Solution of a Coupled Channel Inverse Scattering Problem at Fixed Energy by a Modified Newton-Sabatier Method.

The modiied Newton-Sabatier method is extended to the inverse scattering problem of coupled channels at xed energy. The problem is solved for a system of coupled radial Schrr odinger equations with a potential matrix independent of the angular momentum of the relative motion. The new method is applied to the example of two channels coupled by complex-valued square well potentials. 1 Introduction The inverse scattering problem in quantum mechanics plays an important role in the determination of eeective potentials from measured cross sections. The xed angular momentum inversion problem for one channel can be solved by the Gel'fand-Levitan method 1] or by the Marchenko method 2]. An extension of the Gel'fand-Levitan method to coupled channels was developed by Cox 3]. A rational scheme was proposed by Kohlhoo and von Geramb 4] and applied to nucleon-alpha spin-orbit interactions by Becker 5]. The xed energy inversion problem was solved for one channel by New-ton 6] and Sabatier 7]. Reviews are given in Refs. 8] and 9]. M unchow and Scheid 10] modiied the one channel Newton-Sabatier method under the assumption that the spherical potential V (r) is known from a certain nite distance r 0 up to innnity which is usually the case in practice. With this mod-iication the method is applicable to realistic heavy ion collisions 11,12]. A spin-orbit inversion scheme at xed energy was given by Leeb et al. 13] and applied by Alexander et al. 14] to derive nucleon-alpha spin-orbit potentials. The results of these calculations compare well to the potentials obtained by Becker 5]. In this Letter we extend the modiied Newton-Sabatier method to the case of N coupled channels under the assumption that the potential matrix does not depend on the angular momentum of relative motion.