Exact analytic asymptotic formulae for continuum distorted-wave matrix elements

By parametrizing, and evaluating analytically, an intermediate double integral over radial prolate spheroidal coordinates, we show how the long-range part of a prototype of the two-centre volume integrals which commonly occur in three-body heavy particle rearrangement scattering, may be written in a simple close form as an infinite series of parametric derivatives. Using a complex contour-integral representation of the confluent hypergeometric function, we show how this result may be used to calculate analytically the asymptotic expansions for large positive and large negative time, of continuum distorted-wave matrix elements to all orders of the internuclear distance R. Analysis of the leading term of the resulting infinite series shows that these matrix elements may be adequately represented by only the first few terms of their asymptotic expansions. This is illustrated by comparing graphically the analytical asymptotic expressions with a direct numerical evaluation for specified examples. We demonstrate how the use of these asymptotic expansions in a close-coupling continuum distorted-wave calculation can realize up to a 70% saving in computing time when calculating the probability amplitudes for a given heavy particle trajectory, with a commensurate increase in the accuracy of the calculations. We also indicate how this method includes as a subset the asymptotic forms of eikonal and Born-like matrix elements. (author)