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

## TL;DR

This paper introduces a spectral matrix method using Legendre polynomials for accurately approximating eigenvalues of Sturm-Liouville problems with unbounded potentials at one endpoint, including error analysis and numerical validation.

## Contribution

It develops a corrected spectral method tailored for Sturm-Liouville problems with unbounded potentials, enhancing eigenvalue approximation accuracy.

## Key findings

- The method effectively approximates eigenvalues near singularities.
- Error analysis guides the correction procedures.
- Numerical experiments confirm improved accuracy and efficiency.

## Abstract

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.