Inverse scattering on the line for Schrödinger operators with Miura potentials, II. Different Riccati representatives

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This is the second in a series of papers on scattering theory for one-dimensional Schrödinger operators with Miura potentials admitting a Riccati representation of the form q=u'+u² for some u∈ L²(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 L¹(R,(1+|x|)dx) generating positive Schrödinger 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.