On modified second Paine-de Hoog-Anderssen boundary value problem

TL;DR

This paper introduces a modified Sturm-Liouville boundary value problem, the PdHA problem, and proposes a method to estimate the lowest eigenvalue using landscape and potential functions without solving the eigenproblem.

Contribution

It presents a novel BVP formulation and a new eigenvalue estimation technique that avoids direct eigenproblem solutions, applicable to similar boundary value problems.

Findings

Eigenvalue estimates generally overestimate numerical results.

Qualitative eigenvalue estimates are highly accurate.

Method can be adapted to other boundary value problems.

Abstract

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.