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Boundary Value Problems and Symplectic Algebra for Ordinary Differential and Quasi-differential Operators
W. Norrie Everitt, University of Birmingham, UK, and Lawrence Markus, University of Minnesota, Minneapolis, MN
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Mathematical Surveys and Monographs
1999; 187 pp; hardcover
Volume: 61
ISBN-10: 0-8218-1080-4
ISBN-13: 978-0-8218-1080-4
List Price: US$60 Member Price: US$48
Order Code: SURV/61

In the classical theory of self-adjoint boundary value problems for linear ordinary differential operators there is a fundamental, but rather mysterious, interplay between the symmetric (conjugate) bilinear scalar product of the basic Hilbert space and the skew-symmetric boundary form of the associated differential expression. This book presents a new conceptual framework, leading to an effective structured method, for analyzing and classifying all such self-adjoint boundary conditions. The program is carried out by introducing innovative new mathematical structures which relate the Hilbert space to a complex symplectic space. This work offers the first systematic detailed treatment in the literature of these two topics: complex symplectic spaces--their geometry and linear algebra--and quasi-differential operators.

Features:

• Authoritative and systematic exposition of the classical theory for self-adjoint linear ordinary differential operators (including a review of all relevant topics in texts of Naimark, and Dunford and Schwartz).
• Introduction and development of new methods of complex symplectic linear algebra and geometry and of quasi-differential operators, offering the only extensive treatment of these topics in book form.
• New conceptual and structured methods for self-adjoint boundary value problems.
• Extensive and exhaustive tabulations of all existing kinds of self-adjoint boundary conditions for regular and for singular ordinary quasi-differential operators of all orders up through six.

Research mathematicians and graduate students interested in boundary value problems represented by self-adjoint differential operators, and symplectic linear algebra and geometry for real and complex vector spaces, with applications; mathematical physicists and engineers.

Reviews

"With this monograph Everitt and Markus have produced a major advance in our understanding of the structure of self-adjoint boundary conditions for regular and singular linear ordinary differential equations of arbitrary order $$n$$ and with arbitrary deficiency index $$d$$."

-- Mathematical Reviews, Featured Review

• Introduction: Fundamental algebraic and geometric concepts applied to the theory of self-adjoint boundary value problems
• Maximal and minimal operators for quasi-differential expressions, and GKN-theory
• Symplectic geometry and boundary value problems
• Regular boundary value problems
• Singular boundary value problems
• Appendix A. Constructions for quasi-differential operators
• Appendix B. Complexification of real symplectic spaces, and the real GKN-theorem for real operators
• References