Local enhancement of the mean-field approximation for bosons

We study the quantum many-body dynamics of a Bose-Einstein condensate (BEC) on the lattice in the mean-field regime. We derive a local enhancement of the mean-field approximation: At positive distance $\rho >0$ from the initial BEC, the mean-field approximation error at time $t\le \rho \mathrm{/}v$ is bounded as ${\rho }^{-n}$, for arbitrarily large $n\ge 1$. This is a consequence of new ballistic propagation bounds on the fluctuations around the condensate. To prove this, we develop a variant of the ASTLO (adiabatic spacetime localization observable) method for the particle non-conserving generator of the fluctuation dynamics around Hartree states.