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**Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate.**
*(English)*
Zbl 1204.82028

A system of \(N\) bosons in three dimensions is assumed to interact via a repulsive short range pair potential (mimicking \(1/N\) scaled Dirac delta functional). The corresponding \(N\)-particle Schrödinger equation is considered in the time domain with initial data satisfying a suitable energy condition. Wave functions (pure states) are replaced by density matrices and the \(k\)-particle density matrices of an initial state are assumed to factorize as \(N\rightarrow \infty\).

The major result of the paper is that a resultant solution \(\psi _{N,t}\) supports \(k\)-particle density matrices that asymptotically factorize and that the one-particle orbital wave function solves the Gross-Pitaevskii equation. It was known before that the minimizer of the Gross-Pitaevskii energy functional correctly describes the ground state of an \(N\)-boson system in the large \(N\) limit, provided the length scale of the pair potential is of order \(1/N\).

The present analysis has been motivated by the fact that in experiments on Bose-Einstein condensation one observes the dynamics of the condensate when its initially imposed traps are removed. That takes the system away from the previous ground state. The experimental validity of the G-P equation asserts that the approximation of many body effects by a nonlinear on-site self-interaction of the order parameter applies to a certain class of excited states and their subsequent time evolution as well. A bit surprising observation is that for product initial states the energy behavior is not captured by the G-P evolution.

The major result of the paper is that a resultant solution \(\psi _{N,t}\) supports \(k\)-particle density matrices that asymptotically factorize and that the one-particle orbital wave function solves the Gross-Pitaevskii equation. It was known before that the minimizer of the Gross-Pitaevskii energy functional correctly describes the ground state of an \(N\)-boson system in the large \(N\) limit, provided the length scale of the pair potential is of order \(1/N\).

The present analysis has been motivated by the fact that in experiments on Bose-Einstein condensation one observes the dynamics of the condensate when its initially imposed traps are removed. That takes the system away from the previous ground state. The experimental validity of the G-P equation asserts that the approximation of many body effects by a nonlinear on-site self-interaction of the order parameter applies to a certain class of excited states and their subsequent time evolution as well. A bit surprising observation is that for product initial states the energy behavior is not captured by the G-P evolution.

Reviewer: Piotr Garbaczewski (Opole)

### MSC:

82C22 | Interacting particle systems in time-dependent statistical mechanics |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

82C26 | Dynamic and nonequilibrium phase transitions (general) in statistical mechanics |

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

81V70 | Many-body theory; quantum Hall effect |

81T18 | Feynman diagrams |

### Keywords:

Gross-Pitaevskii equation; Bose-Einstein condensate; N-body Schrödinger equation; repulsive bosons; mean field description; scaling limits; confining traps; condensate release; time evolution problem after trap removal
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\textit{L. Erdős} et al., Ann. Math. (2) 172, No. 1, 291--370 (2010; Zbl 1204.82028)

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