# Gross-Pitaevskii Dynamics for Bose-Einstein Condensates

## TL;DR

This paper rigorously demonstrates that Bose-Einstein condensates evolve according to the Gross-Pitaevskii equation, providing optimal bounds on condensation rates and extending previous analytical methods.

## Contribution

It introduces improved bounds on condensation preservation and combines methods from mean-field and Gross-Pitaevskii regimes for better analysis.

## Key findings

- Condensation is preserved during evolution.
- The condensate wave function follows the Gross-Pitaevskii equation.
- Optimal bounds on the rate of condensation are established.

## 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.