High Density Limit of the Stationary One Dimensional Schrödinger–Poisson System
Raymond El Hajj, Naoufel Ben Abdallah · Multiscale Modeling and Simulation · 2008
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.