Ordinary Differential Operators (Mathematical Surveys and Monographs)

In 1910 Herman Weyl published one of the most widely quoted papers of the 20th century in Analysis, which initiated the study of singular Sturm-Liouville problems. The work on the foundations of Quantum Mechanics in the 1920s and 1930s, including the proof of the spectral theorem for unbounded self-adjoint operators in Hilbert space by von Neumann and Stone, provided some of the motivation for the study of differential operators in Hilbert space with particular emphasis on self-adjoint operators and their spectrum. Since then the topic developed in several directions and many results and applications have been obtained. In this monograph the authors summarize some of these directions discussing self-adjoint, symmetric, and dissipative operators in Hilbert and Symplectic Geometry spaces. Part I of the book covers the theory of differential and quasi-differential expressions and equations, existence and uniqueness of solutions, continuous and differentiable dependence on initial data, adjoint expressions, the Lagrange Identity, minimal and maximal operators, etc. In Part II characterizations of the symmetric, self-adjoint, and dissipative boundary conditions are established. In particular, the authors prove the long standing Deficiency Index Conjecture. In Part III the symmetric and self-adjoint characterizations are extended to two-interval problems. These problems have solutions which have jump discontinuities in the interior of the underlying interval. These jumps may be infinite at singular interior points. Part IV is devoted to the construction of the regular Green's function. The construction presented differs from the usual one as found, for example, in the classical book by Coddington and Levinson. Cover 1 Title page 4 Preface 10 Part 1 . Differential Equations and Expressions 18 Chapter 1. First Order Systems 20 1. Introduction 20 2. Existence and Uniqueness of Solutions 20 3. Variation of Parameters 25 4. The Gronwall Inequality 25 5. Bounds and Extensions to the Endpoints 27 6. Continuous Dependence of Solutions on the Problem 29 7. Differentiable Dependence of Solutions on the Data 31 8. Adjoint Systems 37 9. Inverse Initial Value Problems 38 10. Comments 39 Chapter 2. Quasi-Differential Expressions and Equations 42 1. Introduction 42 2. Classical Symmetric Expressions 42 3. Quasi-Derivative Formulation of the Classical Expressions 43 4. General Quasi-Differential Expressions 44 5. Quasi-Differential Equations 46 6. Comments 47 Chapter 3. The Lagrange Identity and Maximal and Minimal Operators 48 1. Introduction 48 2. Adjoint and Symmetric Expressions 48 3. The Lagrange Identity 51 4. Maximal and Minimal Operators 52 5. Boundedness Below of the Minimal Operator 55 6. Comments 57 Chapter 4. Deficiency Indices 58 1. Introduction 58 2. The Deficiency Index Continued 62 3. Powers of Differential Expressions and Their Deficiency Index 64 4. Complex Parameter Decompositions of the Maximal Domain 69 5. Comments 74 Part 2 . Symmetric, Self-Adjoint, and Dissipative Operators 76 Chapter 5. Regular Symmetric Operators 78 1. Introduction 78 2. Boundary Conditions and Boundary Matrices 78 3. Characterization of Symmetric Domains 79 4. Examples of Symmetric Operators 83 5. Comments 85 Chapter 6. Singular Symmetric Operators 86 1. Introduction 86 2. Singular Boundary Conditions 86 3. Symmetric Domains and Proofs 87 4. Symmetric Domain Characterization with Maximal Domain Functions 92 5. Comments 96 Chapter 7. Self-Adjoint Operators 98 1. Introduction 98 2. LC Solutions and Real Parameter Decompositions of the Maximal Domain 99 3. A Real Parameter Characterization of Self-Adjoint Domains 106 4. The Maximal Deficiency Cases 108 5. Boundary Conditions for the Friedrichs Extension 110 6. Comments 113 Chapter 8. Self-Adjoint and Symmetric Boundary Conditions 114 1. Introduction 114 2. Separated Conditions 115 3. Separated, Coupled, and Mixed Conditions 118 4. Examples and Construction for All Types 122 5. Symmetric Boundary Conditions 129 6. Comments 134 Chapter 9. Solutions and Spectrum 136 1. Introduction 136 2. Only One Singular Endpoint 137 3. Two Singular Endpoints 141 4. Comments 144 Chapter 10. Coefficients, the Deficiency Index, Spectrum 146 1. Introduction 146 2. An Algorithm for the Construction of the Maximal Deficiency Index 147 3. Discreteness Conditions 148 4. Comments 152 Chapter 11. Dissipative Operators 154 1. Introduction 154 2. Concepts for Complex Symplectic Geometry Spaces 154 3. Finite Dimensional Complex Symplectic Spaces and Their Dissipative Subspaces 156 4. Applications of Symplectic Geometry to Ordinary Differential Operators 160 5. LC Representation of Dissipative Operators 166 6. Symplectic Geometry Characterization of Symmetric Operators 172 7. Comments 173 Part 3 . Two-Interval Problems 174 Chapter 12. Two-Interval Symmetric Domains 176 1. Introduction 176 2. Two-Interval Minimal and Maximal Operators 178 3. Two-Interval Symmetric Domains 179 4. Discontinuous Symmetric and Self-Adjoint Boundary Conditions 185 5. Examples 189 6. Comments 196 Chapter 13. Two-Interval Symmetric Domain Characterization with Maximal Domain Functions 198 1. Introduction and Main Theorem 198 2. Comments 203 Part 4 . Other Topics 204 Chapter 14. Green’s Function and Adjoint Problems 206 1. Introduction 206 2. Adjoint Matrices and Green’s Functions for Regular Systems 206 3. Green’s Functions of Regular Scalar Boundary Value Problems 210 4. Regularization of Singular Problems 213 5. Green’s Functions of Singular Boundary Value Problems 215 6. Construction of Adjoint and Self-Adjoint Boundary Conditions 217 7. The Green’s Function of the Legendre Equation 223 8. Comments 228 Chapter 15. Notation 230 Chapter 16. Topics Not Covered and Open Problems 232 1. Topics Not Covered 232 2. Open Problems 235 Bibliography 236 Index 266 Back Cover 269 The work on the foundations of Quantum Mechanics in the 1920s and 1930s provided some of the motivation for the study of differential operators in Hilbert space with particular emphasis on self-adjoint operators and their spectrum. Since then the topic has developed in several directions. This book summarizes some of these directions.