Non-Self-Adjoint Boundary Eigenvalue Problems

Reinhard Mennicken and Manfred Möller (Eds.)0444514473, 9780444514479, 9780080537733

This monograph provides a comprehensive treatment of expansion theorems for regular systems of first order differential equations and n -th order ordinary differential equations. In 10 chapters and one appendix, it provides a comprehensive treatment from abstract foundations to applications in physics and engineering. The focus is on non-self-adjoint problems. Bounded operators are associated to these problems, and Chapter 1 provides an in depth investigation of eigenfunctions and associated functions for bounded Fredholm valued operators in Banach spaces. Since every n -th order differential equation is equivalent to a first order system, the main techniques are developed for systems. Asymptotic fundamental systems are derived for a large class of systems of differential equations. Together with boundary conditions, which may depend polynomially on the eigenvalue parameter, this leads to the definition of Birkhoff and Stone regular eigenvalue problems. An effort is made to make the conditions relatively easy verifiable; this is illustrated with several applications in chapter 10. The contour integral method and estimates of the resolvent are used to prove expansion theorems. For Stone regular problems, not all functions are expandable, and again relatively easy verifiable conditions are given, in terms of auxiliary boundary conditions, for functions to be expandable. Chapter 10 deals exclusively with applications; in nine sections, various concrete problems such as the Orr-Sommerfeld equation, control of multiple beams, and an example from meteorology are investigated.
Key features: • Expansion Theorems for Ordinary Differential Equations • Discusses Applications to Problems from Physics and Engineering • Thorough Investigation of Asymptotic Fundamental Matrices and Systems • Provides a Comprehensive Treatment • Uses the Contour Integral Method • Represents the Problems as Bounded Operators • Investigates Canonical Systems of Eigen- and Associated Vectors for Operator Functions

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Table of contents :
Content:
Preface
Pages v-vi

Introduction
Pages xi-xviii

Chapter I Operator functions in Banach spaces Original Research Article
Pages 1-51

Chapter II First order systems of ordinary differential equations Original Research Article
Pages 53-100

Chapter III Boundary eigenvalue problems for first order systems Original Research Article
Pages 101-127

Chapter IV Birkhoff regular and stone regular boundary eigenvalue problems Original Research Article
Pages 129-201

Chapter V Expansion theorems for regular boundary eigenvalue problems for first order systems Original Research Article
Pages 203-248

Chapter VI n-th order differential equations Original Research Article
Pages 249-278

Chapter VII Regular boundary eigenvalue problems for n-th order equations Original Research Article
Pages 279-320

Chapter VIII The differential equation Kη=λHη Original Research Article
Pages 321-388

Chapter IX n-th order differential equations and n-fold expansions Original Research Article
Pages 389-408

Chapter X Applications Original Research Article
Pages 409-440

Appendix A Exponential sums
Pages 441-474

Bibliography
Pages 475-495

Notations
Pages 497-498

Index
Pages 499-500