Wigner Measure and the Semiclassical Limit of Schrödinger--Poisson Equations

Abstract

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 prove here that in any space dimension, with the initial data of the form $\sqrt{\rho^\epsilon_0(x)}\exp(\frac{i}{\epsilon}S^\epsilon(x))$ 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.