Sturm's theorem on the zeros of sums of eigenfunctions: Gelfand's strategy implemented

Pierre Bérard (1) , Bernard Helffer (2, 3)

1 IF - Institut Fourier
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay

In the second section ``Courant-Gelfand theorem'' of his last published paper (Topological properties of eigenoscillations in mathematical physics, Proc. Steklov Institute Math. 273 (2011) 25--34), Arnold recounts Gelfand's strategy to prove that the zeros of any linear combination of the $n$ first eigenfunctions of the Sturm-Liouville problem $$-\, y''(s) + q(x)\, y(x) = \lambda\, y(x) \mbox{ in } ]0,1[\,, \mbox{ with } y(0)=y(1)=0\,,$$divide the interval into at most $n$ connected components, and concludes that ``the lack of a published formal text with a rigorous proof \dots is still distressing.''\\Inspired by Quantum mechanics, Gelfand's strategy consists in replacing the ana\-lysis of linear combinations of the $n$ first eigenfunctions by that of their Slater determinant which is the first eigenfunction of the associated $n$-particle operator acting on Fermions.\\In the present paper, we implement Gelfand's strategy, and give a complete proof of the above assertion. As a matter of fact, refining Gelfand's strategy, we prove a stronger property taking the multiplicity of zeros into account, a result which actually goes back to Sturm (1836).

Mots clés: Zeros of eigenfunction; Nodal domain; Courant nodal domain theorem; Sturm theorem

Domaines: Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]; Physique mathématique [math-ph]

Pierre Bérard, Bernard Helffer. Sturm's theorem on the zeros of sums of eigenfunctions: Gelfand's strategy implemented. Moscow Mathematical Journal, 2020, 20 (1), pp.1--25. ⟨10.17323/1609-4514-2020-20-1-1-25⟩. ⟨hal-01832618v3⟩