Inverse scattering for Miura potentials

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

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C Frayer¹, R O Hryniv^2,3,4, Ya V Mykytyuk⁴ and P A Perry⁵
¹ Department of Mathematics, University of Wisconsin–Platteville, Platteville, WI 53818, USA
² Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 79601 Lviv, Ukraine
³ Institute of Mathematics, University of Rzeszów, 16 A Rejtana str., 35-959 Rzeszów, Poland
⁴ Department of Mechanics and Mathematics, Lviv Franko National University, 1 Universytets'ka str., 79602, Lviv, Ukraine
⁵ Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA
E-mail: frayerc@uwplatt.edu, rhryniv@iapmm.lviv.ua, yamykytyuk@yahoo.com and perry@ms.uky.edu

Abstract. This is the first in a series of papers on scattering theory for one-dimensional Schrödinger operators with highly singular potentials $q\in H_{\mathrm{loc}}^{-1}(\mathbb {R})$ . 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.

Print publication: Issue 11 (November 2009)
Received 13 July 2009, in final form 4 September 2009
Published 29 October 2009

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

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