The converse of Sturm's Separation Theorem

TL;DR

This paper demonstrates that Sturm's separation theorem, which describes zero interlacing of solutions to certain differential equations, fails when a unique turning point exists, and extends this to multiple turning points.

Contribution

It reveals the limitations of Sturm's theorem in the presence of turning points and extends the analysis to multiple such points.

Findings

Sturm's separation theorem fails with a single turning point.

The failure extends to equations with multiple turning points.

Related results on zero interlacing are discussed.

Abstract

We show that Sturm's classical separation theorem on the interlacing of the zeros of linearly independent solutions of real second order two-term ordinary differential equations necessarily fails in the presence of a unique turning point in the principal part of the equation. Related results are discussed. The last section contains an extension of the main result to a finite number of turning points.