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Unraveling the complexity of inverting the Sturm-Liouville boundary value problem to its canonical form

N. KarjantoP. Sadhani
Feb 2024
The Sturm-Liouville boundary value problem (SLBVP) stands as a fundamental cornerstone in the realm of mathematical analysis and physical modeling. Also known as the Sturm-Liouville problem (SLP), this paper explores the intricacies of this classical problem, particularly the relationship between its canonical and Liouville normal (Schr\"odinger) forms. While the conversion from canonical to Schr\"odinger form using Liouville's transformation is well-known in the literature, the inverse transformation seems to be neglected. Our study attempts to fill this gap by investigating the inverse of Liouville's transformation, that is, given any SLP in the Schr\"odinger form with some invariant function, we seek the SLP in its canonical form. By examining closely the second Paine-de Hoog-Anderson (PdHA) problem, we argue that retrieving the SLP to its canonical form can be notoriously difficult and even impossible to achieve in its exact form. Finding the inverse relationship between the two independent variables seems to be the main obstacle. We confirm this claim by considering four different scenarios depending on the potential and density functions that appear in the corresponding invariant function. In the second PdHA problem, this invariant function takes a reciprocal quadratic binomial form. In some cases, the inverse Liouville's transformation produces an exact expression for the SLP in its canonical form. In other situations, however, while an exact canonical form is not possible to obtain, we have successfully derived the SLP in its canonical form asymptotically.
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