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

By L. Erdős, B. Schlein, Horng-Tzer Yau. 2004

373 citations recorded by the source corpus.

Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N 2 V (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 Schrodinger equation. Suppose that the initial data ψN,0 satisfies the energy condition h ψN,0, H k N ψN,0i ≤ 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 Schrodinger 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. AMS Subject Classification Number: 81V70, 81T18, 35Q55

Subfields: Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/e75d63626fb8a59938caf3cc9d6eee461ad836c6

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics (2007, methodology): “In [3, 5, 6 ], models described by the Hamiltonian”
Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I. (2009, methodology): “roaches totheactualmany-particle Hamiltonian evolution raises fundamental questions. One question is the rigorous derivation and interpretation of the mean ﬁeld limit. Elgart, Erd˝os, Schlein and Yau [6, 7, 8, 9, 10, 11] showed rigorously how meanﬁeld limits for Bosons can be extracted in the limit ”
Kac’s program in kinetic theory (2011, methodology): “The BBGKY hierarchy method revisited The so-called BBGKY hierarchy method (Bogoliubov, Born, Green, Kirkwood and Yvon) is very popular in physics and mathematics for studying many-particle systems: see for instance [3] where this approach is used for Kac’s master equation for hard spheres, or see, a”
Mean–Field Evolution of Fermionic Systems (2013, methodology): “More recently, these techniques have been applied in [10, 11, 12, 13] to systems of bosons in the so called Gross-Pitaevskii limit; in this case, the interaction is scaled so that its range and its scattering length are both of the order 1/N .”
Quantitative Derivation of the Gross‐Pitaevskii Equation (2012, methodology): “This is in contrast with the results of [7, 8, 9, 10], which can also be applied to completely factorized initial data.”
The cubic nonlinear Schr\"odinger equation in two dimensions with radial data (2007, background): “Outline of the proof 11 Acknowledgements 13 2.”
Effective Evolution Equations from Quantum Dynamics (2015, background): “1 (for the case of weak potential treated in [28]; for large potentials, a different energy estimate has been proved and applied in [29]).”
A review of the mean field limits for Vlasov equations (2014, background): “• Quantum mechanics and the derivation of non linear Schrödinger equations (in particular Schrödinger-Poisson and Gross-Pitaevskii) from linear, N -particles Schrödinger; see for example [9], [8], [61], [62] and the references therein.”
Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems (2005, neutral): “Keywords: Feynman diagrams, BBGKY hierarchy, dispersive estimates, propagation of chaos.”
Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials (2014, background): “, 11, 27]. There has been a growing interest in the mathematical theories and numerical methods of GP equations [2]. Several rigorous mathematical veriﬁcations of GP theory were established, see e.g. [8, 20, 21, 22, 23]. It is known that the classical trapping potential used in the study of BEC is t”
Rigorous Derivation of the Cubic NLS in Dimension One (2007, background): “During the final draft of this paper we were made aware of the fact that Erd s, Schlein and Yau had achieved the proof of the uniqueness for the infinite hierarchy in the three-dimensional setting, (5) and in the time between the submission of our paper and the draft of the revised version, they com”