On the Convergence of Born Expansions

Abstract

A study is made of the convergence of the Born expansions in collision problems involving a static central potential. The scattering of each partial wave, characterized by the phase shift ${\ensuremath{\eta}}_{l}$, is treated separately. Estimates of the radii of convergence of $tan{\ensuremath{\eta}}_{l}$ and of ${S}_{l}\ensuremath{\equiv}{e}^{2i{\ensuremath{\eta}}_{l}}$ (as well as of the corresponding wave functions) are established, and the truncation errors due to breaking off the Born expansions are investigated. An appendix deals with the expansion of ${\ensuremath{\eta}}_{l}$ itself.