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 WignerPoisson (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).The mathematically 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 L 2 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 equations", i.e. the "semiconductor V-P system", with the assumption of the initial data to be concentrated in isolated bands.Support by the Austrian START project "Nonlinear Schrödinger and quantum Boltzmann equations" is acknowledged. MSC 2000 : 35L65, 35Q55, 81Q05, 81Q20.

Exact equations for smoothed Wigner transforms and homogenization of wave propagation

“…Physically, the high-frequency -or geometrical optics, or semiclassicalasymptotic regime corresponds to wave propagation problems where the coefficients (soundspeed, potential etc) vary on a length scale much larger than the wavelengths that appear. Semiclassical limits of quantum mechanics [18,29,31,32,40], and fluid mechanics [39], are two rich sources of problems in this regime.…”

Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice – III. From ab initio models to WKB for Schrödinger–Poisson

“…We also want our framework to be fully consistent with the ''ideal case'' e = 0 studied in [7,44,50] …”

“…If we restrict ourselves to models endowed with translational invariance in 2 space dimensions [44], a onedimensional problem like (1) arises. The preceding derivation seems to indicate that only smooth initial…”

“…Relying on the well-known fact that the Hartree term is but the Green function of the Laplace operator in R 3 , we rewrite the mean-field equation as the weakly nonlinear Schrö dingerPoisson system. Then, assuming translational invariance in two independent space directions, [44], we move forward to a one-dimensional model like (1) (see [37,41,48] for examples of its physical realizations) to which it becomes possible to apply a variant of the linear WKB receipe. These derivations are presented in full detail within Section 2.2.…”

The Gaussian Beam Methods for Schrödinger-Poisson Equations

“…-pd.V, (1. There are many papers discussing mathematical analysis and numerical methods for those equations [8,11,12,24,37], such as the existence and uniqueness of suitable weak solution to Vlasov-Poisson equations [9,19,29,32,50] and numerical methods for capturing the multi-valued solutions to the Euler-Poisson equations [14,27,31]. A well-known drawback to the semiclassical approach is that it can not give accurate solutions around caustics.…”