Numerically Absorbing Boundary Conditions for Quantum Evolution Equations

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 1 D 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 1 D scattering model for mixed quantum states is presented.

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