Wigner measure and the semiclassical limit of Schrodinger-Poisson equations

The wave function {psi(epsilon)(t,x)} of single particle approximation, which is used in the study of quantum transportation in some semiconductive devices, satisfies Schrodinger-Poisson equations. It is well known that the Wigner transformation f(epsilon)(t,x,xi) of the corresponding wave function psi(epsilon)(t,x) satisfies the so-called Wigner Poisson equations. We pro e here that in any space dimension, with the initial data of the form rootrho(0)(epsilon)(x) exp(i/epsilon S-epsilon(x)) to the wave function, and before the formation of vortices, the Wigner measure f(t,x,xi), which is the weak limit of f(epsilon)(t,x,xi) as the normalized Planck constant epsilon approaches 0, satisfies Vlasov-Poisson equations, and the limits of the quantum density and momentum to the Schrodinger Poisson equations satisfy the pressureless Euler-Poisson equations.