Multiparameter Eigenvalue Problems: Sturm-Liouville Theory

Contents......Page 10
1.1 Main results of Sturm-Liouville theory......Page 17
1.2 General hypotheses for Sturm-Liouville theory......Page 18
1.3 Transformations of linear second-order equations......Page 20
1.5 The generalized Lamé equation......Page 22
1.6 Klein's problem of the ellipsoidal shell......Page 24
1.7 The theorem of Heine and Stieltjes......Page 25
1.8 The later work of Klein and others......Page 26
1.9 The Carmichael program......Page 27
1.10 Research problems and open questions......Page 29
2.1 The Sturm-Liouville case......Page 32
2.2 The diagonal and triangular cases......Page 34
2.3 Transformations of the parameters......Page 35
2.4 Finite difference equations......Page 36
2.5 Mixed column arrays......Page 38
2.6 The differential operator case......Page 40
2.7 Separability......Page 42
2.8 Problems with boundary conditions......Page 43
2.9 Associated partial differential equations......Page 44
2.10 Generalizations and variations......Page 45
2.11 The half-linear case......Page 47
2.12 A mixed problem......Page 48
2.13 Research problems and open questions......Page 49
3.1 Introduction......Page 52
3.2 Eigenfunctions and multiplicity......Page 55
3.3 Formal self-adjointness......Page 56
3.4 Definiteness......Page 57
3.5 Orthogonalities between eigenfunctions......Page 59
3.6 Discreteness properties of the spectrum......Page 61
3.7 A first definiteness condition, or "right-definiteness"......Page 62
3.8 A second definiteness condition, or "left-definiteness"......Page 64
3.9 Research problems and open questions......Page 66
4.1 Introduction......Page 68
4.2 Multilinear property......Page 70
4.3 Sign-properties of linear combinations......Page 71
4.4 The interpolatory conditions......Page 74
4.6 An alternative restriction......Page 76
4.7 A separation property......Page 80
4.8 Relation between the two main conditions......Page 82
4.9 A third condition......Page 85
4.10 Conditions (A), (C) in the case k = 5......Page 88
4.11 Standard forms......Page 91
4.12 Borderline cases......Page 93
4.13 Metric variants on condition (A)......Page 95
4.14 Research problems and open questions......Page 98
5.1 Introduction......Page 100
5.2 Oscillation numbers and eigenvalues......Page 101
5.3 The generalized Prüfer transformation......Page 102
5.4 A Jacobian property......Page 104
5.5 The Klein oscillation theorem......Page 105
5.6 Oscillations under condition (B), without condition (A)......Page 109
5.7 The Richardson oscillation theorem......Page 110
5.8 Unstandardized formulations......Page 114
5.9 A partial oscillation theorem......Page 115
5.10 Research problems and open questions......Page 118
6.1 Introduction......Page 120
6.2 Eigencurves......Page 122
6.3 Slopes of eigencurves......Page 124
6.4 The Klein oscillation theorem for k = 2......Page 125
6.5 Asymptotic directions of eigencurves......Page 126
6.6 The Richardson oscillation theorem for k = 2......Page 127
6.7 Existence of asymptotes......Page 129
6.8 Research problems and open questions......Page 131
7.1 Introduction......Page 132
7.2 An example......Page 133
7.3 Local definiteness......Page 134
7.5 Orthogonality......Page 135
7.6 Oscillation properties......Page 136
7.7 The curve μ = f(λ, m).......Page 137
7.8 The curve λ = g(μ, n)......Page 140
7.9 Research problems and open questions......Page 143
8.1 Introduction......Page 144
8.2 A lower order-bound for eigenvalues......Page 146
8.3 An upper order-bound under condition (A)......Page 147
8.4 An upper bound under condition (B)......Page 149
8.5 Exponent of convergence......Page 150
8.6 Approximate relations for eigenvalues......Page 151
8.7 Solubility of certain equations......Page 153
8.8 Research problems and open questions......Page 156
9.1 Introduction......Page 158
9.2 The essential spectrum......Page 159
9.3 Some subsidiary point-sets......Page 160
9.4 The essential spectrum under condition (A)......Page 162
9.5 The essential spectrum under condition (B)......Page 165
9.6 Dependence on the underlying intervals......Page 169
9.7 Nature of the essential spectrum......Page 170
9.8 Research problems and open questions......Page 171
10.1 Introduction......Page 172
10.2 Green's function......Page 173
10.3 Transition to a set of integral equations......Page 174
10.4 Orthogonality relations......Page 178
10.5 Discussion of the integral equations......Page 179
10.6 Completeness of eigenfunctions......Page 183
10.7 Completeness via partial differential equations......Page 187
10.8 Preliminaries on the case k = 2......Page 188
10.9 Decomposition of an eigensubspace......Page 190
10.10 Completeness via discrete approximations......Page 193
10.11 The one-parameter case......Page 194
10.12 The finite-difference approximation......Page 196
10.13 The multiparameter case......Page 198
10.14 Finite difference approximations......Page 200
10.15 Research problems and open questions......Page 206
11.1 Introduction......Page 207
11.2 Fundamentals of the Weyl theory......Page 209
11.3 Dependence on a single parameter......Page 213
11.4 Boundary conditions at infinity......Page 216
11.5 Linear combinations of functions......Page 218
11.6 A single equation with several parameters......Page 221
11.7 Several equations with several parameters......Page 223
11.8 More on positive linear combinations......Page 226
11.9 Further integrable-square properties......Page 230
11.10 Research problems and open questions......Page 232
12.1 Introduction......Page 233
12.2 Spectral functions......Page 234
12.3 Rate of growth of the spectral function......Page 236
12.4 Limiting spectral functions......Page 240
12.5 The full limit-circle case......Page 241
12.6 Research problems and open questions......Page 245
A.1 Introduction......Page 247
A.2 The oscillatory case, continuous f......Page 248
A.3 The Lipschitz case......Page 250
A.4 Oscillations in the differentiable case......Page 251
A.5 The Lebesgue integrable case......Page 252
A.6 The nonoscillatory case......Page 255
A.7 Research problems and open questions......Page 257
Bibliography......Page 259