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 form g_N N^{3\beta-1} V(N^\beta x) $, for some \beta\in(0,1/3) and diverging as \to \infty $. prove that there is complete Bose-Einstein condensation at level ground state and, furthermore, that, if \beta\in (0,1/6) preserved by time evolution.