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The eigenfunction expansion for a Dirichlet problem with explosive factor. (English) Zbl 1237.34143

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.

MSC:

34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34B24 Sturm-Liouville theory

Citations:

Zbl 1034.34098
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References:

[1] W. O. Amrein, A. M. Hinz, and D. P. Pearson, Sturm-Liouville Theory: Past and Present, Springer, New York, NY, USA, 2005.
[2] B. M. Levitan, Inverse Sturm-Liouville Problems, VNU Press, 1987. · Zbl 0749.34001
[3] Sh. Alemov, “On the Tikhonov work about the inverse problem for the Sturm-Liouville equation,” YMN, vol. 31, no. 7, pp. 84-88, 1976.
[4] Z. F. A. El-Raheem and A. H. Nasser, “On the spectral property of a Dirichlet problem with explosive factor,” Applied Mathematics and Computation, vol. 138, no. 2-3, pp. 355-374, 2003. · Zbl 1034.34098 · doi:10.1016/S0096-3003(02)00134-0
[5] M. G. Gasymov, A. Sh. Kakhramanov, and S. K. Petrosyan, “On the spectral theory of linear differential operators with discontinuous coefficients,” Akademiya Nauk Azerbaĭdzhanskoĭ SSR. Doklady, vol. 43, no. 3, pp. 13-16, 1987. · Zbl 0701.47029
[6] V. A. Marchenko, Sturm-Liouville Operators and Applications, AMS, 2011. · Zbl 1298.34001
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