**Mathematical Surveys and Monographs**

Volume: 61;
1999;
187 pp;
Hardcover

MSC: Primary 34; 58;
Secondary 11; 47

Print ISBN: 978-0-8218-1080-4

Product Code: SURV/61

List Price: $63.00

Individual Member Price: $50.40

**Electronic ISBN: 978-1-4704-1288-3
Product Code: SURV/61.E**

List Price: $63.00

Individual Member Price: $50.40

# Boundary Value Problems and Symplectic Algebra for Ordinary Differential and Quasi-differential Operators

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*W. Norrie Everitt; Lawrence Markus*

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.

#### Table of Contents

# Table of Contents

## Boundary Value Problems and Symplectic Algebra for Ordinary Differential and Quasi-differential Operators

- Contents vii8 free
- Preface ix10 free
- I. Introduction: Fundamental algebraic and geometric concepts applied to the theory of self- adjoint boundary value problems 114 free
- II. Maximal and minimal operators for quasi-differential expressions, and GKN-theory 1730
- III. Symplectic geometry and boundary value problems 2538
- IV. Regular boundary value problems 6578
- V. Singular boundary value problems 7992
- Appendix A. Constructions for quasi-differential operators 109122
- Appendix B. Complexification of real symplectic spaces, and the real GKN-theory 137150
- References 179192
- Notation Index 181194
- Index 185198

#### Readership

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