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Sturm-Liouville Theory
Anton Zettl, Northern Illinois University, DeKalb, IL
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Mathematical Surveys and Monographs
2005; 328 pp; softcover
Volume: 121
ISBN-10: 0-8218-5267-1
ISBN-13: 978-0-8218-5267-5
List Price: US$88
Member Price: US$70.40
Order Code: SURV/121.S
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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

"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

Part 1. Existence and uniqueness problems
  • First order systems
  • Scalar initial value problems
Part 2. Regular boundary value problems
  • Two-point regular boundary value problems
  • Regular self-adjoint problems
  • Regular left-definite and indefinite problems
Part 3. Oscillation and singular existence problems
  • Oscillation
  • The limit-point, limit-circle dichotomy
  • Singular initial value problems
Part 4. Singular boundary value problems
  • Two-point singular boundary value problems
  • Singular self-adjoint problems
  • Singular indefinite problems
  • Singular left-definite problems
Part 5. Examples and other topics
  • Two intervals
  • Examples
  • Notation
  • Comments on some topics not covered
  • Open problems
  • Bibliography
  • Index
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