 Title Pages
 Epigraph
 Preface
 Acknowledgments

Chapter One Introduction 
Chapter Two Introduction to the “Physics” of Rays 
Chapter Three Introduction to the Mathematics of Rays 
Chapter Four Ray Optics: The Classical Rainbow 
Chapter Five An Improvement over Ray Optics: Airy’s Rainbow 
Chapter Six Diffraction Catastrophes 
Chapter Seven Introduction to the WKB(J) Approximation: All Things Airy 
Chapter Eight Island Rays 
Chapter Nine Seismic Rays 
Chapter Ten Elastic Waves 
Chapter Eleven Surface Gravity Waves 
Chapter Twelve Ocean Acoustics 
Chapter Thirteen Tsunamis 
Chapter Fourteen Atmospheric Waves 
Chapter Fifteen The Classical Connection 
Chapter Sixteen Gravitational Scattering 
Chapter Seventeen Scattering of Surface Gravity Waves by Islands, Reefs, and Barriers 
Chapter Eighteen Acoustic Scattering 
Chapter Nineteen Electromagnetic Scattering: The Mie Solution 
Chapter Twenty Diffraction of Plane Electromagnetic Waves by a Cylinder 
Chapter TwentyOne The ClassicaltoSemiclassical Connection 
Chapter TwentyTwo The WKB(J) Approximation Revisited 
Chapter TwentyThree A SturmLiouville Equation: The TimeIndependent OneDimensional Schrödinger Equation 
Chapter TwentyFour The SMatrix and Its Analysis 
Chapter TwentyFive The Jost Solutions: Technical Details 
Chapter TwentySix OneDimensional Jost Solutions: The SMatrix Revisited 
Chapter TwentySeven MorphologyDependent Resonances: The Effective Potential 
Chapter TwentyEight Back Where We Started 
Appendix A Order Notation: The “Big O,” “Little o,” and “∼” Symbols 
Appendix B Ray Theory: Exact Solutions 
Appendix C Radially Inhomogeneous Spherically Symmetric Scattering: The Governing Equations 
Appendix D Electromagnetic Scattering from a Radially Inhomogeneous Sphere 
Appendix E Helmholtz’s Theorem 
Appendix F Semiclassical Scattering: A Précis (and a Few More Details)  Bibliography
 Index
 Princeton Series in Applied Mathematics
A SturmLiouville Equation: The TimeIndependent OneDimensional Schrödinger Equation
A SturmLiouville Equation: The TimeIndependent OneDimensional Schrödinger Equation
 Chapter:
 (p.459) Chapter TwentyThree A SturmLiouville Equation: The TimeIndependent OneDimensional Schrödinger Equation
 Source:
 Rays, Waves, and Scattering
 Author(s):
John A. Adam
 Publisher:
 Princeton University Press
This chapter examines the mathematical properties of the timeindependent onedimensional Schrödinger equation as they relate to SturmLiouville problems. The regular SturmLiouville theory was generalized in 1908 by the German mathematician Hermann Weyl on a finite closed interval to secondorder differential operators with singularities at the endpoints of the interval. Unlike the classical case, the spectrum may contain both a countable set of eigenvalues and a continuous part. The chapter first considers the onedimensional Schrödinger equation in the standard dimensionless form (with independent variable x) and various relevant theorems, along with the proofs, before discussing bound states, taking into account boundstate theorems and complex eigenvalues. It also describes Weyl's theorem, given the SturmLiouville equation, and looks at two cases: the limit point and limit circle. Four examples are presented: an “eigensimple” equation, Bessel's equation of order ? greater than or equal to 0, Hermite's equation, and Legendre's equation.
Keywords: bound states, Schrödinger equation, boundstate theorems, eigenvalues, Weyl's theorem, SturmLiouville equation, limit point, limit circle, Hermite's equation, Legendre's equation
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 Title Pages
 Epigraph
 Preface
 Acknowledgments

Chapter One Introduction 
Chapter Two Introduction to the “Physics” of Rays 
Chapter Three Introduction to the Mathematics of Rays 
Chapter Four Ray Optics: The Classical Rainbow 
Chapter Five An Improvement over Ray Optics: Airy’s Rainbow 
Chapter Six Diffraction Catastrophes 
Chapter Seven Introduction to the WKB(J) Approximation: All Things Airy 
Chapter Eight Island Rays 
Chapter Nine Seismic Rays 
Chapter Ten Elastic Waves 
Chapter Eleven Surface Gravity Waves 
Chapter Twelve Ocean Acoustics 
Chapter Thirteen Tsunamis 
Chapter Fourteen Atmospheric Waves 
Chapter Fifteen The Classical Connection 
Chapter Sixteen Gravitational Scattering 
Chapter Seventeen Scattering of Surface Gravity Waves by Islands, Reefs, and Barriers 
Chapter Eighteen Acoustic Scattering 
Chapter Nineteen Electromagnetic Scattering: The Mie Solution 
Chapter Twenty Diffraction of Plane Electromagnetic Waves by a Cylinder 
Chapter TwentyOne The ClassicaltoSemiclassical Connection 
Chapter TwentyTwo The WKB(J) Approximation Revisited 
Chapter TwentyThree A SturmLiouville Equation: The TimeIndependent OneDimensional Schrödinger Equation 
Chapter TwentyFour The SMatrix and Its Analysis 
Chapter TwentyFive The Jost Solutions: Technical Details 
Chapter TwentySix OneDimensional Jost Solutions: The SMatrix Revisited 
Chapter TwentySeven MorphologyDependent Resonances: The Effective Potential 
Chapter TwentyEight Back Where We Started 
Appendix A Order Notation: The “Big O,” “Little o,” and “∼” Symbols 
Appendix B Ray Theory: Exact Solutions 
Appendix C Radially Inhomogeneous Spherically Symmetric Scattering: The Governing Equations 
Appendix D Electromagnetic Scattering from a Radially Inhomogeneous Sphere 
Appendix E Helmholtz’s Theorem 
Appendix F Semiclassical Scattering: A Précis (and a Few More Details)  Bibliography
 Index
 Princeton Series in Applied Mathematics