Inverse scattering for Schrödinger 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 Schrödinger operators with highly singular potentials q ε H^-1_loc(ℝ) . In this paper, we study Miura potentials q associated with positive Schrödinger operators that admit a Riccati representation q = u′ + u² for a unique u ε L ¹ (ℝ) ∩ L² (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 : 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.