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 q is an element of H(loc)(-1)(R). In this paper, we study Miura potentials q associated with positive Schrodinger operators that admit a Riccati representation q = u' + u(2) for a unique u is an element of L(1)(R) boolean AND L(2)(R). Such potentials have a well-defined reflection coefficient r(k) that satisfies vertical bar r(k)vertical bar < 1 and determines u uniquely. We show that the scattering map S : u -> 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.