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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 '' +q(x)y=λρ(x)y,0xπ,y(0)=y(π)=0,

where q(x)0 has a second piecewise integrable derivative on [0,π], ρ(x) is the explosive factor defined as ρ(x)=1 on [0,a) with a<π, ρ(x)=-1 on (a,π]. 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.

34L10Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions (ODE)
34B24Sturm-Liouville theory