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THE EXISTENCE OF EIGENVALUES EMBEDDED IN THE CONTINUOUS SPECTRUM OF ORDINARY DIFFERENTIAL OPERATORS

IP.com Disclosure Number: IPCOM000128708D
Original Publication Date: 1976-Dec-31
Included in the Prior Art Database: 2005-Sep-16
Document File: 11 page(s) / 29K

Publishing Venue

Software Patent Institute

Related People

M. S. P. Eastham: AUTHOR [+4]

Abstract

In answer to two questions raised by W. N. Everitt, we show that, given p >.l and any countably infinite set of points on the positive X-axis, there is a [Equation ommitted] for which the set of points constitutes the point -continuous spectrum associated with the equation [Equation ommitted] and some homogeneous boundary condition at x = 0 AMS(MOS) Subject Classification - 34B25~ 47E05P 81. 34 Key Words - Ordinary differential operators, eigenvalues, continuous spectrum. Work Unit Number I - Applied Analysis Sponsored by the United States Army under Contract No. DAAG29-75-C-0024.

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Page 1 of 11

THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

THE EXISTENCE OF EIGENVALUES EMBEDDED IN THE CONTINUOUS SPECTRUM OF ORDINARY DIFFERENTIAL OPERATORS

M. S. P. Eastham and J. B. McLeod

Technical Summary Report #1688 October 197 6

ABSTRACT

In answer to two questions raised by W. N. Everitt, we show that, given p >.l and any countably infinite set of points on the positive X-axis, there is a

(Equation Omitted)

for which the set of points constitutes the point -continuous spectrum associated with the equation

(Equation Omitted)

and some homogeneous boundary condition at x = 0

AMS(MOS) Subject Classification - 34B25~ 47E05P 81. 34

Key Words - Ordinary differential operators, eigenvalues, continuous spectrum. Work Unit Number I - Applied Analysis

Sponsored by the United States Army under Contract No. DAAG29-75-C-0024.

1. Introduction

The purpose of this paper is to answer two questions raised by

W. N. Everitt concerning the spectrum associated with the differential

equation

(Equation Omitted)

and the boundary condition

(Equation Omitted)

where q(x) is real-valued and continuous and a is a real constant. We introduce the usual division Of T a into the point, continuous, and point-continuous spectrum:

(Equation Omitted)

University of Wisconsin Page 1 Dec 31, 1976

Page 2 of 11

THE EXISTENCE OF EIGENVALUES EMBEDDED IN THE CONTINUOUS SPECTRUM OF ORDINARY DIFFERENTIAL

OPERATORS

[1, section 43; 41, and since PC(r a is our concern in this paper, we

repeat the definition that PCu- consists of those eigenvalues in (r a a which are not isolated points of o- If q E L p (O,oo) for some p > 1, it is well-known that

(Equation Omitted)

[10, p. 112, Theorem 25] and that PCT a is empty if

(Equation Omitted)

references). Specific examples have been constructed which show that

Sponsored by the United States Army under Contract No. DAAG29-75-C-0024. PC(r is not necessarily empty when p > I and these are described later a in this section. The questions raised by Everitt [8, section 1, remark 71

are whether PCa- can contain a countably infinite number of points and a

whether PCa- is bounded above when p > I . We answer these questions a in the following theorem.

THEOREM. Let

(Equation Omitted)

(n be any sequence of positive real numbers. Then, given p> 1, there exists a real number a and a real-valued continuous function

(Equation Omitted)

such that

(Equation Omitted)

is precisely

(Equation Omitted)

Here {X n I is permitted to be either bounded or unbounded. Despite the doubts expressed in [8, section 1] about the applicability of the in-verse spectral theory of Gelfand and Levitan [9], we shall nevertheless use this theory in the simplified form of Levitan and Gasymov [11] to establish our theorem. We state the results that we need from [11] in section 2 below and we prove the theorem in section 3.

The first general result on the nature of PC(r was obtained by Wallach [14] as follows.

Let

(Equation Omitted)

University of Wisconsin Page 2 Dec 3...