High density limit of the stationary one dimensional Schrödinger-Poisson system

Raymond El Hajj (1) , Naoufel Ben Abdallah (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 IMT - Institut de Mathématiques de Toulouse UMR5219

The stationary one dimensional Schrödinger-Poisson system on a bounded interval is considered in the limit of a small Debye length (or small temperature). Electrons are supposed to be in a mixed state with the Boltzmann statistics. Using various reformulations of the system as convex minimization problems, we show that only the first energy level is asymptotically occupied. The electrostatic potential is shown to converge towards a boundary layer potential with a profile computed by means of a half space Schrödinger-Poisson system.

Mots clés: convex minimization; min-max theorem; concentration-compactness principle; boundary layer

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Raymond El Hajj, Naoufel Ben Abdallah. High density limit of the stationary one dimensional Schrödinger-Poisson system. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 2008, 7 (1), pp.124-146. ⟨hal-00463943⟩