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. The problems treated are of a mathematical nature, but have important physical applications. This book will thus be of great use to research workers in various branches of theoretical physics, where the problems can be reduced to one-dimensional second-order differential equations of the Schrödinger type for which phase-integral solutions are required. Includes contributions from notable scientists who have already made use of the authors'technique.

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.