A new geometric view on Sturm-Liouville eigenvalue problems

V. Balestro, H. Martini and R. Teixeira, A new geometric view on Sturm-Liouville eigenvalue problems, to appear in Monatsh. Math., 2018. (arXiv link) (journal link)

Comments: In this paper we investigate further the relation of certain Sturm-Liouville equations with geometries given by norms in a plane. We prove that whenever the equation is related to a certain geometry, then any affinely equivalent geometry is associated to this same equation. We also show that, in the particular case of Radon planes, the solutions of the associated Sturm-Liouville eigenvalue problem are given by trigonometric functions (as in the Euclidean case). Further, we study the conditions under which a given Sturm-Liouville problem yields a norm, proving that this occurs precisely when the first positive eigenvalue is double and equals 1.