Adiabatic Approximation of the Schrödinger-Poisson System with a Partial Confinement

Asymptotic quantum transport models of a two-dimensional electron gas are presented. The starting point is a singular perturbation of the threedimensional Schrödinger-Poisson system. The small parameter ε is the scaled width of the electron gas and appears as the lengthscale on which a one dimensional confining potential varies. The rigorous ε→ 0 limit is performed by projecting the three dimensional wavefunction on the eigenfunctions corresponding to the confining potential. This leads to a two-dimensional Schrödinger-Poisson system with a modified Poisson equation keeping track of the third dimension. This limit model is proven to be a first-order approximation of the initial model. An intermediate model, called the “2.5D adiabatic model” is then introduced. It shares the same structure as the limit model but is shown to be a secondorder approximation of the 3D model.