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\centerline{\bf On the mean-field approximation of a Quantum
system}
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\centerline{M. Pulvirenti}
\centerline{Department of Mathematics, Universit\'a di Roma "La Sapienza ",
Roma}
\centerline{e-mail {\tt pulvirenti@giove.mat.uniroma1.it}}

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ABSTRACT:

The purpose of this joint work (in collaboration with S. Graffi and A.
Martinez) is
to characterize the behavior of a Quantum system in the mean field
approximation,
when $N$, the number of particles,
diverges simulteneosly with the inverse Planck constant $h$. It is proven
that,
under suitable assumptions on the two-body potential, the Wigner transform of
the system
factorizes in the limit and that
each factor satisfies the classical Vlasov equation (associated to the same
potential).
The proof is based on a non-conventional decomposition of the WKB method and
on
some hydrodynamical estimates.




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