Phase-Integral Method: Allowing Nearlying Transition Points

ISBN-13: 9781461275114 / Angielski / Miękka / 2011 / 250 str.

Nanny Fr Man; Per O. Fr Man

Opis

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."

Spis treści

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.