Phase-Integral Method

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.- 1.2.1 Derivation of the WKB Approximation.- 1.2.2 Deficiencies of the WKB Approximation in Higher Order.- 1.2.3 Phase-Integral Approximation of Arbitrary Order, Freed from the First Deficiency.- 1.3 Phase-Integral Approximation of Arbitrary Order, Generated from an Unspecified Base Function.- 1.3.1 Direct Procedure.- 1.3.2 Transformation Procedure.- 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.- 1.5.1 Solutions of the Schrödinger Equation and Solutions of the q-Equation Expressed in Terms of Each Other.- 1.5.2 Ermakov―Lewis Invariant.- 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.- 2.2.1 Differential Equation for ?0.- 2.2.2 Determination of the Coefficients An,0 and Bq.- 2.2.3 Differential Equation for ?2N When N > 0.- 2.2.4 Regularity Properties of I2N and ?2N When N > 0.- 2.2.5 Determination of the Coefficients An,2N When N > 0.- 2.2.6 Expressions for ?2 and ?4.- 2.2.7 Behavior of ?2N(z) in the Neighborhood of a First-or Second-Order Pole of Q2(z) When N > 0.- 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.- 3.1 Introduction.- 3.2 Comparison Equation Solution.- 3.3 Phase-Integral Approximation Obtained from the Comparison Equation Solution.- References.- 4 Relations Between Different Nonoscillating Solutions of the q-Equation Close to a Transition Zero.- 4.1 Introduction.- 4.2 Comparison Equation Solutions.- 4.3 Comparison Equation Expressions for Nonoscillating Solutions of the q-Equation.- 4.3.1 The Case When Re ? Increases as z Moves Away from t in the Neighborhood of the Anti-Stokes Line A.- 4.3.2 The Case When Re ? Decreases as z Moves Away from t in the Neighborhood of the Anti-Stokes Line A.- 4.3.3 Summary of the Results for the Two Cases in Sections 4.3.1 and 4.3.2.- 4.3.4 Application Illustrating the Consistency of the Formulas Obtained.- 4.4 Simple First-Order Formulas.- 4.5 Relations Between the a-Coefficients Associated with Different q-Functions, in Terms of Which a Given Solution ?(z) is Expressed.- 4.6 Condition for Determination of Regge Pole Positions.- References.- 5 Cluster of Two Simple Transitions Zeros.- 5.1 Introduction.- 5.2 Wave Equation and Phase-Integral Approximation.- 5.3 Comparison Equation.- 5.4 Comparison Equation Solution.- 5.4.1 Determination of ?0(z) and $${\overline K _0}$$.- 5.4.2 Determination of ?2? and $${\overline K _{2\beta }}$$ for ? > 0.- 5.5 Phase-Integral Solution Obtained from the Comparison Equation Solution.- 5.6 Stokes Constants.- 5.7 Application to Complex Potential Barrier.- 5.8 Application to Regge Pole Theory.- Appendix: Phase-Integral Solution Obtained from the Comparison Equation Solution by Straightforward Calculation.- References.- 6 Phase-Integral Formulas for the Regular Wave Function When There Are Turning Points Close to a Pole of the Potential.- 6.1 Introduction.- 6.2 Definitions and Preparatory Calculations.- 6.2.1 Determination of ?0 and A1,0.- 6.2.2 Determination of ?2? and A1,2? for ? > 0.- 6.3 Comparison Equation Corresponding to Scattering States.- 6.3.1 Comparison Equation Solution.- 6.3.2 Phase-Integral Approximation Obtained from the Comparison Equation Solution.- 6.3.3 Behavior of the Wave Function Close to the Origin.- 6.3.4 Summary of Formulas in Section 6.3.- 6.4 Comparison Equation Corresponding to Bound States.- 6.4.1 Quantization Condition.- 6.4.2 Normalized Wave Function.- Appendix: Calculation of q(z) and ?(2n+1).- References.- 7 Normalized Wave Function of the Radial Schrödinger Equation Close to the Origin.- 7.1 Introduction.- 7.2 ?0 > 0.- 7.3 ?0 = 0, ?0 ? 0.- 7.4 Summary of the Results Obtained in the Present Chapter and Discussion of Results Obtained by Previous Authors.- References.- 8 Phase-Amplitude Method Combined with Comparison Equation Technique Applied to an Important Special Problem.- 8.1 Introduction.- 8.2 Quantization Condition.- 8.3 Solution of the Difficulty at the Origin by Means of Comparison Equation Solutions Expressed in Terms of Coulomb Wave Functions.- 8.4 Application to a Two-Dimensional Anharmonic Oscillator.- References.- 9 Improved Phase-Integral Treatment of the Combined Linear and Coulomb Potential.- 9.1 Introduction.- 9.2 Energy Levels.- 9.3 Expectation Values.- Appendix: Expressions for Phase-Integral Quantities in Terms of Complete Elliptic Integrals.- References.- 10 High-Energy Scattering from a Yukawa Potential.- 10.1 Introduction.- 10.2 Phase Shifts.- 10.3 Probability Density at the Origin.- Appendix: Numerical Solution of the Schrödinger Equation.- References.- 11 Probabilities for Transitions Between Bound States in a Yukawa Potential, Calculated with Comparison Equation Technique.- 11.1 Introduction.- 11.2 Phase-Integral Formulas.- 11.3 Comparison Equation Formulas.- References.- Author Index.