The classical limit of a self-consistent quantum-Vlasov equation in 3D. (English) Zbl 0772.35061

Summary: Under natural assumptions on the initial density matrix of a mixed quantum state (Hermitian, nonnegative definite, uniformly bounded trace, Hilbert-Schmidt norm and kinetic energy) we prove that accumulation points (as the scaled Planck constant tends to zero) of solutions of a corresponding slightly regularized Wigner-Poisson system are distributional solutions of the classical Vlasov-Poisson system. The result holds for the gravitational and repulsive cases. Also, for every phase-space density in \(L^ 1_ +(\mathbb{R}^ 6_{x,v})\cap L^ 2_ +(\mathbb{R}^ 6_{x,v})\) (with bounded kinetic energy) we prepare a sequence of density matrices satisfying the above assumptions, such that the given density is the limit of the Wigner transforms of these density matrices.


35Q40 PDEs in connection with quantum mechanics
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
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