Multidimensional canonical integrals for the asymptotic evaluation of the S-matrix in semiclassical collision theory

Abstract

The uniform asymptotic evaluation of multidimensional integrals for the S-matrix in semiclassical collision theory is considered. A concrete example of a non-separable two-dimensional integral with four coalescing saddle points is chosen since it exhibits many of the features of more general cases. It is shown how a uniform asymptotic approximation can be obtained in terms of a non-separable two-dimensional canonical integral and its derivatives. This non-separable two-dimensional canonical integral plays a similar role to the Airy integral in one-dimensional integrals with two coalescing saddle points. The uniform approximation is obtained by applying to the two-dimensional case the asymptotic techniques introduced by Chester et al. for one-dimensional integrals. An exact series representation is obtained for the canonical integral by means of complex variable techniques. The series representation can be used to evaluate the canonical integral for small to moderate values of its arguments, whilst for large values of its arguments existing asymptotic techniques may be used.