Spectral Theory of Differential Operators: Self-Adjoint Differential Operators

by V.A. Il'in

Overview

In this fully-illustrated textbook, the author examines the spectral theory of self-adjoint elliptic operators. Chapters focus on the problems of convergence and summability of spectral decompositions about the fundamental functions of elliptic operators of the second order. The author's work offers a novel method for estimation of the remainder term of a spectral function and its Riesz means without recourse to the traditional Carleman technique and Tauberian theorem apparatus.

Product Details

ISBN-13:	9780306110375
Publisher:	Springer US
Publication date:	08/31/1995
Edition description:	1995
Pages:	390
Product dimensions:	7.01(w) x 10.00(h) x (d)

Table of Contents

1. Expansion in the Fundamental System of Functions of the Laplace Operator.- 1.1 Fundamental Systems of Functions and Their Properties.- 1.2 Fractional Kernels.- 1.3 Estimate for the Remainder Term of a Spectral Function in the Metric L2 and the Resulting Corollaries.- 1.4 Exact Conditions for the Localization and Uniform Convergence of Expansions with Respect to an Arbitrary FSF in the Sobolev-Liouville Classes.- 1.5 On the Potential Generalization of the Theory.- Comments on Chapter 1.- 2. Spectral Decompositions Corresponding to an Arbitrary Self-Adjoint Nonnegative Extension of the Laplace Operator.- 2.1 Self-Adjoint Nonnegative Extensions of Elliptic Operators. Ordered Spectral Representations of the Space L2. Classes of Differentiate Functions of N Variables.- 2.2 Formulation and Analysis of Main Results.- 2.3 Certain Properties of the Fundamental Functions of an Arbitrary Ordered Spectral Representation in the Space L2.- 2.4 Proof of Negative Theorem 2.1.- 2.5 Proof of Positive Theorem 2.3.- 2.6 Estimate for the Remainder Term of the Riesz Means of a Spectral Function in the Metric L2.- 2.7 Estimate for the Remainder Term of the Riesz Means of a Spectral Function in the Metric L2.- Comments on Chapter 2.- 3. On the Riesz Equisummability of Spectral Decompositions in the Classical and the Generalized Sense.- 3.1 On the Riesz Equisummability of Spectral Decompositions in the Classical Sense.- 3.2 On the Riesz Equisummability of Spectral Decompositions in the Generalized Sense.- Comments on Chapter 3.- 4. Self-Adjoint Nonnegative Extensions of an Elliptic Operator of Second Order.- 4.1 Ancillary Propositions about Fundamental Functions.- 4.2 Theorems of Negative Type.- 4.3 Theorems of Positive Type.- Comments on Chapter 4.- Appendix 1. Conditions for the Uniform Convergence of Multiple Trigonometric Fourier Series with Spherical Partial Sums.- Appendix 2. Conditions for the Uniform Convergence of Decompositions in Eigenfunctions of the First, Second, and Third Boundary-Value Problems for an Elliptic Operator of Second Order.- Epilogue.- References.

Editorial Reviews

Demonstrates the efficiency of the author's novel techniques based on the mean-value theorems for regular solutions of elliptic equations with a spectral parameter. Four chapters discuss functions of the Laplace operator and its arbitrary self-adjoint nonnegative extensions, the Reisz equisummability of spectral decompositions, and the technique's application to a general self-adjoint elliptic operator L of second order with smooth coefficients. Appendices deal with situations in which the function to be expanded is not compactly supported. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Booknews