Inverse scattering on the line for Schr\"odinger operators with Miura potentials, II. Different Riccati representatives

This is the second in a series of papers on scattering theory for one-dimensional Schr"odinger operators with Miura potentials admitting a Riccati representation of the form $q=u^{\mathrm{\prime }}+u^2$ for some $u\in L^2(R)$. 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 $L^1(R,(1+\mathrm{\mid }x\mathrm{\mid })dx)$ generating positive Schr"odinger 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 $r$.