Connection Between Time- and Frequency-Domain Three-Dimensional Inverse Problems for the Schrödinger Equation

The use of inverse scattering methods in electromagnetic remote sensing, seismic exploration and ultrasonic imaging is rapidly expanding. For these cases which involve classical wave equations with variable velocity,1 no exact inversion methods exists for general three-dimensional (3d) scatterers. However, exact inversion methods (for example, those based on the Born series2 and the Newton-Marchenko equation2) do exist for the 3d Schrödinger equation. In this paper, these inversion methods for Schrödinger’s equation will be rewritten in a form which brings out certain analogies with classical wave equations. It is hoped these analogies will eventually contribute to a common exact inversion method for both types of equations