Phase-Integral Method

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.