TRANSFORMATION OF STURM–LIOUVILLE PROBLEMS WITH DECREASING AFFINE BOUNDARY CONDITIONS

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...

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Bibliographic Details
Published in:Proceedings of the Edinburgh Mathematical Society 2004-10, Vol.47 (3), p.533-552
Main Authors: Binding, Paul A., Browne, Patrick J., Code, Warren J., Watson, Bruce A.
Format: Article
Language:English
Subjects:
Darboux transformation
eigenparameter-dependent boundary conditions
Sturm–Liouville
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Summary: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
ISSN:0013-0915
1464-3839
DOI:10.1017/S0013091504000197