Non-boundedness of the number of nodal domains of a sum of eigenfunctions

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

1 IF - Institut Fourier
2 UdeM - Université de Montréal
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay

Generalizing Courant's nodal domain theorem, the ``Extended Courant property'' is the statement that a linear combination of the first $n$ eigenfunctions has at most $n$ nodal domains. In the first part of the paper, we prove that the Extended Courant property is false for the subequilateral triangle and for regular $N$-gons ($N$ large), with the Neumann boundary condition. More precisely, we prove that there exists a Neumann eigenfunction $u_k$ of the $N$-gon, with index $4 \le k \le 6$, such that the set $\{u_k \not = 1\}$ has $(N+1)$ connected components. In the second part, we prove that there exist metrics $g$ on $\mathbb{T}^2$ (resp. on $\mathbb{S}^2$) which are arbitrarily close to the flat metric (resp. round metric), and an eigenfunction $f$ of the associated Laplace-Beltrami operator such that the set $\{ f \not = 1 \}$ has infinitely many connected components. In particular the Extended Courant property is false for these closed surfaces. These results are strongly motivated by a recent paper by Buhovsky, Logunov and Sodin (arXiv:1811.03835). As for the positive direction, in Appendix B, we prove that the Extended Courant property is true for the isotropic quantum harmonic oscillator in $\mathbb{R}^2$.

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

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

Pierre Bérard, Philippe Charron, Bernard Helffer. Non-boundedness of the number of nodal domains of a sum of eigenfunctions. 2019. ⟨hal-02176094v1⟩