Completeness of eigenfunctions of Sturm-Liouville problems with   discontinuities at three points
  In this work, we study discontinuous Sturm-Liouville type problems with eigenparameter dependent boundary condition and transmission conditions at three interior points. 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 eigenvalues of the problem are analytically simple, and the eigenfunctions of A are complete in H. 
Erdo\u{g}an \c{S}en
math.FA math.CA