A corrected spectral method for Sturm–Liouville problems with unbounded potential at one endpoint

In this paper, we shall derive a spectral matrix method for the approximation of the eigenvalues of (weakly) regular and singular Sturm–Liouville problems in normal form with an unbounded potential at the left endpoint. The method is obtained by using a Galerkin approach with an approximation of the eigenfunctions given by suitable combinations of Legendre polynomials. We will study the errors in the eigenvalue estimates for problems with unsmooth eigenfunctions in proximity of the left endpoint. The results of this analysis will be then used conveniently to determine low-cost and effective procedures for the computation of corrected numerical eigenvalues. Finally, we shall present and discuss the results of several numerical experiments which confirm the effectiveness of the approach.