Development and numerical analysis of "black-box" counterpropagating wave algorithm for exact quantum scattering calculations

In a recent series of papers,^1-3 a bipolar counter-propagating wave decomposition, Ψ = Ψ ₊ + Ψ _-, was presented for stationary bound states Ψ of the one-dimensional Schrödinger equation, such that the components Ψ_± approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well-behaved, even when Ψ has many nodes, or is wildly oscillatory. In this paper, the earlier results are used to construct an universal "black-box" algorithm, numerically robust, stable and efficient, for computing accurate scattering quantities of any quantum dynamical system in one degree of freedom.