Collisions and umbilic catastrophes

Abstract

The theory of uniform approximations in semiclassical scattering theory is reviewed in order to emphasize the role of the unfolding of the relevant catastrophe type. Application of the theory to situations involving hyperbolic and elliptic umbilic catastrophes is shown to involve a mapping from four physically determined semiclassical phases {f_k } to three control (a, b, c) and one translational parameter d. The paper is devoted to solution of this mapping problem. The final reduction is to a ninth degree polynomial equation in a ³, each root of which gives rise to six symmetry related combinations (b, c). Algebraic and numerical investigations demonstrate the general existence of three physically acceptable (real a, (b + c)) combinations for any set of four phases. Given the four associated jacobians {J_k }, this ambiguity may be definitively removed by following the evolution of (a, b, c) between members of a family of related scattering transitions. Numerical experiments indicate that the correct choice will lead to a real set of expansion coefficients in the uniform approximation, a condition requiring knowledge of a single set {f_k , J_k }.