  This article deals with a special case of the Sturm-Liouville boundary value
problem (BVP), an eigenvalue problem characterized by the Sturm-Liouville
differential operator with unknown spectra and the associated eigenfunctions.
By examining the BVP in the Schr\"odinger form, we are interested in the
problem where the corresponding invariant function takes the form of a
reciprocal quadratic form. We call this BVP the modified second Paine-de
Hoog-Anderssen (PdHA) problem. We estimate the lowest-order eigenvalue without
solving the eigenvalue problem but by utilizing the localized landscape and
effective potential functions instead. While for particular combinations of
parameter values that the spectrum estimates exhibit a poor quality, the
outcomes are generally acceptable although they overestimate the numerical
computations. Qualitatively, the eigenvalue estimate is strikingly excellent,
and the proposal can be adopted to other BVPs.
