Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-31T16:51:23.879Z Has data issue: false hasContentIssue false
  • English
  • Français

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])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Atkinson, F. V.. Discrete and continuous boundary problems (London and New York: Academic Press, 1964).Google Scholar
2Bailey, P. B., Everitt, W. N., Weidmann, J. and Zettl, A.. Approximating the spectrum of singular Sturm—Liouville problems with eigenvalues of regular problems. Results Math. 23 (1993), 2–22.CrossRefGoogle Scholar
3Bailey, P. B., Everitt, W. N. and Zettl, A.. Computing eigenvalues of singular Sturm—Liouville problems. Results Math. 20 (1991), 391–423.CrossRefGoogle Scholar
4Bailey, P. B., Everitt, W. N. and Zettl, A.. Sleign2: a FORTRAN code for the numerical approximation of eigenvalues, eigenfunctions and continuous spectrum of Sturm–Liouville problems. (Available on the Web at ftp://ftp.math.niu.edu/pub/papers/Zettl/Sleign2.)Google Scholar
5Everitt, W. N., Möller, M. and Zettl, A.. Discontinous dependence of the n-th Sturm—Liouville eigenvalue. In General Inequalities 7: Proc. Int. Conf. on General Inequalities 7, Oberwolfach, 1995 (ed. Bandle, C., Everitt, W. N., Losonczi, L. and Walter, W.). International Series of Numerical Mathematics, vol. 123, pp. 147–150 (Basel: Birkhauser, 1997).Google Scholar
6Jörgens, K.. Spectral theory of second-order differential operators. Notes from lectures delivered at Aarhus Universitet, 1962/63. Matematisk Institut, Aarhus Universitet, Aarhus, Denmark, 1964.Google Scholar
7Möller, M. and Zettl, A.. Differentiable dependence of simple eigenvalues of operators in Banach spaces. J. Operator Theory 36 (1996), 335–355.Google Scholar
8Weidmann, J.. Spectral theory of ordinary differential operators. Lecture Notes in Mathematics, vol. 1258 (Heidelberg: Springer, 1987).CrossRefGoogle Scholar