Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2025-01-06T04:05:16.964Z Has data issue: false hasContentIssue false
  • English
  • Français

The asymptotic distribution of the eigenvalues of right definite multiparameter Sturm-Liouville systems

Published online by Cambridge University Press:  20 January 2009

Bryan P. Rynne
Bryan P. Rynne
Affiliation:
Department of MathematicsHeriot-Watt UniversityRiccarton Edinburgh EH14 4AS, Scotland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper studies the asymptotic distribution of the multiparameter eigenvalues of a right definite multiparameter Sturm–Liouville eigenvalue problem. A uniform asymptotic analysis of the oscillation number of solutions of a single Sturm–Liouville type equation with potential depending on a general parameter is given; these results are then applied to the system of multiparameter Sturm–Liouville equations to give the asymptotic eigenvalue distribution for the system as a function of a “multi-index” oscillation number.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 36 , Issue 1 , February 1993 , pp. 35 - 47
Copyright
Copyright © Edinburgh Mathematical Society 1993

References

REFERENCES

1.Ahlfors, L. V., Complex Analysis, 2nd Edition (McGraw-Hill, New York, 1966).Google Scholar
2.Binding, P., Multiparameter definiteness conditions, Proc. Roy. Soc. Edinburgh 89 A (1981), 319–322.CrossRefGoogle Scholar
3.Binding, P. and Browne, P. J., Comparison cones for multiparameter eigenvalue problems, J. Math. Anal. Appl. 77 (1980), 132–149.CrossRefGoogle Scholar
4.Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).Google Scholar
5.Dorodnicyn, A. A., Asymptotic laws of distribution of the characteristic values for certain special forms of differential equations of the second order, Amer. Math. Soc. Transl. (2) 16 (1960), 1–101.Google Scholar
6.Faierman, M., On the distribution of the eigenvalues of a two-parameter system of ordinary differential equations of the second order, SIAM J. Math. Anal. 8 (1977), 854–870.CrossRefGoogle Scholar
7.Faierman, M., Distribution of eigenvalues of a two-parameter system of differential equations, Trans. Amer. Math. Soc. 247 (1979), 45–86.CrossRefGoogle Scholar
8.Ince, E., Ordinary Differential Equations (Dover reprint, New York, 1956).Google Scholar
9.Kato, T., Perturbation Theory for Linear Operators, 2nd Edition (Springer-Verlag, New York, 1984).Google Scholar
10.Olver, F. W. J., Asymptotics and Special Functions (Academic Press, New York, 1974).Google Scholar
11.Schafke, R. and Volkmer, H., Bounds for the eigenfunctions of multiparameter Sturm-Liouville systems, Asymptotic Analysis 2 (1989), 139–159.CrossRefGoogle Scholar