Finite-difference proof of the completeness of eigenfunctions of the Sturm–Liouville operator in conservative form

A finite-difference method is used to prove the completeness of the eigenfunctions of the Sturm–Liouville operator in conservative form. The finite-difference schemes corresponding to the conservative Sturm–Liouville equation with various boundary conditions are shown to be self-adjoint. The accuracy and convergence of the method are analyzed, and the properties of eigenvalues and eigenvectors of the difference scheme approximating the differential equation and the boundary conditions are examined.