10-95 F. Pezzotti

Mean-field Limit and Semiclassical Approximation for Quantum Particle Systems (326K, LATeX 2e) Jun 17, 10

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Abstract. We consider a system constituted by $N$ identical particles interacting by means of a mean-field Hamiltonian. The evolution of the above system is taken into account both in the classical and in the quantum framework, as well as the corresponding effective dynamics when $N o\infty$. It is well known that, in the limit $N o\infty$, the one-particle state obeys to the Vlasov equation, in the classical case, and to the Hartree equation, in the quantum framework. Moreover, in both situations propagation of chaos holds. In this work, we present an overview over known results concerning the problem, with particular attention to the case of smooth pair-interaction potentials, and we analize the link between the asymptotics $N o\infty$ (Mean-Field Limit) and the semiclassical aprroximation ($\hbar o 0$). We dicuss and present in a unified way some known results, highlighting some open problems on that topic. In particular, we discuss in a wider way the result presented in [29] (see References), giving even an outlook on generalizations and possible applications.

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