  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.
