Non-boundedness of the number of super level domains of eigenfunctions

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

1 IF - Institut Fourier
2 UdeM - Université de Montréal
3 LMJL - Laboratoire de Mathématiques Jean Leray

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. A related question is to estimate the number of connected components of the (super) level sets of a Neumann eigenfunction $u$. Indeed, in this case, the first eigenfunction is constant, and looking at the level sets of $u$ amounts to looking at the nodal sets $\{u-a=0\}$, where $a$ is a real constant. 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 labelling $k$, $4 \le k \le 6$, such that the set $\{u_k \not = 1\}$ has $(N+1)$ connected components.In the second part, we prove that there exists a metric $g$ on $\mathbb{T}^2$ (resp. on $\mathbb{S}^2$), which can be chosen arbitrarily close to the flat metric (resp. round metric), and an eigenfunction $u$ of the associated Laplace-Beltrami operator, such that the set $\{u \not = 1\}$ 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. 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: Eigenfunction; Nodal domain; Courant nodal domain theorem.

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 super level domains of eigenfunctions. Journal d'analyse mathématique, 2021, https://doi.org/10.1007/s11854-021-0189-9. ⟨10.1007/s11854-021-0189-9⟩. ⟨hal-02176094v2⟩