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Indefinite Sturm–Liouville problems

Published online by Cambridge University Press:  12 July 2007

Q. Kong ,
H. Wu ,
A. Zettl  and
M. Möller
Q. Kong
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA
H. Wu
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA
A. Zettl
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA
M. Möller
Affiliation:
School of Mathematics, University of the Witwatersrand, WITS, 2050, South Africa
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Abstract

We study the spectrum of regular and singular Sturm–Liouville problems with real-valued coefficients and a weight function that changes sign. The self-adjoint boundary conditions may be regular or singular, separated or coupled. Sufficient conditions are found for (i) the spectrum to be real and unbounded below as well as above and (ii) the essential spectrum to be empty. Also found is an upper bound for the number of non-real eigenvalues. These results are achieved by studying the interplay between the indefinite problems (with weight function which changes sign) and the corresponding definite problems. Our approach relies heavily on operator theory of Krein space.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 133 , Issue 3 , June 2003 , pp. 639 - 652
Copyright
Copyright © Royal Society of Edinburgh 2003

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