Phase-Integral Method

The efficiency of the phase-integral method developed by the present au thors has been shown both analytically and numerically in many publica tions. With the inclusion of supplementary quantities, closely related to new Stokes constants and obtained with the aid of comparison equation technique, important classes of problems in which transition points may approach each other become accessible to accurate analytical treatment. The exposition in this monograph is of a mathematical nature but has important physical applications, some examples of which are found in the adjoined papers. Thus, we would like to emphasize that, although we aim at mathematical rigor, our treatment is made primarily with physical needs in mind. To introduce the reader into the background of this book, we start by de scribing the phase-integral approximation of arbitrary order generated from an unspecified base function. This is done in Chapter 1, which is reprinted, after minor changes, from a review article. Chapter 2 is the result of re search work that was pursued during more than two decades, interrupted at times. It started in the sixties, when we were still using a phase-integral approximation, which in our present terminology corresponds to a special choice of the base function. At the time our primary aim was to derive expressions for the supplementary quantities needed in order to obtain an accurate connection formula for a real potential barrier, when the energy lies in the neighborhood of the top of the barrier.

The result of two decades spent developing and refining the phase-integral method to a high level of precision, the authors have applied this method to problems in various fields of theoretical physics.

1 Phase-Integral Approximation of Arbitrary Order Generated from an Unspecified Base Function.- 1.1 Introduction.- 1.2 The So-Called WKB Approximation, Its Deficiencies in Higher Order, and Early Attempts to Remedy These Deficiencies.- 1.3 Phase-Integral Approximation of Arbitrary Order, Generated from an Unspecified Base Function.- 1.4 Advantage of Phase-Integral Approximation Versus WKB Approximation in Higher Order.- 1.5 Relations Between Solutions of the Schrödinger Equation and the q-Equation.- 1.6 Phase-Integral Method.- Appendix: Phase-Amplitude Relation.- References.- 2 Technique of the Comparison Equation Adapted to the Phase-Integral Method.- 2.1 Background.- 2.2 Comparison Equation Technique.- 2.3 Derivation of the Arbitrary-Order Phase-Integral Approximation from the Comparison Equation Solution.- 2.4 Summary of the Procedure and the Results.- References.- Adjoined Papers.- 3 Problem Involving One Transition Zero.- 4 Relations Between Different Nonoscillating Solutions of the q-Equation Close to a Transition Zero.- 5 Cluster of Two Simple Transitions Zeros.- 6 Phase-Integral Formulas for the Regular Wave Function When There Are Turning Points Close to a Pole of the Potential.- 7 Normalized Wave Function of the Radial Schrödinger Equation Close to the Origin.- 8 Phase-Amplitude Method Combined with Comparison Equation Technique Applied to an Important Special Problem.- 9 Improved Phase-Integral Treatment of the Combined Linear and Coulomb Potential.- 10 High-Energy Scattering from a Yukawa Potential.- 11 Probabilities for Transitions Between Bound States in a Yukawa Potential, Calculated with Comparison Equation Technique.- Author Index.