Ordinary Differential Operators
American Mathematical Society (Verlag)
978-1-4704-5366-4 (ISBN)
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.
Aiping Wang, North China Electric Power University, Beijing, China. Anton Zettl, Northern Illinois University, DeKalb, IL.
Differential equations and expressions: First order systems
Quasi-differential expressions and equations
The Lagrange identity and maximal and minimal operators
Deficiency indices
Symmetric, self-adjoint, and dissipative operators: Regular symmetric operators
Singular symmetric operators
Self-adjoint operators
Self-adjoint and symmetric boundary conditions
Solutions and spectrum
Coefficients, the deficiency index, spectrum
Dissipative operators
Two-interval problems: Two-interval symmetric domains
Two-interval symmetric domain characterization with maximal domain functions
Other topics: Green's function and adjoint problems
Notation
Topics not covered and open problems
Bibliography
Index.
| Erscheinungsdatum | 03.01.2020 |
|---|---|
| Reihe/Serie | Mathematical Surveys and Monographs |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 679 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| ISBN-10 | 1-4704-5366-5 / 1470453665 |
| ISBN-13 | 978-1-4704-5366-4 / 9781470453664 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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