On a converse of Sturm's comparison theorem

TL;DR

This paper investigates the limitations of Sturm's comparison theorem, demonstrating that its classical conditions are nearly optimal and providing criteria for its failure under certain coefficient violations.

Contribution

It establishes necessary and sufficient conditions for the failure of Sturm's comparison theorem and explores the boundaries of its applicability.

Findings

Conditions for failure of SCT are characterized precisely.

The classical conditions on coefficients are nearly optimal.

The converse of SCT does not generally hold.

Abstract

We show that Sturm's classical comparison theorem (SCT) on the interlacing of zeros of solutions of pairs of real second order two-term ordinary differential equations necessarily fails if the usual Sturmian-type conditions on the coefficients are violated. We also show that the conditions on the coefficients are, in some sense, best possible. We give a necessary and sufficient condition for the failure of SCT and we note that its converse is false, and that the comparison theorem may still hold with very mild hypotheses on the coefficients.