Justification of local potentials in coupled channel problems

It is often stated that for reactions between two nuclei, the exchange terms lead to serious nonlocalities in the potential. The resonating group function u/sub E/ indeed obeys a strongly nonlocal equation. However, u/sub E/ is not a wave function and it is impossible to approximate its equation by a Schroedinger equation with a local potential. As stated by several authors, one has to consider Ω/sub E/= (1-K)/sup 1/2/ u/sub E/ instead of u/sub E/ where 1-K is the norm kernel. In this paper, these statements are verified in a simple model in which the Hamiltonian and norm kernels are given analytically for both u and Ω. We then investigate the corresponding equations for several (in general, coupled) channels and the possibility of using local potentials in these equations. Alpha decay into more than one channel is discussed as an example