Eigenvalue approximations for linear periodic differential equations with a singularity
We consider the second order, linear differential equation \begin{equation*}y''(x) + (\lambda.q(x)) y(x) = 0 \tag{1}\end{equation*} where $q$ is a real-valued, periodic function with period a. Our object in this paper is to derive asymptotic estimates for the eigenvalues of (1) on $[0;a]$...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
1999-01-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=16 |
| Summary: | We consider the second order, linear differential equation
\begin{equation*}y''(x) + (\lambda.q(x)) y(x) = 0 \tag{1}\end{equation*}
where $q$ is a real-valued, periodic function with period a. Our object in this paper is to derive asymptotic estimates for the eigenvalues of (1) on $[0;a]$ with periodic and semiperiodic boundary conditions. Our approach to regularizing (1) follows that used by Atkinson [1], Everitt and Race [4], and Harris and Race [6]. We illustrate our methods by calculating asymptotic estimates for the periodic and semiperiodic eigenvalues of (1) in the case where $q(x) = 1/|1-x|$. |
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| ISSN: | 1417-3875 1417-3875 |