Time-independent and time-dependent potential scattering without angular momentum decomposition (Q1102091)

This paper presents methods of treating potential scattering by direct solution of the Lippmann-Schwinger equation avoiding decomposition of the solution in partial waves with fixed angular momentum. The standard time independent treatment gives rise to singularities which are avoided in the time dependent methods. However the problem of large time and energy dependent oscillation exists. One interesting method presented for dealing with this problem is the introduction of complex momenta although this sometimes produces singularities in the potential. Once a matrix \(R((1- i\alpha)\ell,t)=R(x,t)\) is calculated numerically a transformation is used to obtain R(y,t) for real y. The numerical procedures presented depend strongly on the theory of B-splines.