Spectral Theory of Differential Operators

Self-Adjoint Differential Operators

verfasst von: V. A. Il’in

Verlag: Springer US

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Expansion in the Fundamental System of Functions of the Laplace Operator

Abstract

In this chapter, we introduce the concept of a fundamental system of functions (FSF) for the simplest elliptic operator, the Laplace operator, defined in an arbitrary (not necessarily bounded) N-dimensional domain. The FSF encompasses the eigenfunction systems of all self-adjoint boundary-value problems for the Laplace operator; for such systems, the spectrum is a pure point spectrum, admitting of an infinite multiplicity and every where dense set of limit points for the eigenvalues — quite a realistic situation, as we shall see later.

V. A. Il’in

Chapter 2. Spectral Decompositions Corresponding to an Arbitrary Self-Adjoint Nonnegative Extension of the Laplace Operator

Abstract

In this chapter we establish exact conditions for the convergence of the spectral decompositions corresponding to an arbitrary self-adjoint nonnegative extension of the Laplace operator in the domain G (not necessarily a bounded one) of the space E^N.

V. A. Il’in

Chapter 3. On the Riesz Equisummability of Spectral Decompositions in the Classical and the Generalized Sense

Abstract

In proving Theorem 2.3 (Section 2.2) we have established a uniform (on an arbitrary compact set of domain G) equiconvergence of the Riesz means of order s of two arbitrary self-adjoint nonnegative extensions of the Laplace operator in domain G for a finite-in-G function f(x) belonging to one of the four classes [Eq3] with order of differentiability α > (N- 1)/2-s, where 0 ≤ s < (N-1)/2 (and in the case of the Besov class for any θ ≥ 1). This fact ensures a uniform (on an arbitrary compact set K of domain G) tendency to zero of the difference of the Riesz means of order s of the spectral decompositions of this function which correspond to two arbitrary self-adjoint nonnegative extensions of the Laplace operator (in domain G, or in a domain to which G is interior).

V. A. Il’in

Chapter 4. Self-Adjoint Nonnegative Extensions of an Elliptic Operator of Second Order

Abstract

Our intention in this chapter is to show that the theorems on the exact conditions of uniform convergence and localization of spectral decompositions that have been established by us in Chapter 2 for an arbitrary self-adjoint nonnegative extension of the Laplace operator remain valid also for arbitrary self-adjoint nonnegative extensions of a general elliptic operator of second order L.

V. A. Il’in

Backmatter

Titel: Spectral Theory of Differential Operators
verfasst von: V. A. Il’in
Verlag: Springer US
Electronic ISBN: 978-1-4615-1755-9
Print ISBN: 978-0-306-11037-5
DOI: https://doi.org/10.1007/978-1-4615-1755-9

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Expansion in the Fundamental System of Functions of the Laplace Operator

Chapter 2. Spectral Decompositions Corresponding to an Arbitrary Self-Adjoint Nonnegative Extension of the Laplace Operator

Chapter 3. On the Riesz Equisummability of Spectral Decompositions in the Classical and the Generalized Sense

Chapter 4. Self-Adjoint Nonnegative Extensions of an Elliptic Operator of Second Order

Backmatter