Atkinson / Mingarelli Multiparameter Eigenvalue Problems

Sturm-Liouville Theory
1. Auflage 2011
ISBN: 978-1-4398-1623-3
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 301 Seiten

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.

Zielgruppe

Graduate students and researchers in differential equations; mathematicians and physicists.

Autoren/Hrsg.

Atkinson, F.V.

Mingarelli, Angelo B.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Algebra Zahlentheorie

Inhaltsverzeichnis

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

S

Über Autor(innen)

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.