Derivation of the Gross-Pitaevskii Dynamics through Renormalized Excitation Number Operators

We revisit the time evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. We show that the system continues to exhibit BEC once the trap has been released and that the dynamics of the condensate is described by the time-dependent Gross-Pitaevskii equation. Like the recent work \cite{BS}, we obtain optimal bounds on the number of excitations orthogonal to the condensate state. In contrast to \cite{BS}, however, whose main strategy consists of controlling the number of excitations with regards to a suitable fluctuation dynamics $t\mapsto e^{-B_t} e^{-iH_Nt}$ with renormalized generator, our proof is based on controlling renormalized excitation number operators directly with regards to the Schr\"odinger dynamics $t\mapsto e^{-iH_Nt}$.