Accurate numerical bounds for the spectral points of singular Sturm–Liouville problems over 0<x<∞

The eigenvalues of singular Sturm–Liouville problems defined over the semi-infinite positive real axis are examined on a truncated interval 0<x<ℓ as functions of the boundary point ℓ. As a basic theoretical result, it is shown that the eigenvalues of the truncated interval problems satisfying Dirichlet and Neumann boundary conditions provide, respectively, upper and lower bounds to the eigenvalues of the original problem. Moreover, the unperturbed system in a perturbation problem, where ℓ remains sufficiently small, admits analytical solutions in terms of the Bessel functions of the first kind. Applications to the Schrödinger equations of diatomic molecules and a harmonic oscillator confirm the practical implementation of this approach in calculating highly accurate numerical eigenvalue enclosures. It is worth mentioning that this study is, therefore, a completion of the paper (J. Comput. Appl. Math. 115 (2000) 535) where similar problems on the whole real axis −∞<x<∞ were treated along the same lines.