On Bose-Einstein condensates in the Thomas-Fermi regime

We study a system of $N$ trapped bosons in the Thomas-Fermi regime with an interacting pair potential of the form $g_N N^{3\beta-1} V(N^\beta x)$, for some $\beta\in(0,1/3)$ and $g_N$ diverging as $N \to \infty$. We prove that there is complete Bose-Einstein condensation at the level of the ground state and, furthermore, that, if $\beta\in (0,1/6)$, condensation is preserved by the time evolution.