AN ASYMPTOTIC MODEL FOR THE TRANSPORT OF AN ELECTRON GAS IN A SLAB

Abstract

We study the limiting behavior of a Schrödinger–Poisson system describing a three-dimensional quantum gas that is confined along the vertical z-direction in a fine slab. The starting point is the three-dimensional Schrödinger–Poisson system with Dirichlet conditions on two horizontal planes z = 0 and z = ε, where the small parameter ε is the scale width of the slab. The limit ε → 0 appears to be an infinite system of two-dimensional nonlinear Schrödinger equations. Our strategy combines a refined analysis of the Poisson kernel acting on strongly confined densities and a time-averaging process that allows us to deal with the fast time oscillations.