Multiparameter Eigenvalue Problems : Sturm-Liouville Theory

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. 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 With Special Attention To The Sturm-liouville Theory, This Book Discusses The Full Multiparameter Theory As Applied To Second-order Linear Equations. It Considers The Spectral Theory Of These Multiparameter Problems In Detail For Both The Regular And Singular Cases. The Text Covers Eignencurves, The Essential Spectrum, Eigenfunctions, Oscillation Theorems, The Distribution Of Eigencurves, The Limit Point, Limit Circle Theory, And More. This Text Is The Culmination Of More Than Two Decades Of Research By F.v. Atkinson, One Of The Masters In The Field, And His Successors, Who Continued His Work After He Passed Away In 2002-- 1. Preliminaries And Early History -- 2. Some Typical Multiparameter Problems -- 3. Definiteness Conditions And The Spectrum -- 4. Determinants Of Functions -- 5. Oscillation Theorems -- 6. Eigencurves -- 7. Oscillations For Other Multiparameter Systems -- 8. Distribution Of Eigenvalues -- 9. The Essential Spectrum -- 10. The Completeness Of Eigenfunctions -- 11. Limit-circle, Limit-point Theory -- 12. Spectral Functions. F.v. Atkinson, Angelo B. Mingarelli. Includes Bibliographical References (p. 245-278) And Index. "Based on the many approaches available for dealing with large-scale systems (LSS), this book provides a rigorous framework for studying the analysis, stability, and control problems of LSS while addressing the dominating sources of difficulties due to dimensionality, information structure constraints, uncertainties, and time delays. The author reviews past methods and results from a contemporary perspective, examines current trends and approaches, and offers future improvements to approaches. Providing an overall assessment of current LSS theories, he presents key concepts supported by proofs as well as efficient computational methods"-- Provided by publisher "With special attention to the Sturm-Liouville theory, this book discusses the full multiparameter theory as applied to second-order linear equations. It considers the spectral theory of these multiparameter problems in detail for both the regular and singular cases. The text covers eignencurves, the essential spectrum, eigenfunctions, oscillation theorems, the distribution of eigencurves, the limit point, limit circle theory, and more. This text is the culmination of more than two decades of research by F.V. Atkinson, one of the masters in the field, and his successors, who continued his work after he passed away in 2002"-- Provided by publisher Based on the many approaches available for dealing with large-scale systems (LSS), Decentralized Control and Filtering in Interconnected Dynamical Systems supplies a rigorous framework for studying the analysis, stability, and control problems of LSS. Providing an overall assessment of LSS theories, it addresses model order reduction, parametric uncertainties, time delays, and control estimator gain perturbations. Taking readers on a guided tour through LSS, the book examines recent trends and approaches and reviews past methods and results from a contemporary perspective. It traces the progre