Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate.” Annals of Mathematics 172 (1) (June 27): 291–370. doi:10.4007/annals.2010.172.291.
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Abstract: Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N²V (N(xi − xj)), where x = (x1,..., xN) denotes the positions of the particles. Let HN denote the Hamiltonian of the system and let ψN,t be the solution to the Schrödinger equation. Suppose that the initial data ψN,0 satisfies the energy condition 〈ψN,0, H k NψN,0 〉 ≤ C k N k for k = 1, 2,.... We also assume that the k-particle density matrices of the initial state are asymptotically factorized as N → ∞. We prove that the k-particle density matrices of ψN,t are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant given by the scattering length of the potential V. We also prove the same conclusion if the energy condition holds only for k = 1 but the factorization of ψN,0 is assumed in a stronger sense.
Published Version: doi:10.4007/annals.2010.172.291
Other Sources: https://arxiv.org/abs/math-ph/0606017
Terms of Use: This article is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:32706729
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