On the "movement" of the zeros of eigenfunctions of the Sturm-Liouville problem

TL;DR

This paper investigates how the zeros of Sturm-Liouville eigenfunctions change with boundary condition parameters, leading to a proof of the Sturm oscillation theorem that relates the number of zeros to the eigenfunction index.

Contribution

It provides a new analysis of zero movement in eigenfunctions and derives the Sturm oscillation theorem as a consequence.

Findings

Zeros of eigenfunctions depend continuously on boundary parameters

The n-th eigenfunction has exactly n zeros

The Sturm oscillation theorem is established

Abstract

We study the dependence of the zeros of eigenfunctions of Sturm-Liouville problem on the parameters that define the boundary conditions. As a corollary, we obtain Sturm oscillation theorem, which states that the $n$-th eigenfunction has $n$ zeros.