Discrete Transparent Boundary Conditions for transient kp-Schrödinger Equations with Application to Quantum-Heterostructures

This work is concerned with transparent boundary conditions (TBCs) for systems of Schrödinger-type equations, namely the time-dependent kp-Schrödinger equations. These TBCs are constructed for the fully discrete scheme (Crank-Nicolson, finite differences), in order to maintain unconditional stability of the scheme and to avoid numerical reections. The discrete transparent boundary conditions (DTBCs) are discrete convolutions in time and are constructed using the Z-transformed solution of the exterior problem. We will analyse the numerical error of these convolution coefcients caused by the inverse Z-transformation. Since the DTBCs are non-local in time and thus very costly to evaluate, we present approximate DTBCs of a sum-of-exponentials form that allow for a fast calculation of the boundary terms