Two-Variable Galilei-Group Expansions of Nonrelativistic Scattering Amplitudes

Abstract

Two-variable Galilei-group expansions are derived for the two-particle nonrelativistic scattering amplitude. These expansions contain the usual partial-wave and eikonal expansions and supplement them by further expansions in the remaining kinematic variable. The expansions are written both for square-integrable and for asymptotically increasing amplitudes and are shown to correspond to the nonrelativistic limit of previously considered relativistic two-variable expansions. Dynamical singularities (Breit-Wigner resonances, bound states, poles in the impact-parameter plane, etc.) are investigated and related to the asymptotic behavior of the expansion coefficients (or Galilei amplitudes). The threshold and high-energy limits of the expansions are discussed. As a mathematical by-product we give a classification of the subgroups of E(3) and also some results on the representation theory of this group; in particular we study the Clebsch-Gordan coefficients.