On Courant's nodal domain property for linear combinations of eigenfunctions, Part~{II}

Pierre Bérard (1) , Bernard Helffer (2)

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

This paper is about the extension of Courant's nodal domain theorem to sums of eigenfunctions. \\ The second section in Arnold's last published paper ``Topological properties of eigenoscillations in mathematical physics'' (Proceedings of the Steklov Institute of Mathematics 2011, vol. 273, pp. 25-34) is entitled \emph{Courant-Gelfand theorem}, and states 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. Arnold describes the idea of proof suggested by Gelfand, and concludes that ``the lack of a published formal text with a rigorous proof \dots is still distressing.''\\ In the first part of our paper, we discuss this \emph{Courant-Gelfand theorem}, which actually goes back to Sturm (1836), and give a complete proof following Gelfand's ideas. In the second part, we prove that the extension of Courant's nodal domain theorem to sums of eigenfunctions is false for the equilateral rhombus with Neumann boundary condition, and we explain how to construct counterexamples in higher dimension. This paper is a companion to arXiv:1705.03731, which contains several counterexamples to the extension of Courant's theorem to sums of eigenfunctions.

Mots clés: Courant nodal domain theorem.; Nodal domain; Eigenfunction

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

Pierre Bérard, Bernard Helffer. On Courant's nodal domain property for linear combinations of eigenfunctions, Part~{II}. 2018. ⟨hal-01718768v2⟩