Numerically stable solution of coupled channel equations: The wave function

In two earlier papers a numerically stable solution of the stationary Schrödinger equation for coupled channels was presented. The Schrödinger function and its first derivative were expressed in terms of two matrices: A so-calledlocal reflection matrix (LORE) and aninverse local transmission matrix (INTRA). These matrices obey very simple boundary conditions: They approach asymptotically zero (one) on one side of the reaction path and the reflection (transmission) matrix on the other side. Hence by propagating both matrices along the reaction path one can determine directly the observable scattering matrix elements without ever having to calculate wave functions. On the other hand it is often useful to know the wave functions, for instance in order to interpret scattering data in terms of flow patterns etc. Although the relation between theINTRA-LORE and the wave function is simple, a straight forward calculation is not possible. It would involve an inversion of theINTRA which is numerically ill behaved. In this paper we describe a numerically stable method of computing the wave function and illustrate by two examples of surface reactions.Dedicated to Herbert Wagner on the occasion of his 60th birthday. One of the authors (W.B.) also recalls with great pleasure the stimulating discussions he had with his first graduate student