The modern theory of linear differential systems dates from the Levinson Theorem of 1948. It is only in more recent years, however, following the work of Harris and Lutz in 1974-7, that the significance and range of applications of the theorem have become appreciated. This book gives the first coherent account of the extensive developments of the last 15 years. The main results and techniques are clearly identified and earlier results, obtained originally by specialized methods in particular situations, are placed in a wider context. Some of the material is new, some is a re-presentation of existing theory in line with the main theme of the book, and much of it has appeared only recently in the research literature. By drawing together diverse materials in an organized, comprehensive account, the book should provide a stimulus to further research.
Author(s): M. S. P. Eastham
Series: London Mathematical Society Monographs New Series
Publisher: Oxford University Press, USA
Year: 1989
Language: English
Pages: 253
1. Asymptotically diagonal systems
1.1 Introduction
1.2 Notation and basic theory
1.3 The Levinson theorem
1.4 Proof of the Levinson theorem
1.5 The Hartman-Wintner theorem
1.6 A pointwise condition on R (x)
1.7 Conditions on higher derivatives of R (x)
1.8 Asymptotically constant systems
1.9 Higher-order differential equations
1.10 Coefficient matrices of Jordan type
1.11 Integral conditions with non-absolute convergence
Notes and references
2. Two-term differential equations
2.1 The second-order equation
2.2 The Liouville-Green asymptotic formulae
2.3 Repeated diagonalization
2.4 Extended Liouville-Green asymptotic formulae
2.5 The Liouville-Green transformation
2.6 Equations of Euler type
2.7 Subdominant coefficient q
2.8 Application of the Hartman- Wintner theorem
2.9 Higher-order equations
2.10 Higher-order equations of type
2.11 Subdominant coefficient q
Notes and references
3. Equations of self-adjoint type
3.1 Introduction
3.2 Eigenvalues of the same magnitude
3.3 Eigenvalues of differing magnitudes
3.4 Small eigenvalues
3.5 The fourth-order equation
3.6 Odd-order equations
3.7 Equations of generalized hypergeometric type
3.8 Integration of asymptotic formulae
3.9 Estimation of error terms
3.10 The deficiency index problem
3.11 Evaluation of deficiency indices
Notes and references
4. Resonance and non-resonance
4.1 Introduction
4.2 Perturbations of harmonic oscillation
4.3 Resonance and embedded eigenvalues
4.4 Two examples
4.5 Higher-order equations
4.6 Non-resonance for systems
4.7 The matrices A1' A2' A3
4.8 The form of P
4.9 Values of A in the resonant set
4.10 Resonance for M
4.11 Perturbations without periodicity
4.12 Systems with 1'\) == 0
4.13 Slowly decaying oscillatory coefficient
Notes and references
Bibliography
Index
