Numerically Absorbing Boundary Conditions for Quantum Evolution Equations

Abstract

Transparent boundary conditions for the transient Schrödinger equation on a domain Ω can be derived explicitly under the assumption that the given potential V is constant outside of this domain. In 1D these boundary conditions are non-local in time (of memory type). For the Crank-Nicolson finite difference scheme, discrete transparent boundary conditions are derived, and the resulting scheme is proved to be unconditionally stable. A numerical example illustrates the superiority of discrete transparent boundary conditions over existing ad-hoc discretizations of the differential transparent boundary conditions. As an application of these boundary conditions to the modeling of quantum devices, a transient 1D scattering model for mixed quantum states is presented.