Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator

Abstract

Given a compact manifold $\mathcal{M}$ with boundary of dimension $n\ge 3$ and any integers $K$ and $N$, we show that there exists a metric on $\mathcal{M}$ for which the first $K$ nonconstant eigenfunctions of the Dirichlet-to-Neumann map on $\partial \mathcal{M}$ have at least $N$ nodal components. This provides a negative answer to the question of whether the number of nodal domains of Dirichlet-to-Neumann eigenfunctions satisfies a Courant-type bound, which has been featured in recent surveys by Girouard and Polterovich [21, Open problem 9] and by Colbois, Girouard, Gordon and Sher [9, Open question 10.14].