^{1}Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, 53115, Bonn, Germany

^{2}SFB 611, HCM, and IZKS, Bonn, Germany

^{3}BiBoS, Bielefeld, Germany

^{4}CERFIM, Locarno, Switzerland

^{5}Accademia di Architettura, Mendrisio, Switzerland

^{6}Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova st., 79601 Lviv, Ukraine

^{7}Institute of Mathematics, the University of Rzeszów, 16A Rejtana al., 35-959 Rzeszów, Poland

^{8}Lviv National University, 1 Universytetska st., 79602 Lviv, Ukraine

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2012, Issue 666, Pages 83–113, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2011.115, July 2011

## Abstract

We show that for the Schrödinger operators on the semi-axis with Bessel-type potentials *κ*(*κ* + 1)/*x*^{2},
, there exists a meaningful direct and inverse scattering theory. Several new phenomena not observed in the “classical case” of Faddeev–Marchenko potentials arise here; in particular, for *κ* ≠ 0 the scattering function *S* takes two different values on the positive and negative semi-axes and is thus discontinuous both at the origin and at infinity.

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