Sign-Changing Points of Solutions of Homogeneous Sturm-Liouville Equations with Measure-Valued Coefficients

TL;DR

This paper studies the properties of sign-changing points of solutions to homogeneous Sturm-Liouville equations with measure-valued coefficients, establishing their isolation, and proving separation and comparison theorems.

Contribution

It introduces a framework for analyzing sign-changing points of solutions with measure-valued coefficients, extending classical Sturm theory.

Findings

Sign-changing points are isolated in the interval.

A Sturm-type separation theorem for solutions is established.

A Sturm-type comparison theorem for equations with different potentials is proved.

Abstract

In this paper we investigate sign-changing points of nontrivial real-valued solutions of homogeneous Sturm-Liouville differential equations of the form $-d(du/d\alpha)+ud\beta=0$, where $d\alpha$ is a positive Borel measure supported everywhere on $(a,b)$ and $d\beta$ is a locally finite real Borel measure on $(a,b)$. Since solutions for such equations are functions of locally bounded variation, sign-changing points are the natural generalization of zeros. We prove that sign-changing points for each nontrivial real-valued solution are isolated in $(a,b)$. We also prove a Sturm-type separation theorem for two nontrivial linearly independent solutions, and conclude the paper by proving a Sturm-type comparison theorem for two differential equations with distinct potentials.