Effective Dynamics for Weakly Interacting Bosons in an Iterated High-Density Thermodynamic Limit

We study the time evolution of weakly interacting Bose gases on a three-dimensional torus of arbitrary volume. The coupling constant is supposed to be inversely proportional to the density, which is considered to be large and independent of the particle number. We take into account a class of initial states exhibiting quasi-complete Bose-Einstein condensation. For each fixed time in a finite interval, we prove the convergence of the one-particle reduced density matrix towards the projection onto the normalised order parameter describing the condensate - evolving according to the Hartree equation - in the iterated limit where the volume (and therefore the particle number), and subsequently the density go to infinity. The rate of convergence depends only on the density and on the decay of both the expected number of particles and the energy of the initial quasi-vacuum state.