Mean-field limit for the dynamics of quantum particle systems

We will introduce the mean-field asymptotic of a solution of the initial-value problem of the BBGKY hierarchy of quantum many-particle systems. We construct a solution as an expansion over particle clusters whose evolution are governed by the corresponding-order cumulant (semi-invariant) of the evolution operators of finitely many particles. We prove that in the mean-field limit the constructed solution converges to the sequence of marginal operators satisfying the corresponding initial-value problem of the quantum Vlasov hierarchy in the sense of the norm convergence of the space of sequences of trace class operators. The solution of the initial-value problem of the limiting hierarchy possesses of the chaos property which make it possible to substantiate the derivation of the suitable nonlinear kinetic equation - quantum Vlasov equation and as the consequece - the nonlinear Schrödinger equation.