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# 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 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.

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The following ISBNs are associated with this title:

ISBN - 10:1439816220

ISBN - 13:9781439816226

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### Extra Content

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

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 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

**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

*'*=

*f(',m)*

The curve

*'*=

*g(', 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.*

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