Inverse scattering for Schrodinger operators with Miura potentials, I. Unique Riccati representatives and ZS-AKNS systems

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 admit 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.