## Sturm-Liouville TheoryIn 1836-1837 Sturm and Liouville published a series of papers on second order linear ordinary differential operators, which started the subject now known as the Sturm-Liouville problem. In 1910 Hermann Weyl published an article which started the study of singular Sturm-Liouville problems. Since then, the Sturm-Liouville theory remains 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 knowledge about some aspects of this theory. To use the book, only a basic knowledge of advanced calculus and a rudimentary knowledge of Lebesgue integration and operator theory are assumed. An extensive list of references and examples is provided and numerous open problems are given. The list of examples includes those classical equations and functions associated with the names of Bessel, Fourier, Heun, Ince, Jacobi, Jorgens, Latzko, Legendre, Littlewood-McLeod, Mathieu, Meissner, Morse, as well as examples associated with the harmonic oscillator and the hydrogen atom. Many special functions of applied mathematics and mathematical physics occur in these examples. |

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### Contents

3 | |

Comments | 21 |

TwoPoint Regular Boundary Value Problems | 43 |

Regular SelfAdjoint Problems | 69 |

Canonical Forms of SelfAdjoint Boundary Conditions | 71 |

Existence of Eigenvalues | 72 |

Dependence of Eigenvalues on the Problem | 76 |

The Prüfer Transformation | 81 |

Comments | 169 |

Singular SelfAdjoint Problems | 171 |

The Minimal and Maximal Domains and SelfAdjoint Operators | 173 |

Operator Theory Characterization and SelfAdjoint Boundary Conditions | 174 |

The Friedrichs Extension | 193 |

Nonoscillatory Endpoints | 194 |

Oscillatory Endpoints | 200 |

Behavior of Eigenvalues near a Singular Boundary | 201 |

Separated Boundary Conditions | 86 |

Coupled Boundary Conditions | 90 |

An Elementary Existence Proof for Coupled Boundary Conditions | 91 |

Monotonicity of Eigenvalues | 95 |

Multiplicity of Eigenvalues | 96 |

Finite Real Spectrum | 98 |

Comments | 104 |

Regular LeftDefinite and Indefinite Problems | 107 |

Definition and Characterization of LeftDefinite Problems | 108 |

Existence of Eigenvalues | 112 |

Continuous Dependence of Eigenvalues on the Problem | 114 |

Eigenvalue Inequalities | 116 |

Differentiability of Eigenvalues | 118 |

TLeftDefinite Problems | 121 |

Indefinite Problems and Complex Eigenvalues | 122 |

Comments | 125 |

Oscillation and Singular Existence Problems | 129 |

Oscillation | 131 |

Oscillation Criteria | 133 |

Oscillatory Characterizations | 137 |

Comments | 140 |

The LimitPoint LimitCircle Dichotomy | 143 |

R LC LP O NO LCNO LCO | 145 |

LP and LC Conditions | 146 |

Comments | 151 |

Singular Initial Value Problems | 155 |

Factorization of Solutions near an LCNO Endpoint | 156 |

LimitCircle Initial Value Problems | 158 |

Comments | 159 |

Singular Boundary Value Problems | 161 |

TwoPoint Singular Boundary Value Problems | 163 |

Transcendental Characterization of Eigenvalues | 165 |

Greens Function | 167 |

Approximating a Singular Problem with Regular Problems | 204 |

Greens Function | 205 |

Multiplicity of Eigenvalues | 206 |

Summary of Spectral Properties | 208 |

Comments | 211 |

Singular Indefinite Problems | 215 |

Krein Spaces | 219 |

SelfAdjoint Operators in Krein Spaces | 220 |

A Construction of LeftDefinite Krein Spaces | 223 |

Proof of Theorems 11 1 1 and 11 1 3 | 225 |

Comments | 227 |

Singular LeftDefinite Problems | 229 |

An Associated One Parameter Family of Right Definite Operators | 232 |

Existence of Eigenvalues | 235 |

Lemmas and Proofs | 240 |

LC NonOscillatory Problems | 243 |

Further Eigenvalue Properties in the LCNO Case | 248 |

Approximating a Singular Problem with Regular Problems | 249 |

Floquet Theory of LeftDefinite Problems | 257 |

Comments | 259 |

Examples and other Topics | 263 |

Two Intervals | 265 |

Notation and Basic Assumptions | 266 |

Comments | 275 |

Examples | 277 |

Notation | 293 |

Comments on Some Topics not Covered | 295 |

Open Problems | 299 |

Bibliography | 303 |

25 | 304 |

327 | |

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### Common terms and phrases

adjoint Assume BC basis boundary value problems bounded changes sign Chapter characterization coefficients continuous function coupled boundary conditions deﬁned denote differential equations Dirichlet Dmax domain functions eigenfunction essential spectrum exists finite follows Friedrichs extension fundamental matrix given Green’s function Hence Hilbert space hold hypotheses and notation identically zero inequalities infinite number Kong Krein space LCNO LD problems Lemma Let the hypotheses limit-circle linear linearly independent LP endpoint Math Mathematics maximal domain functions minimal operator multiplicity Niessen and Zettl nonreal eigenvalues notation of Theorem Note number of eigenvalues number of zeros oscillation oscillatory parameter positive principal solution Proc quasi-derivatives real-valued regular problems Remark right-definite satisfying Section self-adjoint boundary conditions self-adjoint domain self-adjoint extensions self-adjoint operators self-adjoint realization separated boundary conditions singular problem SLEIGN2 SLP consisting Smax space H spectral theory Sturm-Liouville problems subinterval weight function Wu and Zettl