Explicit inverse scattering for the one-dimensional Schrodinger equation

Working with the one-dimensional Schrödinger equation in impedance form, we derive an exact inverse scattering formula that expresses impedance in terms of the reflection coefficient, and we prove injectivity of the scattering map for impedance functions of lower regularity than previously analyzed. The inverse scattering formula translates directly into an efficient numerical algorithm that accurately transforms digital scattering data into impedance. The results apply to acoustic imaging of layered media, as well as to inverse quantum scattering.