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The interlacing of eigenvalues for periodic multi-parameter problems*

Published online by Cambridge University Press:  14 November 2011

Patrick J. Browne
Patrick J. Browne
Affiliation:
University of Calgary, Calgary, Alberta, Canada
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Synopsis

This paper studies a linked system of second order ordinary differential equations

where xx ∈ [ar, br] and the coefficients qrars are continuous, real valued and periodic of period (br − ar), 1 ≤ r,s ≤ k. We assume the definiteness condition det{ars(xr)} > 0 and 2k possible multiparameter eigenvalue problems are then formulated according as periodic or semi-periodic boundary conditions are imposed on each of the equations of (*). The main result describes the interlacing of the 2k possible sets of eigentuples thus extending to the multiparameter case the well known theorem concerning 1-parameter periodic equation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 80 , Issue 3-4 , 1978 , pp. 357 - 362
Copyright
Copyright © Royal Society of Edinburgh 1978

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References

1Binding, P. A. and Browne, P. J. A Variational Approach to multiparameter Eigenvalue Problems in Hilbert Space. S.I.A.M. J. Math. Anal., to appear.Google Scholar
2Eastham, M. S. P.The Spectral Theory of Periodic Differential Equations (Edinburgh: Scottish Academic Press, 1973).Google Scholar
3Weinstein, A. and Stenger, W.Methods of Intermediate Problems for Eigenvalues (New York: Academic Press, 1972).Google Scholar