Miscellaneous Approaches to Inverse Problems at Fixed Energy

Abstract

As we have seen throughout this book, the key to the solution of inverse problems usually is some machinery centered on a “transformation kernel.” But how can this machinery be obtained? We give here two general approaches which have proved useful. The first one makes use of the interpolation properties of the wave functions. It has been applied either directly (Miodek’s approach to the inverse plasma problem) or as a tool to obtain the machinery (Sabatier-Hooshyar construction of spin-orbit potentials, and generalizations), which in turn enabled the development, by analogy, of the Jaulent-Jean method for inverse problems at fixed l with potentials depending linearly on. $\sqrt E$