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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Spectral asymptotics for Sturm-Liouville equations with indefinite weight


Authors: Paul A. Binding, Patrick J. Browne and Bruce A. Watson
Journal: Trans. Amer. Math. Soc. 354 (2002), 4043-4065
MSC (2000): Primary 34L20, 34B09, 34B24; Secondary 47E05
Published electronically: May 22, 2002
MathSciNet review: 1926864
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Abstract: The Sturm-Liouville equation

\begin{displaymath}-(py')' + qy =\lambda ry \text{\rm on} [0,l]\end{displaymath}

is considered subject to the boundary conditions

\begin{displaymath}y(0)\cos\alpha = (py')(0)\sin\alpha,\end{displaymath}


\begin{displaymath}y(l)\cos\beta = (py')(l)\sin\beta.\end{displaymath}

We assume that $p$ is positive and that $pr$ is piecewise continuous and changes sign at its discontinuities. We give asymptotic approximations up to $O(1/\sqrt{n})$for $\sqrt{\lambda_n}$, or equivalently up to $O(\sqrt{n})$ for $\lambda_n$, the eigenvalues of the above boundary value problem.


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Additional Information

Paul A. Binding
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4
Email: binding@ucalgary.ca

Patrick J. Browne
Affiliation: Mathematical Sciences Group, Department of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5E6
Email: browne@snoopy.usask.ca

Bruce A. Watson
Affiliation: Department of Mathematics, University of the Witwatersrand, Private Bag 3, P O WITS 2050, South Africa
Email: watson-ba@e-math.ams.org

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03023-4
PII: S 0002-9947(02)03023-4
Keywords: Eigenvalue asymptotics, indefinite weight, turning point
Received by editor(s): January 12, 2002
Published electronically: May 22, 2002
Article copyright: © Copyright 2002 American Mathematical Society