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Sturm—Liouville problems and discontinuous eigenvalues

Published online by Cambridge University Press:  14 November 2011

W. N. Everitt ,
M. Möller  and
A. Zettl
W. N. Everitt
Affiliation:
School of Mathematics and Statistics, University of Birmingham, Egdbaston, Birmingham B15 2TT, UK ([email protected])
M. Möller
Affiliation:
Department of Mathematics, University of the Witwatersrand, Wits 2050, Republic of South Africa ([email protected])
A. Zettl
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115-2888, USA ([email protected])
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Extract

If a Sturm—Liouville problem is given in an open interval of the real line, then regular boundary value problems can be considered on compact sub-intervals. For these regular problems, all with necessarily discrete spectra, the eigenvalues depend on both the end-points of the compact intervals, and upon the choice of the real separated boundary conditions at these end-points. These eigenvalues are not, in general, continuous functionsof the end-points and boundary conditions. The paper shows the surprising form of these discontinuities. The results have applications to the approximations of singular Sturm—Liouville problems by regular problems, and to the theoretical aspects of the Sleign2 Computer program.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 4 , 1999 , pp. 707 - 716
Copyright
Copyright © Royal Society of Edinburgh 1999

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