Sturm-Liouville Theory

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American Mathematical Soc., Sep 23, 2010 - Differential equations - 328 pages
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In 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

First Order Systems
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
Index
327
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