Multiparameter Eigenvalue Problems

Preliminaries and Early History Main results of Sturm-Liouville theory General hypotheses for Sturm-Liouville theory Transformations of linear second-order equations Regularization in an algebraic case The generalized Lame equation Klein's problem of the ellipsoidal shell The theorem of Heine and Stieltjes The later work of Klein and others The Carmichael program Some Typical Multiparameter Problems The Sturm-Liouville case The diagonal and triangular cases Transformations of the parameters Finite difference equations Mixed column arrays The differential operator case Separability Problems with boundary conditions Associated partial differential equations Generalizations and variations The half-linear case A mixed problem Definiteness Conditions and the Spectrum Introduction Eigenfunctions and multiplicity Formal self-adjointness Definiteness Orthogonalities between eigenfunctions Discreteness properties of the spectrum A first definiteness condition, or "right-definiteness" A second definiteness condition, or "left-definiteness" Determinants of Functions Introduction Multilinear property Sign-properties of linear combinations The interpolatory conditions Geometrical interpretation An alternative restriction A separation property Relation between the two main conditions A third condition Conditions (A), (C) in the case k = 5 Standard forms Borderline cases Metric variants on condition (A) Oscillation Theorems Introduction Oscillation numbers and eigenvalues The generalized Prufer transformation A Jacobian property The Klein oscillation theorem Oscillations under condition (B), without condition (A) The Richardson oscillation theorem Unstandardized formulations A partial oscillation theorem Eigencurves Introduction Eigencurves Slopes of eigencurves The Klein oscillation theorem for k = 2 Asymptotic directions of eigencurves The Richardson oscillation theorem for k = 2 Existence of asymptotes Oscillation Properties for Other Multiparameter Systems Introduction An example Local definiteness Sufficient conditions for local definiteness Orthogonality Oscillation properties The curve I = f(I",m) The curve I" = g(I , n) Distribution of Eigenvalues Introduction A lower order-bound for eigenvalues An upper order-bound under condition (A) An upper bound under condition (B) Exponent of convergence Approximate relations for eigenvalues Solubility of certain equations The Essential Spectrum Introduction The essential spectrum Some subsidiary point-sets The essential spectrum under condition (A) The essential spectrum under condition (B) Dependence on the underlying intervals Nature of the essential spectrum The Completeness of Eigenfunctions Introduction Green's function Transition to a set of integral equations Orthogonality relations Discussion of the integral equations Completeness of eigenfunctions Completeness via partial differential equations Preliminaries on the case k = 2 Decomposition of an eigensubspace Completeness via discrete approximations The one-parameter case The finite-difference approximation The multiparameter case Finite difference approximations Limit-Circle, Limit-Point Theory Introduction Fundamentals of the Weyl theory Dependence on a single parameter Boundary conditions at infinity Linear combinations of functions A single equation with several parameters Several equations with several parameters More on positive linear combinations Further integrable-square properties Spectral Functions Introduction Spectral functions Rate of growth of the spectral function Limiting spectral functions The full limit-circle case Appendix on Sturmian Lemmas Bibliography Index Research problems and open questions appear at the end of each chapter.