On the non-existence of oscillation numbers in Sturm-Liouville theory

TL;DR

This paper proves a longstanding conjecture linking non-real eigenvalues in Sturm-Liouville problems to the absence of oscillation numbers in real eigenfunctions, extending classical oscillation theorems with new estimates.

Contribution

It establishes the equivalence between non-real eigenvalues and the non-existence of oscillation numbers, generalizing previous results and providing new bounds on oscillation indices.

Findings

Non-real eigenvalues imply no oscillation numbers for real eigenfunctions

Provides estimates on Haupt and Richardson indices and numbers

Extends Sturm oscillation theorem with new oscillation bounds

Abstract

We prove an old conjecture that relates the existence of non-real eigenvalues of Sturm-Liouville Dirichlet problems on a finite interval to the non-existence of oscillation numbers of its real eigenfunctions, [[6], p.104, Problems 3 and 5]. This extends to the general case, a previous result in [1], [2] where it was shown that the presence of even one pair of non-real eigenvalues implies the non-existence of a positive eigenfunction (or ground state). We also provide estimates on the Haupt and Richardson indices and Haupt and Richardson numbers thereby complementing the original Sturm oscillation theorem with the Haupt-Richardson oscillation theorem discovered over 100 years ago with estimates on the missing oscillation numbers of the real eigenfunctions observed.