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Multiparameter Eigenvalue Problems

Sturm-Liouville Theory

By F.V. Atkinson, Angelo B. Mingarelli

Published December 7th 2010 by CRC Press – 301 pages

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Description

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.

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

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 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

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

Author Bio

F.V. Atkinson was 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 published more than 100 papers on subjects ranging from the theory of the Riemann zeta function to operator theory. He earned his Ph.D. from the University of Oxford, under the guidance of E.C. Titchmarsh.

Angelo B. Mingarelli is a professor of mathematics at Carleton University. He previously taught at the Pennsylvania State University and the University of Ottawa. Dr. Mingarelli has been an NSERC University Research Fellow for many years and has won numerous awards for excellence in teaching. He earned his Ph.D. from the University of Toronto, under the supervision of F.V. Atkinson.

Name: Multiparameter Eigenvalue Problems: Sturm-Liouville Theory (Hardback) – CRC Press 
Description: By F.V. Atkinson, Angelo B. Mingarelli. 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...
Categories: Differential Equations, Computational Numerical Analysis, Mathematical Physics