Non-Self-Adjoint Boundary Eigenvalue Problems
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Table of contents
- Cover
- Contentsvii
- Introductionxi
- Chapter I. Operator functions in Banach spaces1
- 1.1. Banach spaces2
- 1.2. Holomorphic vector valued functions6
- 1.3. The inverse of a Fredholm operator valued function9
- 1.4. Root functions of holomorphic operator functions13
- 1.5. Representation of the principal part of a finitely meromorphic oper- ator function18
- 1.6. Eigenvectors and associated vectors27
- 1.7. Semi-simple eigenvalues31
- 1.8. Local factorizations33
- 1.9. The completion of biorthogonal systems of root functions37
- 1.10. The operator function A + λB41
- 1.11. Abstract boundary eigenvalue operator functions46
- 1.12. Notes50
- Chapter II. First order systems of ordinary differential equations53
- 2.1. Sobolev spaces on intervals53
- 2.2. The dual of Wkp(a,b) for p <∞59
- 2.3. Multiplication in Sobolev spaces on the interval (a,b)65
- 2.4. Compact inclusion maps in Sobolev spaces on (a, b)67
- 2.5. Fundamental matrices69
- 2.6. Regularity of solutions of differential equations74
- 2.7. Estimates of integrals with a complex parameter76
- 2.8. Asymptotic matrices81
- 2.9. Notes99
- Chapter III. Boundary eigenvalue problems for first order systems101
- 3.1. The boundary eigenvalue problem102
- 3.2. The inhomogeneous boundary eigenvalue problem105
- 3.3. The adjoint boundary eigenvalue problem108
- 3.4. The adjoint boundary eigenvalue problem in parametrized form110
- 3.5. Two-point boundary eigenvalue problems in (Lp (a, b) )n119
- 3.6. Notes126
- Chapter IV. Birkhoff regular and Stone regular boundary eigenvalue problems129
- 4.1. Definitions and basic results130
- 4.2. Examples of Birkhoff regular problems139
- 4.3. Estimates of the characteristic determinant148
- 4.4. Estimates of the Green's matrix160
- 4.5. A special case of the Hilbert transform171
- 4.6. Improved estimates of the Green's matrix181
- 4.7. Uniform estimates of the Green's matrix189
- 4.8. Notes201
- Chapter V. Expansion theorems for regular boundary eigenvalue problems for first order systems203
- 5.1. First order systems which are linear in the eigenvalue parameter204
- 5.2. Birkhoff regular first order systems206
- 5.3. Expansion theorems for Birkhoff regular problems211
- 5.4. Examples for expansions in eigenfunctions and associated functions214
- 5.5. Stone regular boundary eigenvalue problems221
- 5.6. Expansion theorems for Stone regular problems232
- 5.7. Improved expansion theorems for Stone regular problems241
- 5.8. Notes247
- Chapter VI. n-th order differential equations249
- 6.1. Differential equations and systems250
- 6.2. Boundary conditions255
- 6.3. The boundary eigenvalue operator function257
- 6.4. The inverse of the boundary eigenvalue operator function260
- 6.5. The adjoint of the boundary eigenvalue problem262
- 6.6. The adjoint boundary eigenvalue problem in parametrized form263
- 6.7. Two-point boundary eigenvalue problems in Lp (a, b)271
- 6.8. Notes278
- Chapter VII. Regular boundary eigenvalue problems for n-th order equations279
- 7.1. General assumptions280
- 7.2. Asymptotic linearizations283
- 7.3. Birkhoff regular problems295
- 7.4. Expansion theorems for Birkhoff regular n-th order differential equa- tions297
- 7.5. An example for a Birkhoff regular problem with X-dependent bound- ary conditions300
- 7.6. Stone regular problems301
- 7.7. Boundary eigenvalue problems for ή + P1 ή p0 + p0ή =λ2 ή310
- 7.8. The Regge problem316
- 7.9. Almost Birkhoff regular problems318
- 7.1 0. Notes319
- Chapter VIII. The differential equation Kη=λHη321
- 8.1. The eigenvalue problem and general assumptions322
- 8.2. An asymptotic fundamental system for Kη =λHη326
- 8.3. The asymptotic fundamental system in the general case340
- 8.4. The inverse of the asymptotic fundamental matrix344
- 8.5. Almost Birkhoff regular boundary eigenvalue problems351
- 8.6. Estimates of the characteristic determinant357
- 8.7. Asymptotic estimates of the Green's function361
- 8.8. Expansion theorems373
- 8.9. The differential equation Kη(4) – α η=λη"376
- 8.10. The differential equation Kη(4) – Kη=λHη377
- 8.11. A boundary eigenvalue problem with associated functions at each eigenvalue381
- 8.1 2. Notes387
- Chapter IX. n-th order differential equations and n-fold expansions389
- 9.1. Shkalikov's linearization389
- 9.2. A first convergence result397
- 9.3. The expansion theorem404
- 9.4. Notes408
- Chapter X. Applications409
- 10.1. The clamped-free elastic bar409
- 10.2. Control of beams411
- 10.3. Control of one beam412
- 10.4. Control of multiple beams413
- 10.5. An example from meteorology417
- 10.6. The Orr-Sommerfeld equation428
- 10.7. A system of differential equations in the theory of viscous fluds429
- 10.8. Heat-conducting viscous fluid429
- 10.9. Motions of an incompressible magnetized plasma436
- Appendix A. Exponential sums441
- A.1. The convex hull of sums of complex numbers441
- A.2. Estimates of exponential sums450
- A.3. hnproved estimates for exponential sums473
- Bibliography475
- Notations497
- Index499
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.
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