“Single-sided” focusing of the time-dependent Schrödinger equation

The problem of focusing is considered for the one-dimensional time-dependent Schrödinger equation with a local potential. Focusing is also considered for the closely related plasma wave equation. It is supposed that an experimenter can send in an incident wave from one side of the potential. Then to what degree can the wave be focused? It will be shown that an incident wave can be found so that the real part of the wave collapses to a delta function δ(x-x_o) when t=0. The equation that governs this previously unsolved “single-sided” focusing problem is derived and shown to be Marchenko’s equation—the canonical equation of one-dimensional inverse-scattering theory.