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

Erdös L, Schlein B, Yau H. 2010. Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. Annals of Mathematics. 172(1), 291–370.

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Author

Erdös, László

^{ISTA}^{}; Schlein, Benjamin; Yau, Horng-TzerAbstract

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 〈 ψ N,0,Hk N ψN,0〉 ≤ CkNk for k =1, 2, ....We also assume that the k-particle density matrices of the initial state are asymptotically factorized as N →∞1. 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 nonlinear 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.

Publishing Year

Date Published

2010-07-01

Journal Title

Annals of Mathematics

Volume

172

Issue

1

Page

291 - 370

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### Cite this

Erdös L, Schlein B, Yau H. Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate.

*Annals of Mathematics*. 2010;172(1):291-370. doi:10.4007/annals.2010.172.291Erdös, L., Schlein, B., & Yau, H. (2010). Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate.

*Annals of Mathematics*. Princeton University Press. https://doi.org/10.4007/annals.2010.172.291Erdös, László, Benjamin Schlein, and Horng Yau. “Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate.”

*Annals of Mathematics*. Princeton University Press, 2010. https://doi.org/10.4007/annals.2010.172.291.L. Erdös, B. Schlein, and H. Yau, “Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate,”

*Annals of Mathematics*, vol. 172, no. 1. Princeton University Press, pp. 291–370, 2010.Erdös, László, et al. “Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate.”

*Annals of Mathematics*, vol. 172, no. 1, Princeton University Press, 2010, pp. 291–370, doi:10.4007/annals.2010.172.291.