GROSS-PITAEVSKII DYNAMICS FOR BOSE-EINSTEIN CONDENSATES

We study the time-evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. 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 Lewin, Nam and Schlein (2015), who analyzed fluctuations around the Hartree dynamics for N-particle initial data in the mean-field regime, with ideas of Benedikter, de Oliveira and Schlein (2015), who considered the evolution of Fock-space initial data in the Gross-Pitaevskii regime.