Mathematics of the Bose Gas in the Thomas-Fermi Regime

In this Thesis we discuss the Bose-Einstein condensation (BEC) in the Thomas-Fermi limit for a many-body system, both in the stationary and in the dynamical framework. In particular, we prove that if the pair interaction is of the form $ g_N N^{3beta-1}(N^beta x) $, with $ g_N $ growing as $ N $ goes to infinity and $ beta $ smaller than 1/3 there is BEC in the ground state of a system in a box. We also prove that in a trapped system, if there is BEC in the initial datum, BEC is preserved, at least for $ beta $ smaller than 1/6.