A NEW APPROACH TO THE INVERSE SCATTERING AND SPECTRAL PROBLEMS FOR THE STURM-LIOUVILLE EQUATION

New proofs of the known uniqueness theorems for the one-dimensional in verse spectral and scattering problems are given. Proof of the inverti bility of all of the steps in the inversion procedures of Gelfand-Levi tan and Marchenko is given. The proposed method of investigation yield s some new results, for example, a Marchenko-type equation at x = 0 wh ich holds on the whole axis, rather than on a half-axis, as usual for the scattering theory on half-axis. It also yields a new method, short er and simpler than earlier published, for proving that the potential in the class L-1,L- 1, obtained by the Marchenko reconstruction proced ure, generates the scattering data from which it was reconstructed.