The reconstruction of a differential operator form discrete spectra is reduced to its reconstruction from an S-matrix. This method makes it possible to solve the singular Sturm–Liouville problems which determine certain modes of a sphere. The results pave the way for handling studies in which information on modes and scattering results would all be taken into account. They are applied to the earth inverse problem and partial answers are given to a well-known conjecture. Finally the relevance of the JWKB approximation in this kind of problem is briefly discussed.