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.


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


Zbl 1034.34098
Full Text: DOI


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