Inverse Scattering for Schrodinger Operators with Miura Potentials, II. Different Riccati Representatives

This is the second in a series of papers on scattering theory for one-dimensional Schrodinger operators with Miura potentials admitting a Riccati representation of the form q = u' + u(2) for some u is an element of L-2 (R). We consider potentials for which there exist 'left' and 'right' Riccati representatives with prescribed integrability on half-lines. This class includes all Faddeev-Marchenko potentials in L-1 (R, (1 + |x|) dx) generating positive Schrodinger operators as well as many distributional potentials with Dirac delta-functions and Coulomb-like singularities. We completely describe the corresponding set of reflection coefficients r and justify the algorithm reconstructing q from r.