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Zbl 1237.34143
El-Raheem, Zaki F.A.; Nasser, A.H.
The eigenfunction expansion for a Dirichlet problem with explosive factor.
(English)
[J] Abstr. Appl. Anal. 2011, Article ID 828176, 16 p. (2011). ISSN 1085-3375; ISSN 1687-0409/e

The authors consider the following Dirichlet eigenvalue problem for the Sturm-Liouville operator with explosive factor: $$-y^{\prime \prime} + q(x)y = \lambda \rho(x) y, ~0 \leq x \leq \pi, ~y(0) = y(\pi) = 0,$$ where $q(x) \geq 0$ has a second piecewise integrable derivative on $[0, \pi]$, $\rho(x)$ is the explosive factor defined as $\rho(x) = 1$ on $[0, a)$ with $a < \pi$, $\rho(x) = -1$ on $(a, \pi]$. By menas of the method of Green's function, they prove that the eigenfunction expansion formula is true both pointwise and in the $L^2$ norm.
[Chie-Ping Chu (Taipei)]
MSC 2000:
*34L10 Eigenfunction expansions, etc. (ODE)
34B24 Sturm-Liouville theory

Keywords: eigenfunction expansion; Dirichlet problem with explosive factor

Citations: Zbl 1034.34098

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