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 β - 1 V ( N β x ) , for some β ∈ ( 0 , 1 / 3 ) and g N diverging as N → ∞ . We prove that there is complete Bose–Einstein condensation at the level of the ground state and, furthermore, that, if β ∈ ( 0 , 1 / 6 ) , condensation is preserved by the time evolution.