# Sturm's theorem on zeros of linear combinations of eigenfunctions

## TL;DR

This paper revisits Sturm's 1836 theorem on zeros of linear combinations of eigenfunctions, highlighting its historical context and potential relevance to modern spectral theory and Courant's nodal domain theorem.

## Contribution

It brings renewed attention to Sturm's classical theorem, emphasizing its historical significance and potential applications in current spectral analysis.

## Key findings

- Historical analysis of Sturm's theorem
- Discussion on relevance to Courant's nodal domain theorem
- Highlighting overlooked classical results

## Abstract

Motivated by recent questions about the extension of Courant's nodal domain theorem, we revisit a theorem published by C. Sturm in 1836, which deals with zeros of linear combination of eigenfunctions of Sturm-Liouville problems. Although well known in the nineteenth century, this theorem seems to have been ignored or forgotten by some of the specialists in spectral theory since the second half of the twentieth-century. Although not specialists in History of Sciences, we have tried to put these theorems into the context of nineteenth century mathematics.