Inverse Problem in the Coupling Constant
We show how to solve the inverse problem in the coupling constant by reducing it, with the help of Liouville transformation, to the classical case of Gel’fand-Levitan theory for discrete spectra. Once the Gel’fand-Levitan integral equation is solved, one has three more steps to get the potential, the first two being integrations, and the third the inversion of a function and differentiation. By generalizing to the case of two potentials, we show that one can solve the inverse problem in the coupling constant in the general case, at any (fixed) energy, and for finite or infinite intervals. Several soluble examples are presented.