Exact and Glauber amplitudes in multi-channel scattering

A model for one-dimensional many-channel scattering of a particle from a composite system is considered. It is assumed that the target cannot break up. Using Bellman's method of invariant embedding, a system of non-linear first-order differential equations with one-point boundary conditions is obtained for the transmission and reflection coefficients. These equations have stable numerical solutions. An alternative invariant embedding with the approximation of no reflection at any stage of the scattering process yields the Glauber approximation for the transmission coefficients. These formulations are used to compare the exact and the Glauber amplitudes for a projectile scattering from a bound two-particle system. It is found that unlike the inelastic Glauber amplitudes, the elastic Glauber amplitudes are very good approximations to the exact results.