Galoisian approach for a Sturm-Liouville problem on the infinite interval

TL;DR

This paper applies differential Galois theory to analyze a Sturm-Liouville eigenvalue problem on an infinite interval, demonstrating integrability at discrete eigenvalues and illustrating with quantum and reaction-diffusion examples.

Contribution

It introduces a Galoisian framework for understanding the integrability of Sturm-Liouville problems on infinite domains, extending classical methods.

Findings

Eigenfunctions decay exponentially at infinity.

Differential equations are integrable at discrete eigenvalues.

Applications include Schrödinger and Allen-Cahn equations.

Abstract

We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval. Here the eigenfunctions are nonzero solutions exponentially decaying at infinity. We prove that at any discrete eigenvalue the differential equations are integrable in the setting of differential Galois theory under general assumptions. Our result is illustrated with two examples for a stationary Schroedinger equation having a generalized Hulthen potential and an eigenvalue problem for a traveling front in the Allen-Cahn equation.