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

doi:10.1088/0266-5611/25/11/115007

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Inverse Problems Volume 25 Number 11 Create an alert RSS this journal

C Frayer et al 2009 Inverse Problems 25 115007 doi:10.1088/0266-5611/25/11/115007

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

C Frayer¹, R O Hryniv^2,3,4, Ya V Mykytyuk⁴ and P A Perry⁵

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This is the first in a series of papers on scattering theory for one-dimensional Schrödinger operators with highly singular potentials $q\in H_{\mathrm{loc}}^{-1}(\mathbb {R})$ . In this paper, we study Miura potentials q associated with positive Schrödinger operators that admit a Riccati representation q = u' + u² for a unique $u\in L^{1}(\mathbb {R})\cap L^{2}(\mathbb {R})$ . 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 $\mathcal {S}\,:\, u\mapsto r$ 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.

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