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

Abstract

This is the first in a series of papers on scattering theory for one-dimensional Schrödinger operators with highly singular potentials $q\in H^{-1}(R)$. In this paper, we study Miura potentials $q$ associated to positive Schrödinger operators that admit a Riccati representation $q=u^{\prime }+u^2$ for a unique $u\in L^1(R)\cap L^2(R)$. 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 $S:u\mapsto r$ 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.

29 pages; requires IOP "iopart.cls" style file; to appear in Inverse Problems 25 (2009)

References in corpus (1)

• Inverse spectral problems for Dirac operators with summable potentials