New method for rapid numerical solution of the one-dimensional Schroedinger equation

One can generate certain special solutions to the one-dimensional Schr\"odinger equation, $exp\left[\pm i{S}_{\pm }\right(x\left)\right]$, such that $S_{\pm }\left(x\right)$ and derivatives are computationally unique, slowly varying, and do not have oscillatory or steplike behavior associated with the de Broglie wavelength. This leads to a method for accurate numerical solution which has all the advantages of high computational speed and conceptual simplicity of the JWKB approximation---of which it is the extension to an exact solution.