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

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 $\mathbb{R}^2$.

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