All the ingredients for the solution of the inverse scattering problem via the Gel’fand-Levitan procedure have now been assembled. What is actually going to be solved is an inverse spectral problem posed for the regular solution. This inverse spectral problem is of relatively little intrinsic interest because in dimensions higher than one (for noncentral potentials) the regular solution is not a natural solution of the Schrodinger equation. As we saw in Chapter 3, it has to be defined in a very indirect manner and the inverse spectral problem for it does not arise naturally. However, solving this inverse spectral problem, in which the spectral function, defined in Section 3.3, is the input, solves at the same time the inverse scattering problem because the Jost function forms a direct link from the scattering data, that is, the S matrix, to the spectral function. Thus, solving the problem of finding the potential that underlies a given spectral function also solves the problem of finding the potential that underlies a given S matrix. Once the Wiener-Hopf factorization problem has been solved and the Jost function has been constructed from the S matrix, the spectral function is known.