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

Laurent Gosse (1) , Norbert Mauser (2)

1 IAC - Istituto per le Applicazioni del Calcolo "Mauro Picone"
2 WPI - Wolfgang Pauli Institute

This work is concerned with the semiclassical approximation of the Schrödinger-Poisson equation modeling ballistic transport in a 1D periodic potential by means of WKB techniques. It is derived by considering the mean-field limit of a $N$-body quantum problem, then $K$-multivalued solutions are adapted to the treatment of this weakly nonlinear system obtained after homogenization without taking into account for Pauli's exclusion principle. Numerical experiments display the behaviour of self-consistent wave packets and screening effects.

Mots clés: Hartree ansatz; semiclassical limit; periodic potential; homogenization; Vlasov-Poisson equation; moment method; nonstrictly hyperbolic systems.; nonstrictly hyperbolic systems

Domaines: Analyse numérique [math.NA]; Physique Quantique [quant-ph]; Equations aux dérivées partielles [math.AP]

Laurent Gosse, Norbert Mauser. Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice: III. From {\it ab-initio} models to WKB for Schrödinger-Poisson. Journal of Computational Physics, 2006, 211 (1), pp.326-346. ⟨10.1016/j.jcp.2005.05.020⟩. ⟨hal-00432235⟩