Corrected finite difference eigenvalues of periodic Sturm–Liouville problems

Computation of eigenvalues of regular Sturm–Liouville problems with periodic boundary conditions is considered. We show that a proof similar to that given by Andrew (1989) can be used to prove that a correction technique applied to a finite difference scheme given by Vanden Berghe et al. (1995) reduces the error in the kth eigenvalue estimate from O(k⁴h²) to O(kh²), where h is the uniform mesh length. We also provide a significantly shorter proof of a slightly weaker result.