**Mathematical Surveys and Monographs**

Volume: 73;
2000;
252 pp;
Hardcover

MSC: Primary 34; 47;

**Print ISBN: 978-0-8218-2049-0
Product Code: SURV/73**

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**Electronic ISBN: 978-1-4704-1300-2
Product Code: SURV/73.E**

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# Spectral Theory of Non-Self-Adjoint Two-Point Differential Operators

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*John Locker*

This monograph develops the spectral theory of an
\(n\)th order non-self-adjoint two-point differential operator
\(L\) in the Hilbert space \(L^2[0,1]\). The
mathematical foundation is laid in the first part, where the spectral
theory is developed for closed linear operators and Fredholm
operators. An important completeness theorem is established for the
Hilbert-Schmidt discrete operators. The operational calculus plays a
major role in this general theory.

In the second part, the spectral theory of the differential
operator \(L\) is developed by expressing \(L\) in the
form \(L = T + S\), where \(T\) is the principal part
determined by the \(n\)th order derivative and \(S\) is
the part determined by the lower-order derivatives. The spectral
theory of \(T\) is developed first using operator theory, and
then the spectral theory of \(L\) is developed by treating
\(L\) as a perturbation of \(T\). Regular and irregular
boundary values are allowed for \(T\), and regular boundary
values are considered for \(L\). Special features of the
spectral theory for \(L\) and \(T\) include the
following: calculation of the eigenvalues, algebraic multiplicities
and ascents; calculation of the associated family of projections which
project onto the generalized eigenspaces; completeness of the
generalized eigenfunctions; uniform bounds on the family of all finite
sums of the associated projections; and expansions of functions in
series of generalized eigenfunctions of \(L\) and \(T\).

#### Readership

Graduate students and research mathematicians interested in ordinary differential equations.

#### Reviews & Endorsements

An up-to-date account of the spectral theory of non-self-adjoint ordinary differential equations on a compact interval of the real line … This book is well written and is accessible to all who have a rudimentary knowledge of functional analysis. It is well suited both to graduate students working in two-point boundary value problems and to other scientists seeking further information concerning them.

-- Bulletin of the LMS

Detailed proofs are given; this, together with the introductory material of the first two chapters, should make the book accessible to a good graduate student with some background in functional analysis and differential equations. The monograph will probably appeal mainly to specialists.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Spectral Theory of Non-Self-Adjoint Two-Point Differential Operators

- Contents vii8 free
- Preface ix10 free
- Chapter 1. Unbounded Linear Operators 114 free
- Chapter 2. Fredholm Operators 4154
- Chapter 3. Introduction to the Spectral Theory of Differential Operators 8396
- Chapter 4. Principal Part of a Differential Operator 97110
- 1. The Principal Part T 97110
- 2. The Characteristic Determinant of T 98111
- 3. The Green's Function of λI T 103116
- 4. Alternate Representations 107120
- 5. The Boundary Values: Case n = 2v 110123
- 6. The Boundary Values: Case n = 2v – 1 118131
- 7. The Eigenvalues: Case n = 2v 128141
- 8. The Eigenvalues: Case n = 2v – 1 146159
- 9. Completeness of the Generalized Eigenfunctions 181194

- Chapter 5. Projections and Generalized Eigenfunction Expansions 193206
- Chapter 6. Spectral Theory for General Differential Operators 211224
- Bibliography 247260
- Index 249262