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

Given a compact manifold M with boundary of dimension n\geq 3 and any integers K and N, we show that there exists a metric on M for which the first K nonconstant eigenfunctions of the Dirichlet-to-Neumann map on \partial 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 (2017) and by Colbois, Girouard, Gordon and Sher (2024)