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Multiparameter Sturm theory

Published online by Cambridge University Press:  14 November 2011

Paul Binding  and
Patrick J. Browne
Paul Binding
Affiliation:
Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, CanadaT2N 1N4
Patrick J. Browne
Affiliation:
Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, CanadaT2N 1N4
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Extract

Let Sturm–Liouville problems

with continuous coefficients and appropriate boundary conditions, be coupled by the eigenvalue λ = (λ1, … λk). When k = 1, there are various oscillation, perturbation and comparison theorems concerning existence and continuous or monotonic dependence of eigenvalues, eigenfunctions and their zeros (i.e. focal points).

We attempt a unified theory for such results, valid for general fc, under conditions known as "left" and “right” definiteness. A representative result may be stated loosely as follows: if LD holds then (elementwise) monotonic dependence of p, q and the matrix [ars] forces monotonic dependence of λ. LD is a generalisation of the “polar” case for k = 1, and was originally conceived for a quite different purpose, viz. completeness of eigenfunctions via elliptic partial differential equation theory.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 99 , Issue 1-2 , 1984 , pp. 173 - 184
Copyright
Copyright © Royal Society of Edinburgh 1984

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