General Method for the Inverse Scattering Problem at Fixed Energy

Abstract

A method for studying the inverse scattering problem at fixed energy is given; it enables one to get at a fairly large class of potentials $\mathcal{C}$ . It is shown that the problem has in $\mathcal{C}$ an infinity of solutions, depending on an infinity of parameters. If the study is restricted to the potentials of $\mathcal{C}$ which can be continued as even analytic functions in a circle centered at the origin, the problem has only one solution. A linear approximation is given and is shown to yield inverse formulas of the Born approximation. For even potentials of $\mathcal{C}$ , the relevance of this approximation is studied as follows: Comparison is made between a given static potential and the potential obtained from the scattering amplitude through the inverse formula. The standard deviation between these two potentials is shown to go to zero as the energy goes to ∞. Miscellaneous properties of potentials, wavefunctions, and Jost functions are given in the framework of this method.