# Weakly regular Sturm-Liouville problems: a corrected spectral matrix method

## TL;DR

This paper introduces a spectral matrix method for solving weakly regular Sturm-Liouville eigenproblems with unbounded potentials, providing accurate eigenvalue approximations and validated through numerical experiments.

## Contribution

It proposes a novel Galerkin spectral matrix approach with error analysis and a correction formula for eigenvalues in weakly regular Sturm-Liouville problems.

## Key findings

- Convergence analysis confirms the method's accuracy.
- The correction formula improves eigenvalue estimates.
- Numerical experiments validate the effectiveness of the approach.

## Abstract

In this paper, we consider weakly regular Sturm-Liouville eigenproblems with unbounded potential at both endpoints of the domain. We propose a Galerkin spectral matrix method for its solution and we study the error in the eigenvalue approximations it provides. The result of the convergence analysis is then used to derive a low-cost and very effective formula for the computation of corrected numerical eigenvalues. Finally, we present and discuss the results of several numerical experiments which confirm the validity of the approach.