Gross-Pitaevskii Dynamics for Bose-Einstein Condensates

Abstract

We study the time-evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. Under a physically motivated assumption on the energy of the initial data, we show that condensation is preserved by the many-body evolution and that the dynamics of the condensate wave function can be described by the time-dependent Gross-Pitaevskii equation. With respect to previous works, we provide optimal bounds on the rate of condensation (i.e. on the number of excitations of the Bose-Einstein condensate). To reach this goal, we combine the method of \cite{LNS}, where fluctuations around the Hartree dynamics for $N$-particle initial data in the mean-field regime have been analyzed, with ideas from \cite{BDS}, where the evolution of Fock-space initial data in the Gross-Pitaevskii regime has been considered.

92 pages, typos corrected

References in corpus (5)

• Rigorous Derivation of the Gross-Pitaevskii Equation
• On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction
• On the uniqueness of solutions to the Gross-Pitaevskii hierarchy
• Complete Bose-Einstein condensation in the Gross-Pitaevskii regime
• A simple proof of convergence to the Hartree dynamics in Sobolev trace norms

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• Another proof of BEC in the GP-limit
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• Derivation of the Landau-Pekar equations in a many-body mean-field limit
• Central limit theorem for Bose gases interacting through singular potentials
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• Large deviation estimates for weakly interacting bosons
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• The Excitation Spectrum of the Bose Gas in the Gross-Pitaevskii Regime
• Derivation of the 2d Gross-Pitaevskii equation for strongly confined 3d bosons
• Bogoliubov theory for dilute bose gases: the Gross-Pitaevskii Regime
• Rate of convergence towards equations of Hartree type for mixture condensates with factorized initial data
• Convergence rate towards the fractional Hartree-equation with singular potentials in higher Sobolev norms
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• A Short Proof of Bose-Einstein Condensation in the Gross-Pitaevskii Regime and Beyond
• Third order corrections to the ground state energy of a Bose gas in the Gross-Pitaevskii regime
• Bosonization of Fermionic Many-Body Dynamics
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• The excitation spectrum of a Bose gas with an impurity in the Gross-Pitaevskii regime
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• Bose-Einstein condensation for two dimensional bosons in the Gross-Pitaevskii regime
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• The universe as a nonlinear quantum simulation: Large $n$ limit of the central spin model
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