Multiparameter eigenvalue problems: Sturm-Liouville theory

✍ Scribed by Atkinson F.V., Mingarelli A.B.

• Publisher: CRC
• Year: 2010
• Tongue: English
• Leaves: 291
• Category: Library

✦ Synopsis

One of the masters in the differential equations community, the late F.V. Atkinson contributed seminal research to multiparameter spectral theory and Sturm-Liouville theory. His ideas and techniques have long inspired researchers and continue to stimulate discussion. With the help of co-author Angelo B. Mingarelli, Multiparameter Eigenvalue Problems: Sturm-Liouville Theory reflects much of Dr. Atkinson’s final work. After covering standard multiparameter problems, the book investigates the conditions for eigenvalues to be real and form a discrete set. It gives results on the determinants of functions, presents oscillation methods for Sturm-Liouville systems and other multiparameter systems, and offers an alternative approach to multiparameter Sturm-Liouville problems in the case of two equations and two parameters. In addition to discussing the distribution of eigenvalues and infinite limit-points of the set of eigenvalues, the text focuses on proofs of the completeness of the eigenfunctions of a multiparameter Sturm-Liouville problem involving finite intervals. It also explores the limit-point, limit-circle classification as well as eigenfunction expansions. A lasting tribute to Dr. Atkinson’s contributions that spanned more than 40 years, this book covers the full multiparameter theory as applied to second-order linear equations. It considers the spectral theory of multiparameter problems in detail for both regular and singular cases.

✦ Table of Contents

Contents......Page 9
1.1 Main results of Sturm-Liouville theory......Page 16
1.2 General hypotheses for Sturm-Liouville theory......Page 17
1.3 Transformations of linear second-order equations......Page 19
1.5 The generalized Lamé equation......Page 21
1.6 Klein's problem of the ellipsoidal shell......Page 23
1.7 The theorem of Heine and Stieltjes......Page 24
1.8 The later work of Klein and others......Page 25
1.9 The Carmichael program......Page 26
1.10 Research problems and open questions......Page 28
2.1 The Sturm-Liouville case......Page 31
2.2 The diagonal and triangular cases......Page 33
2.3 Transformations of the parameters......Page 34
2.4 Finite difference equations......Page 35
2.5 Mixed column arrays......Page 37
2.6 The differential operator case......Page 39
2.7 Separability......Page 41
2.8 Problems with boundary conditions......Page 42
2.9 Associated partial differential equations......Page 43
2.10 Generalizations and variations......Page 44
2.11 The half-linear case......Page 46
2.12 A mixed problem......Page 47
2.13 Research problems and open questions......Page 48
3.1 Introduction......Page 51
3.2 Eigenfunctions and multiplicity......Page 54
3.3 Formal self-adjointness......Page 55
3.4 Definiteness......Page 56
3.5 Orthogonalities between eigenfunctions......Page 58
3.6 Discreteness properties of the spectrum......Page 60
3.7 A first definiteness condition, or "right-definiteness"......Page 61
3.8 A second definiteness condition, or "left-definiteness"......Page 63
3.9 Research problems and open questions......Page 65
4.1 Introduction......Page 67
4.2 Multilinear property......Page 69
4.3 Sign-properties of linear combinations......Page 70
4.4 The interpolatory conditions......Page 73
4.6 An alternative restriction......Page 75
4.7 A separation property......Page 79
4.8 Relation between the two main conditions......Page 81
4.9 A third condition......Page 84
4.10 Conditions (A), (C) in the case k = 5......Page 87
4.11 Standard forms......Page 90
4.12 Borderline cases......Page 92
4.13 Metric variants on condition (A)......Page 94
4.14 Research problems and open questions......Page 97
5.1 Introduction......Page 99
5.2 Oscillation numbers and eigenvalues......Page 100
5.3 The generalized Prüfer transformation......Page 101
5.4 A Jacobian property......Page 103
5.5 The Klein oscillation theorem......Page 104
5.6 Oscillations under condition (B), without condition (A)......Page 108
5.7 The Richardson oscillation theorem......Page 109
5.8 Unstandardized formulations......Page 113
5.9 A partial oscillation theorem......Page 114
5.10 Research problems and open questions......Page 117
6.1 Introduction......Page 119
6.2 Eigencurves......Page 121
6.3 Slopes of eigencurves......Page 123
6.4 The Klein oscillation theorem for k = 2......Page 124
6.5 Asymptotic directions of eigencurves......Page 125
6.6 The Richardson oscillation theorem for k = 2......Page 126
6.7 Existence of asymptotes......Page 128
6.8 Research problems and open questions......Page 130
7.1 Introduction......Page 131
7.2 An example......Page 132
7.3 Local definiteness......Page 133
7.5 Orthogonality......Page 134
7.6 Oscillation properties......Page 135
7.7 The curve μ = f(λ, m).......Page 136
7.8 The curve λ = g(μ, n)......Page 139
7.9 Research problems and open questions......Page 142
8.1 Introduction......Page 143
8.2 A lower order-bound for eigenvalues......Page 145
8.3 An upper order-bound under condition (A)......Page 146
8.4 An upper bound under condition (B)......Page 148
8.5 Exponent of convergence......Page 149
8.6 Approximate relations for eigenvalues......Page 150
8.7 Solubility of certain equations......Page 152
8.8 Research problems and open questions......Page 155
9.1 Introduction......Page 157
9.2 The essential spectrum......Page 158
9.3 Some subsidiary point-sets......Page 159
9.4 The essential spectrum under condition (A)......Page 161
9.5 The essential spectrum under condition (B)......Page 164
9.6 Dependence on the underlying intervals......Page 168
9.7 Nature of the essential spectrum......Page 169
9.8 Research problems and open questions......Page 170
10.1 Introduction......Page 171
10.2 Green's function......Page 172
10.3 Transition to a set of integral equations......Page 173
10.4 Orthogonality relations......Page 177
10.5 Discussion of the integral equations......Page 178
10.6 Completeness of eigenfunctions......Page 182
10.7 Completeness via partial differential equations......Page 186
10.8 Preliminaries on the case k = 2......Page 187
10.9 Decomposition of an eigensubspace......Page 189
10.10 Completeness via discrete approximations......Page 192
10.11 The one-parameter case......Page 193
10.12 The finite-difference approximation......Page 195
10.13 The multiparameter case......Page 197
10.14 Finite difference approximations......Page 199
10.15 Research problems and open questions......Page 205
11.1 Introduction......Page 206
11.2 Fundamentals of the Weyl theory......Page 208
11.3 Dependence on a single parameter......Page 212
11.4 Boundary conditions at infinity......Page 215
11.5 Linear combinations of functions......Page 217
11.6 A single equation with several parameters......Page 220
11.7 Several equations with several parameters......Page 222
11.8 More on positive linear combinations......Page 225
11.9 Further integrable-square properties......Page 229
11.10 Research problems and open questions......Page 231
12.1 Introduction......Page 232
12.2 Spectral functions......Page 233
12.3 Rate of growth of the spectral function......Page 235
12.4 Limiting spectral functions......Page 239
12.5 The full limit-circle case......Page 240
12.6 Research problems and open questions......Page 244
A.1 Introduction......Page 246
A.2 The oscillatory case, continuous f......Page 247
A.3 The Lipschitz case......Page 249
A.4 Oscillations in the differentiable case......Page 250
A.5 The Lebesgue integrable case......Page 251
A.6 The nonoscillatory case......Page 254
A.7 Research problems and open questions......Page 256
Bibliography......Page 258