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

Markowich, Peter A.; Mauser, Norbert J.

Math. Models Methods Appl. Sci. 3, No. 1, 109-124 (1993).

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.

MSC:

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

Keywords:

Schrödinger equation; Wigner-Poisson equations; Vlasov-Poisson equations; semiconductor equations