Bounds on Elements of the S Matrix for Elastic Scattering: One-Dimensional Scattering

Abstract
For the quantum‐mechanical scattering of a particle by a static potential, a method due to Kato determines upper and lower bounds on cotη, where η is the phase shift for a given angular momentum. The method is readily generalizable to the case of elastic multi‐channel scattering if one can a priori diagonalize the scattering matrix S to determine the standing‐wave eigenmodes (by symmetry considerations, for example). We here consider the case when this can not be done. Both bounds are obtained on the independent elements Bij of B, where B=i(l+S)(l−S)−1, which then determine bounds on the elements of S, and on the eigenphase shifts and mixing parameters. One‐dimensional scattering by V(x)≠V(−x) provides a concrete two‐channel example. The numerical results obtained are very encouraging. The Kato method is useful only if one can obtain solutions of related scattering problems. The most significant feature of the present approach is that the related scattering problems can be single‐channel scattering problems.