Multiparameter Eigenvalue Problems: Sturm-Liouville Theory

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 Lamé 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 Prüfer 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 μ = f(λ,m)
The curve λ = g(μ, 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