The one-dimensional Schrödinger equation is considered when the potential is real valued, integrable, has a finite first moment, and contains no bound states. From either of the two reflection coefficients of such a potential the right and left reflection coefficients are extracted corresponding to the left and right halves of the potential, respectively, and such half-line potentials are readily constructed from the extracted reflection coefficients. A computational procedure is described for such extractions and the construction of the two halves of the potential, and some applications are considered such as a numerical solution of the initial value problem for the Korteweg–de Vries equation. The theory is illustrated with some explicit examples. Copyright © 2002 John Wiley & Sons, Ltd.