For inverse scattering theories, it is well known that in order to guarrantee the existence and uniquness of the solution, drastic restrictions must be imposed on the scattering potential or on the scattering data collected from experiments (see for example Newton 1966). The existing theories in their original form such as the Geldfan Levitan or the Marehenko equations therefore cannot be used in a straightforward manner and some further steps toward simplifications have to be considered.

In inelastic scattering problems where many coupled channels must be taken into account to construct various elements of the scattering matrix the problem become much more complicated both from the theoretical and experimental point of view. The present work deal only with the energy fixed case and show that a solution can nevertheless be obtained by using on one hand the conventional technique of the Abelian transformation which has been extensively applied in the JWKB approach to the one channel problem (see for example a general review by Buck 1971) with on the other some results obtained recently concerning a system of coupled differential equations (Cao 1982 1).