Sturm's theorem on zeros of linear combinations of eigenfunctions

Pierre Bérard (1) , Bernard Helffer (2)

1 IF - Institut Fourier
2 LMJL - Laboratoire de Mathématiques Jean Leray

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.

Mots clés: Sturm-Liouville eigenvalue problem; Sturm's theorems.

Domaines: Théorie spectrale [math.SP]; Analyse classique [math.CA]; Histoire et perspectives sur les mathématiques [math.HO]

Pierre Bérard, Bernard Helffer. Sturm's theorem on zeros of linear combinations of eigenfunctions. Expositiones Mathematicae, 2020, 38 (1), 27--50 / http://dx.doi.org/10.1016/j.exmath.2018.10.002. ⟨10.1016/j.exmath.2018.10.002⟩. ⟨hal-01541645v4⟩