A new geometric viewpoint on Sturm-Liouville eigenvalue problems

TL;DR

This paper explores the relationship between Sturm-Liouville eigenvalue problems and geometry in normed planes, revealing connections to Hill equations and conditions for Euclidean geometry based on solutions.

Contribution

It introduces a geometric interpretation of Sturm-Liouville problems in normed planes, linking eigenvalues to properties of Minkowski and Euclidean geometries.

Findings

Eigenvalue λ=1 corresponds to a Hill equation related to the norm's geometry.

Solutions are trigonometric functions when the norm is defined by a Radon curve.

Certain eigenvalues induce reparametrizations that characterize Euclidean geometry.

Abstract

In Euclidean plane geometry, cycloids are curves which are homothetic to their respective bi-evolutes. In smooth normed planes, cycloids can be similarly defined, and they are characterized by their radius of curvature functions being solutions to eigenvalue problems of certain Sturm-Liouville equations. In this paper, we show that, for the eigenvalue {\lambda} = 1, this equation is a previously studied Hill equation which is closely related to the geometry given by the norm. We also investigate which geometric properties can be derived from this equation. Moreover, we prove that if the considered norm is defined by a Radon curve, then the solutions to the Hill equation are given by trigonometric functions. Further, we give conditions under which a given Hill equation induces a planar Minkowski geometry, and we prove that in this case the geometry is Euclidean if an eigenvalue other than {\lambda} = 1 induces a reparametrization of the original unit circle.