Asymptotic expansions for parabolic cylinder functions of large order and argument

Abstract

Using the method of steepest descent an exact and practicable asymptotic expansion for the parabolic cylinder function Dp(z) is derived in the case when mod z mod 2 and mod p mod are both large and of the same order. Limitations and defects in the existing literature on the subject are reviewed. Two-term approximations of considerable importance in the phase-integral analysis of four-transition-point problems are obtained. Several examples required in the analysis of double underdense-potential-barrier problems are given. It is pointed out that such potential barrier problems are associated with perturbed symmetric resonance strong rotational coupling and curve-crossing in atom-atom collision theory. New model transition probabilities are presented.