Completeness of Eigenfunctions of Sturm-Liouville Problems with Transmission Conditions

In this paper, we investigate a class of Sturm-Liouville problems with eigenparameter-dependent boundary conditions and transmission conditions at an interior point. A self-adjoint linear operator $A$ is defined in a suitable Hilbert space $H $such that the eigenvalues of such a problem coincide with those of $A$. We show that the operator $A$ has only point spectrum, the eigenvalues of the problem are algebraically simple, and the eigenfunctions of $A$ are complete in $H$.

Keywords

Eigenparameter-dependent boundary conditions, transmission conditions, eigenvalues, eigenfunctions, completeness

2010 Mathematics Subject Classification

34L10, 47E05

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Published 1 January 2009