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

N. Karjanto, P. Sadhani

Abstract

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 (Schrödinger) forms. While the conversion from canonical to Schrödinger 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 Schrödinger 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.

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

Abstract

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 (Schrödinger) forms. While the conversion from canonical to Schrödinger 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 Schrödinger 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.
Paper Structure (8 sections, 16 theorems, 161 equations)

This paper contains 8 sections, 16 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Sturm-Liouville boundary value problem
  3. Reciprocal quadratic invariant function
  4. Vanishing potential and constant density functions
  5. Constant potential and quadratic density functions
  6. Both nonzero constant potential and density functions
  7. Reciprocal linear function for $w\left(t\right)$
  8. Conclusion

Key Result

Lemma 1

When both $p$ and $r$ are constants but nonzero, the ODE EquSLP can be expressed in the Liouville normal form using the change of variable $t = \eta x$, where $\eta = \sqrt{r/p} \neq 0$:

Theorems & Definitions (34)

  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 1: Liouville's transformation liouville1837second
  • ...and 24 more