The Schwarzian Approach in Sturm-Liouville Problems

TL;DR

This paper introduces a new method using the Schwarzian derivative to efficiently find eigenvalues in Sturm-Liouville problems, applicable across various physics fields, by transforming the problem into a nonlinear first-order differential equation.

Contribution

The paper presents a novel approach that simplifies Sturm-Liouville eigenvalue problems through a Schwarzian derivative-based transformation, enabling easier solution construction.

Findings

Method successfully finds eigenvalues in example problems

Applicable to quantum mechanics and fluid dynamics

Simplifies the solution process for Sturm-Liouville problems

Abstract

A novel method for finding the eigenvalues of a Sturm-Liouville problem is developed. Following the minimalist approach the problem is transformed to a single first-order differential equation with appropriate boundary conditions. Although the resulting equation is nonlinear, its form allows to find the general solution by adding a second part to a particular solution. This splitting of the general solution in two parts involves the Schwarzian derivative, hence the name of the approach. The eigenvalues that correspond to acceptable solutions asymptotically can be found by requiring the second part to correct the diverging behavior of the particular solution. The method can be applied to many different areas of physics, such as the Schr\"odinger equation in quantum mechanics and stability problems in fluid dynamics. Examples are presented.