Semi-classical limits of Schrödinger-Poisson systems via Wigner transforms

Abstract

We deal with classical and “semiclassical limits” , i.e. vanishing Planck constant ? ? ? → 0, eventually combined with a homogenization limit of a crystal lattice, of a class of "weakly nonlinear" NLS. The Schrödinger-Poisson (S-P) system for the wave functions {ψ? j(t,x)} is transformed to the Wigner- Poisson (W-P) equation for a “phase space function” f?(t,x, ξ), the Wigner function. The weak limit of f?(t,x, ξ), as ? tends to 0, is called the “Wigner measure” f(t,x, ξ) (also called "semiclassical measure" by P. Gérard). Themathematically rigorous classical limit from S-P to the Vlasov-Poisson (V-P) system has been solved first in 1993 by P.L. Lions and T. Paul in [21] and, independently, by P.A. Markowich and N.J. Mauser in [23]. There the case of the so called “completely mixed state”, i.e. j = 1, 2, . . . ,∞, was considered where strong additional assumptions can be posed on the initial data. For the so called “pure state” case where only one (or a finite number) of wave functions {ψ? j(t,x)} is considered, recently P. Zhang, Y. Zheng and N.J. Mauser [33] have given the limit from S-P to V-P in one space dimension for a very weak class of measure valued solutions of V-P that are not unique. For the setting in a crystal, as it occurs in semiconductor modeling, we consider Schrödinger equations with an additional periodic potential. This allows for the use of the concept of “energy bands”, Bloch decomposition of L2 etc. On the level of the Wigner transform the Wigner function f?(t,x, ξ) is replaced by the “Wigner series” f?(t,x, k), where the “kinetic variable” k lives on the torus (“Brioullin zone”) instead of the whole space. Recently P. Bechouche, N.J. Mauser and F. Poupaud [7] have given the rigorous “semiclassical” limit from S-P in a crystal to the “semiclassical equa- tions”,i.e. the “semiconductor V-P system”, with the assumption of the initial data to be concentrated in isolated bands.