The Schwarzian Approach in 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.