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

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.