Multiparameter Eigenvalue Problems: Sturm-liouville Theory by F.v. AtkinsonMultiparameter Eigenvalue Problems: Sturm-liouville Theory by F.v. Atkinson

Multiparameter Eigenvalue Problems: Sturm-liouville Theory

byF.v. Atkinson, Angelo B. MingarelliEditorF.v. Atkinson

Hardcover | July 12, 2010

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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 Theoryreflects 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.

F.V. Atkinsonwas a professor emeritus of mathematics at the University of Toronto. A Fellow of the Royal Society of Canada and an Honorary Fellow of the Royal Society of Edinburgh, Dr. Atkinson was awarded the Makdougall-Brisbane Prize of the Royal Society of Edinburgh for his enduring paper on limit-n criteria of integral type. He pub...
Title:Multiparameter Eigenvalue Problems: Sturm-liouville TheoryFormat:HardcoverDimensions:301 pages, 9.25 × 6.14 × 0.98 inPublished:July 12, 2010Publisher:Taylor and FrancisLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:1439816220

ISBN - 13:9781439816226


Table of Contents

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 Lam¿quation
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
Problems with boundary conditions
Associated partial differential equations
Generalizations and variations
The half-linear case
A mixed problem

Definiteness Conditions and the Spectrum
Eigenfunctions and multiplicity
Formal self-adjointness
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
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 casek= 5
Standard forms
Borderline cases
Metric variants on condition (A)

Oscillation Theorems
Oscillation numbers and eigenvalues
The generalized Pr¿fer 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

Slopes of eigencurves
The Klein oscillation theorem fork= 2
Asymptotic directions of eigencurves
The Richardson oscillation theorem fork= 2
Existence of asymptotes

Oscillation Properties for Other Multiparameter Systems
An example
Local definiteness
Sufficient conditions for local definiteness
Oscillation properties
The curve¿=f(¿,m)
The curve¿=g(¿, n)

Distribution of Eigenvalues
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
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
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 casek= 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
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
Spectral functions
Rate of growth of the spectral function
Limiting spectral functions
The full limit-circle case

Appendix on Sturmian Lemmas



Research problems and open questions appear at the end of each chapter.

Editorial Reviews

The book reads well and is accessible to everyone with a background in one-parameter Sturm-Liouville theory. The second author is successful in maintaining Atkinson¿s admirable style of writing.
¿Hans W. Volkmer, Mathematical Reviews, Issue 2011k