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

Abstract

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 labeling k, 4 ≤ k ≤ 6, such that the set {u_k ≠ 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 ≠ 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 ℝ².