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TRANSFORMATION OF STURM–LIOUVILLE PROBLEMS WITH DECREASING AFFINE BOUNDARY CONDITIONS

Published online by Cambridge University Press:  09 November 2004

Paul A. Binding ,
Patrick J. Browne ,
Warren J. Code  and
Bruce A. Watson
Paul A. Binding
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, Canada ([email protected])
Patrick J. Browne
Affiliation:
Mathematical Sciences Group, Department of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E6, Canada
Warren J. Code
Affiliation:
Mathematical Sciences Group, Department of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E6, Canada
Bruce A. Watson
Affiliation:
Department of Mathematics, University of the Witwatersrand, Private Bag 3, PO WITS 2050, South Africa
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Abstract

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We consider Sturm–Liouville boundary-value problems on the interval $[0,1]$ of the form $-y''+qy=\lambda y$ with boundary conditions $y'(0)\sin\alpha=y(0)\cos\alpha$ and $y'(1)=(a\lambda+b)y(1)$, where $a\lt0$. We show that via multiple Crum–Darboux transformations, this boundary-value problem can be transformed ‘almost’ isospectrally to a boundary-value problem of the same form, but with the boundary condition at $x=1$ replaced by $y'(1)\sin\beta=y(1)\cos\beta$, for some $\beta$.

AMS 2000 Mathematics subject classification: Primary 34B07; 47E05; 34L05

Keywords

Sturm–Liouvilleeigenparameter-dependent boundary conditionsDarboux transformation
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 47 , Issue 3 , October 2004 , pp. 533 - 552
Copyright
Copyright © Edinburgh Mathematical Society 2004