In this paper, we establish (1) the classical limit of the Hartree equation
leading to the Vlasov equation, (2) the classical limit of the N-body linear
Schr\"{o}dinger equation uniformly in N leading to the N-body Liouville
equation of classical mechanics and (3) the simultaneous mean-field and
classical limit of the N-body linear Schr\"{o}dinger equation leading to the
Vlasov equation. In all these limits, we assume that the gradient of the
interaction potential is Lipschitz continuous. All our results are formulated
as estimates involving a quantum analogue of the Monge-Kantorovich distance of
exponent 2 adapted to the classical limit, reminiscent of, but different from
the one defined in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343
(2016), 165-205]. As a by-product, we also provide bounds on the quadratic
Monge-Kantorovich distances between the classical densities and the Husimi
functions of the quantum density matrices.Comment: 33 page
