Combined Mean-Field and Semiclassical Limits of Large Fermionic Systems

We study the time dependent Schrödinger equation for large spinless fermions with the semiclassical scale $$\hbar = N^{-1/3}$$ ħ = N - 1 / 3 in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schrödinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition uniform estimates for the remainder terms in the hierarchy are derived in order to show that in the semiclassical regime the weak limit of the Husimi measure is exactly the solution of the Vlasov equation.