Multiscale simulation of transport in an open quantum system: Resonances and WKB interpolation

A numerical scheme for the one dimensional stationary Schrödinger-Poisson model is described. The scheme is used to simulate a Resonant Tunneling Diode and provides an important reduction of the simulation time. The improvement is twofold. First the grid spacing in the position variable is made coarser by using oscillating interpolation functions derived from the WKB asymptotics. Then the discretization of the energy variable, which is a parameter for the Schrödinger equation, is improved by approaching the wavefunctions in the double barrier region by its projection on the resonant states (following the work of Presilla-Sjöstrand and Jona-Lasinio, [On Schrödinger equations with concentrated nonlinearities, Ann. Phys. 240, 1–21 (1995)]).