Numerically Absorbing Boundary Conditions for Quantum Evolution Equations

Transparent boundary conditions for the transient Schrödinger equation on a domain Ω can bederived explicitly under the assumption that the given potential V is constant outside of thisdomain. In 1D these boundary conditions are non-local in time (of memory type). For theCrank-Nicolson finite difference scheme, discrete transparent boundary conditions arederived, and the resulting scheme is proved to be unconditionally stable. A numerical exampleillustrates the superiority of discrete transparent boundary conditions over existing ad-hocdiscretizations of the differential transparent boundary conditions.As an application of these boundary conditions to the modeling of quantum devices, a transient1D scattering model for mixed quantum states is presented