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

Abstract

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 $t\mapsto e^{-B_t}{e}^{-i{H}_{N}t}$ with renormalized generator, our proof is based on controlling renormalized excitation number operators directly with regards to the Schrödinger dynamics $t\mapsto e^{-i{H}_{N}t}$.

21 pages + appendix

References in corpus (4)

• Rigorous Derivation of the Gross-Pitaevskii Equation
• Bose-Einstein Condensation with Optimal Rate for Trapped Bosons in the Gross-Pitaevskii Regime
• Quantum Fluctuations of Many-Body Dynamics around the Gross-Pitaevskii Equation
• A Short Proof of Bose-Einstein Condensation in the Gross-Pitaevskii Regime and Beyond