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 $uin L^2(R)$. 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 $r$.
This is the first in a series of papers on scattering theory for one-dimensional Schrodinger operators with highly singular potentials $qin H^{-1}(R)$. In this paper, we study Miura potentials $q$ associated to positive Schrodinger operators that adm
it a Riccati representation $q=u+u^2$ for a unique $uin L^1(R)cap L^2(R)$. Such potentials have a well-defined reflection coefficient $r(k)$ that satisfies $|r(k)|<1$ and determines $u$ uniquely. We show that the scattering map $S:umapsto r$ is real-analytic with real-analytic inverse. To do so, we exploit a natural complexification of the scattering map associated with the ZS-AKNS system. In subsequent papers, we will consider larger classes of potentials including singular potentials with bound states.
