Technique of the Comparison Equation Adapted to the Phase-Integral Method
For a cluster consisting of an unspecified number of transition points we develop a convenient comparison equation technique for obtaining the wave function within the cluster and the Stokes constants that determine the phase-integral solution outside the cluster. The resulting formulas remain valid even if the transition points approach each other. We perform the rather lengthy calculations in a general way and obtain formulas which can easily be particularized to the specific situations encountered in applications. The final analytic formulas for the Stokes constants are expressed in terms of phase integrals, in which the contributions to the integrands in successive orders of approximation are to be integrated along contours enclosing pairs of transition points. With the aid of the comparison equation technique we thus obtain supplementary quantities which, when incorporated into the general formulas and the connection formulas of the phase-integral method, are of decisive importance when the transition points lie close to each other, but which become very small when those points recede sufficiently far away from each other.
It is shown explicitly that, after appropriate asymptotic expansion in terms of a “small” bookkeeping parameter, the comparison equation solution constructed here yields locally, when there are no transition points in the region under consideration (i.e., sufficiently far away from transition points) the arbitrary-order phase-integral approximation generated from an unspecified base function, which has been described in Chapter 1 of the present monograph.
The comparison equation technique used is based on general ideas that are already known; however, their adaptation to our purpose requires a number of new steps in the procedure, and the resulting phase-integral formulas are new.
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