Advances in Differential Equations

Decomposition of spectral asymptotics for Sturm-Liouville equations with a turning point

Paul A. Binding, Patrick J. Browne, and Bruce A. Watson

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Abstract

The Sturm-Liouville equation $-y'' + qy = \lambda ry,$ on $ [0,l],$ is considered subject to the boundary conditions \noindent $y(0)\cos\alpha(0) = y'(0)\sin\alpha(0),$ $y(l)\cos\alpha(l) = y'(l)\sin\alpha(l).$ We assume that $r$ is piecewise continuous with a variety of behaviours at $0, l$ and an interior turning point. We give asymptotic approximations for the eigenvalues $\lambda_n$ of the above boundary value problem in forms equivalent to $\lambda_n = an^2+bn+O(n^\tau)$, where $\tau < 1$.

Article information

Source
Adv. Differential Equations Volume 8, Number 4 (2003), 491-511.

Dates
First available in Project Euclid: 19 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355926851

Mathematical Reviews number (MathSciNet)
MR1972598

Zentralblatt MATH identifier
1039.34020

Subjects
Primary: 34L20: Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions
Secondary: 34B09: Boundary eigenvalue problems 34B24: Sturm-Liouville theory [See also 34Lxx]

Citation

Binding, Paul A.; Browne, Patrick J.; Watson, Bruce A. Decomposition of spectral asymptotics for Sturm-Liouville equations with a turning point. Adv. Differential Equations 8 (2003), no. 4, 491--511. https://projecteuclid.org/euclid.ade/1355926851.


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