Inversion of reflectivity data for nondecaying potentials

The recovery of material properties of thin lms is considered by probing them with neutron beams or X-rays. The interaction between the beam and the thin lm is described by the one-dimensional Schrodinger equation with an optical potential that is supported in the right half-line and asymptotic to a positive constant. The reconstruction of such a potential is studied in terms of the scattering data consisting of the magnitude of the reflection coefficient from the left, a known potential placed to the left of the unknown potential, and the magnitude of the reflection coefficient for the combined potential. A previous method utilizing three sets of reflectivity measurements is generalized to potentials not decaying at infinity, and the precise conditions are indicated for the validity of this method. It is shown that two sets of reflectivity measurements, instead of three, are sufficient for the unique reconstruction. Some analytical and computational methods are provided for the recovery of the unknown potential and the phase of the corresponding reflection coefficient. The recovery is illustrated with some numerical examples.