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
Year: 1989
Language: English
Pages: 256
Tags: Математика;Дифференциальные уравнения;
Contents
1. Asymptotically diagonal systems 1
1.1 Introduction 1
1.2 Notation and basic theory 3
1.3 The Levinson theorem 8
1.4 Proof of the Levinson theorem 11
1.5 The Hartman-Wintner theorem 16
1.6 A pointwise condition on $R(x)$ 21
1.7 Conditions on higher derivatives of $R(x)$ 28
1.8 Asymptotically constant systems 34
1.9 Higher-order differential equations 36
1.10 Coefficient matrices of Jordan type 41
1.11 Integral conditions with non-absolute convergence 46
Notes and references 51
2. Two-term differential equations 55
2.1 The second-order equation 55
2.2 The Liouville-Green asymptotic formulae 57
2.3 Repeated diagonalization 60
2.4 Extended Liouville-Green asymptotic formulae 63
2.5 The Liouville-Green transformation 68
2.6 Equations of Euler type 74
2.7 Subdominant coefficient $q$ 77
2.8 Application of the Hartman-Wintner theorem 84
2.9 Higher-order equations 90
2.10 Higher-order equations of Euler type 93
2.11 Subdominant coefficient $q$ 95
Notes and references 101
3. Equations of self-adjoint type 105
3.1 Introduction 105
3.2 Eigenvalues of the same magnitude 108
3.3 Eigenvalues of differing magnitudes 113
3.4 Small eigenvalues 121
3.5 The fourth-order equation 126
3.6 Odd-order equations 134
3.7 Equations of generalized hypergeometric type 138
3.8 Integration of asymptotic formulae 143
3.9 Estimation of error terms 147
3.10 The deficiency index problem 152
3.11 Evaluation of deficiency indices 155
Notes and references 162
4. Resonance and non-resonance 166
4.1 Introduction 166
4.2 Perturbations of harmonic oscillation 170
4.3 Resonance and embedded eigenvalues 174
4.4 Two examples 178
4.5 Higher-order equations 180
4.6 Non-resonance for systems 183
4.7 The matrices $\Lambda_1$, $\Lambda_2$, $\Lambda_3$ 188
4.8 The form of $P$ 196
4.9 Values of $\lambda$ in the resonant set 199
4.10 Resonance for $M\geq2$ 203
4.11 Perturbations without periodicity 211
4.12 Systems with $\Lamnda_0=0$ 217
4.13 Slowly decaying oscillatory coefficient 221
Notes and references 226
Bibliography 231
Index 241
