The topic of the thesis is the mathematical analysis and the numerical simulation of quantum transport in nanostructures. Two types of structures were investigated: open systems and confined systems in stationary and transient pictures. In an open device, the semiconductor is treated as an active region connected to electron leads. In order to decrease the numerical cost, the computational domain is limited to the active region through the introduction of open boundary conditions at the interfaces lead-active region. Within the active region, electrons generate a self-consistent potential with which they interact. The computation of this potential is done thanks to the resolution of a high number of Schrödinger equations. After having mathemaically analyzed this system, we developed several methods of resolution: an adaptative stepsize method for energy discretization, derivation of oscillating basis functions (WKB) for the discretization of the Schrödinger equation and a projection on the resonant states method.
The second part of the thesis concerns confined structures. The starting point is a 3D Schrödinger-Poisson system. Thanks to asymptotic approximations, 2D models keeping track of the third dimensions were rigorously derived. The first one couples a 2D Schrödinger equation to a 2D equation for the self-consistent potential. This model is shown to be a first order approximation. The second model couples a 2D Schrödinger equation to a 3D equation for the self-consistent potential and is a second order approximation. Numerical simulations realized in a recent thesis showed a spectacular reduction of the numerical cost while keeping a very good precision compared with the initial model. The asymptotics were performed in both stationary and time-dependent frames. They required the derivation of original anisotropic Strichartz and elliptic regularity estimates.