Publisher:

Chapman & Hall/CRC

Number of Pages:

283

Price:

99.95

ISBN:

9781439816226

Date Received:

Monday, January 10, 2011

Reviewable:

No

Reviewer Email Address:

Publication Date:

2011

Format:

Hardcover

Audience:

Category:

Monograph

**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

The curve

**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**

Publish Book:

Modify Date:

Monday, January 10, 2011

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