This graduate textbook offers an introduction to the spectral theory of ordinary differential equations, focusing on Sturm–Liouville equations.
Sturm–Liouville theory has applications in partial differential equations and mathematical physics. Examples include classical PDEs such as the heat and wave equations. Written by leading experts, this book provides a modern, systematic treatment of the theory. The main topics are the spectral theory and eigenfunction expansions for Sturm–Liouville equations, as well as scattering theory and inverse spectral theory. It is the first book offering a complete account of the left-definite theory for Sturm–Liouville equations.
The modest prerequisites for this book are basic one-variable real analysis, linear algebra, as well as an introductory course in complex analysis. More advanced background required in some parts of the book is completely covered in the appendices. With exercises in each chapter, the book is suitable for advanced undergraduate and graduate courses, either as an introduction to spectral theory in Hilbert space, or to the spectral theory of ordinary differential equations. Advanced topics such as the left-definite theory and the Camassa–Holm equation, as well as bibliographical notes, make the book a valuable reference for experts.
Author(s): Christer Bennewitz, Malcolm Brown, Rudi Weikard
Series: Universitext
Edition: 1
Publisher: Springer
Year: 2020
Language: English
Pages: 379
Tags: Hilert Spaces, Spectral Theory, Sturm-Liouville, Scattering
Preface
Contents
Chapter 1 Introduction
1.1 Background
1.2 Linear spaces
1.3 Spaces with scalar product
1.4 The equations of Sturm and Liouville
1.5 Notes and remarks
Chapter 2 Hilbert space
2.1 Complete spaces
2.2 Operators and relations
2.3 Resolvents
2.4 Extension of symmetric relations
2.5 Notes and remarks
Chapter 3 Abstract spectral theory
3.1 The spectral theorem
3.2 The spectrum
3.3 Compactness
3.4 Quadratic forms and the Minimax principle
3.5 Notes and remarks
Chapter 4 Sturm–Liouville equations
4.1 Introduction
4.2 Boundary conditions
4.3 Expansion in eigenfunctions
4.4 Two singular endpoints
4.5 The spectrum
4.6 Notes and remarks
Chapter 5 Left-definite Sturm–Liouville equations
5.1 Left-definite equations
5.2 Expansion in eigenfunctions
5.3 Two limit-point endpoints
5.4 Equations that are both left- and right-definite
5.5 Generalized left-definite Sturm–Liouville equations
5.6 Spectral theory
5.7 Defect indices and boundary conditions
5.8 Eigenfunction expansions
5.9 Notes and remarks
Chapter 6 Oscillation, spectral asymptotics and special functions
6.1 The zeros of solutions
6.2 Order of magnitude estimates for m(lambda)
6.3 Asymptotic estimates for m(lambda)
6.4 Asymptotics of solutions
6.5 Other spectral quantities
6.6 Special functions
6.7 Notes and remarks
Chapter 7 Uniqueness of the inverse problem
7.1 The Borg-Marchenko theorem
7.2 Liouville transforms
7.3 Right-definite Sturm–Liouville equations
7.4 Left-definite Sturm–Liouville equations
7.5 Two singular endpoints
7.6 Notes and remarks
Chapter 8 Scattering
8.1 Introduction
8.2 Jost solutions
8.3 The Jost transform
8.4 Inverse theory
8.5 The case of a compact resolvent
8.6 Notes and remarks
Appendix A Functional analysis
Appendix B Stieltjes integrals
B.1 Riemann–Stieltjes integrals
B.2 Functions of bounded variation
B.3 Radon measures
B.4 Integrable functions
B.5 Convergence theorems
B.6 Measurability
B.7 Some Banach spaces
B.8 Relations between measures
B.9 Product measures
Notes and remarks
Appendix C Schwartz distributions
C.1 Notes and remarks
Appendix D Ordinary differential equations
Appendix E Analytic functions
E.1 Nevanlinna functions
E.2 Entire functions
E.3 Bounded type
E.4 A lemma of de Branges
E.5 Notes and remarks
Appendix F The Camassa–Holm equation
References
Symbol Index
Subject Index
