**Mathematical Surveys and Monographs**

Volume: 121;
2005;
328 pp;
Softcover

MSC: Primary 34;
Secondary 47

**Print ISBN: 978-0-8218-5267-5
Product Code: SURV/121.S**

List Price: $94.00

AMS Member Price: $75.20

MAA Member Price: $84.60

**Electronic ISBN: 978-1-4704-1348-4
Product Code: SURV/121.S.E**

List Price: $88.00

AMS Member Price: $70.40

MAA Member Price: $79.20

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#### Supplemental Materials

# Sturm-Liouville Theory

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*Anton Zettl*

In 1836 and 1837, Sturm and Liouville published a series of
papers on second order linear ordinary differential operators, which
began the subject now known as the Sturm–Liouville theory. In
1910, Hermann Weyl published an article which started the study of
singular Sturm–Liouville problems. Since then,
Sturm–Liouville theory has remained an intensely active field of
research with many applications in mathematics and mathematical
physics.

The purpose of the present book is (a) to provide a modern survey
of some of the basic properties of Sturm-Liouville theory and (b) to
bring the reader to the forefront of research on some aspects of this
theory. Prerequisites for using the book are a basic knowledge of
advanced calculus and a rudimentary knowledge of Lebesgue integration
and operator theory. The book has an extensive list of references and
examples and numerous open problems. Examples include classical
equations and functions associated with Bessel, Fourier, Heun, Ince,
Jacobi, Jörgens, Latzko, Legendre, Littlewood-McLeod, Mathieu,
Meissner, and Morse; also included are examples associated with the
harmonic oscillator and the hydrogen atom. Many special functions of
applied mathematics and mathematical physics occur in these
examples.

This book offers a well-organized viewpoint on some basic features
of Sturm–Liouville theory. With many useful examples treated in
detail, it will make a fine independent study text and is suitable for
graduate students and researchers interested in differential
equations.

#### Readership

Graduate students and research mathematicians interested in differential equations.

#### Reviews & Endorsements

In summary, this monograph offers a wealth of information on Sturm-Liouville theory and is an ideal textbook for a course in this field, serves as an indispensible source for every researcher working in this area, is ideally suited for self-study due to its detailed proofs and comprehensive bibliography, and is recommended to any applied scientist who wants to use Sturm-Liouville theory.

-- Zentralblatt MATH

...this monograph is a valuable and important work, useful for all people interested in the theory of second-order linear differential operators.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Sturm-Liouville Theory

- Contents v6
- Preface ix10
- Part 1. Existence and Uniqueness Problems 114
- Chapter 1. First Order Systems 316
- 1. Introduction 316
- 2. Existence and Uniqueness of Solutions 316
- 3. Variation of Parameters 821
- 4. The Gronwall Inequality 821
- 5. Bounds and Extensions to the Endpoints 1023
- 6. Continuous Dependence of Solutions on the Problem 1225
- 7. Differentiable Dependence of Solutions on the Data 1427
- 8. Adjoint Systems 1932
- 9. Inverse Initial Value Problems 2033
- 10. Comments 2134

- Chapter 2. Scalar Initial Value Problems 2538
- 1. Introduction 2538
- 2. Existence and Uniqueness 2538
- 3. Continuous Extensions to the Endpoints 2639
- 4. Continuous Dependence of Solutions on the Problem 2740
- 5. Differentiable Dependence of Solutions on the Data 2841
- 6. Sturm Separation and Comparison Theorems 3245
- 7. Periodic Coefficients 3548
- 8. Comments 3851

- Part 2. Regular Boundary Value Problems 4154
- Chapter 3. Two-Point Regular Boundary Value Problems 4356
- 1. Introduction 4356
- 2. Transcendental Characterization of the Eigenvalues 4356
- 3. The Fourier Equation 4760
- 4. The Space of Regular Boundary Value Problems 5366
- 5. Continuity of Eigenvalues and Eigenfunctions 5467
- 6. Differentiability of Eigenvalues 5669
- 7. Finite Spectrum 5770
- 8. Green's Function 6174
- 9. Comments 6578

- Chapter 4. Regular Self-Adjoint Problems 6982
- 1. Introduction 6982
- 2. Canonical Forms of Self-Adjoint Boundary Conditions 7184
- 3. Existence of Eigenvalues 7285
- 4. Dependence of Eigenvalues on the Problem 7689
- 5. The Prüfer Transformation 8194
- 6. Separated Boundary Conditions 8699
- 7. Coupled Boundary Conditions 90103
- 8. An Elementary Existence Proof for Coupled Boundary Conditions 91104
- 9. Monotonicity of Eigenvalues 95108
- 10. Multiplicity of Eigenvalues 96109
- 11. Green's Function 96109
- 12. Finite Real Spectrum 98111
- 13. Comments 104117

- Chapter 5. Regular Left-Definite and Indefinite Problems 107120
- 1. Introduction 107120
- 2. Definition and Characterization of Left-Definite Problems 108121
- 3. Existence of Eigenvalues 112125
- 4. Continuous Dependence of Eigenvalues on the Problem 114127
- 5. Eigenvalue Inequalities 116129
- 6. Differentiability of Eigenvalues 118131
- 7. T-Left-Definite Problems 121134
- 8. Indefinite Problems and Complex Eigenvalues 122135
- 9. Comments 125138

- Part 3. Oscillation and Singular Existence Problems 129142
- Part 4. Singular Boundary Value Problems 161174
- Chapter 9. Two-Point Singular Boundary Value Problems 163176
- Chapter 10. Singular Self-Adjoint Problems 171184
- 1. Introduction 171184
- 2. The Lagrange Form 171184
- 3. The Minimal and Maximal Domains and Self-Adjoint Operators 173186
- 4. Operator Theory Characterization and Self-Adjoint Boundary Conditions 174187
- 5. The Friedrichs Extension 193206
- 6. Nonoscillatory Endpoints 194207
- 7. Oscillatory Endpoints 200213
- 8. Behavior of Eigenvalues near a Singular Boundary 201214
- 9. Approximating a Singular Problem with Regular Problems 204217
- 10. Green's Function 205218
- 11. Multiplicity of Eigenvalues 206219
- 12. Summary of Spectral Properties 208221
- 13. Comments 211224

- Chapter 11. Singular Indefinite Problems 215228
- Chapter 12. Singular Left-Definite Problems 229242
- 1. Introduction 229242
- 2. An Associated One Parameter Family of Right Definite Operators 232245
- 3. Existence of Eigenvalues 235248
- 4. Lemmas and Proofs 240253
- 5. LC Non-Oscillatory Problems 243256
- 6. Further Eigenvalue Properties in the LCNO Case 248261
- 7. Approximating a Singular Problem with Regular Problems 249262
- 8. Floquet Theory of Left-Definite Problems 257270
- 9. Comments 259272

- Part 5. Examples and other Topics 263276
- Bibliography 303316
- Index 327340