On the Stokes Phenomenon for the Differential Equations Which Arise in the Problem of Inelastic Atomic Collisions

Abstract

The problem of inelastic atomic collisions, involving exchange of energy between the atoms, depends upon the solution of a pair of coupled differential equations equivalent to a single differential equation of the fourth order, in which the independent variable is the distance, r, between the two atoms. Asymptotic forms for the solutions of this pair of equations may be found. The probability of energy exchange may be shown to depend upon the connections between the solutions for values of r which correspond to regions of space where the relative kinetic energy of the two atoms is positive and the solutions for values of r corresponding to regions where the kinetic energy is negative. The finding of these connections involves a study of the asymptotic forms of the solutions when these forms are considered as functions of a complex variable. Such a study has been made in this paper, and the corresponding Stokes phenomenon investigated. The desired end was not achieved, but results of interest have nevertheless been obtained. A brief discussion is given of the previous work of Stueckelberg.