Recovery of the Potential from I-Function

The inverse scattering problem on half-axis, in other words, in the spherically symmetric case, consists of finding the unknown potential from a suitable class from the scattering data S. The other problem of practical interest is to find this potential from the spectral data dρ(λ). In the literature there are recovery procedures for finding the potential from the spectral or from the scattering data. Define the I  -function: I(k):=f′(0,k)f(k), where f(x, k) is the Jost solution. Constructive ways to find I(k) from dρ(λ) and vice versa, and I(k) from S and vice versa are given. The theory of the Riemann problem is used as an essential tool. If I(k) is found, then our methods allow one to construct the scattering data and from these to recover the potential by the known procedure. Alternatively, one can construct from I(k) the spectral data and from these find the potential by the known procedure.