# Neumann Laplace eigenfunctions

Let $$u_k, u_m$$ be two Neumann Laplace eigenfunctions on a bounded domain $$\Omega \subset \mathbb{R}^n$$ with smooth boundary $$\partial \Omega$$, corresponding to eigenvalues $$\mu_k, \mu_m$$ respectively. My question is, can it also happen that their Dirichlet data also agree on $$\partial \Omega$$, that is, can we also have $$u_k|_{\partial \Omega} = u_m|_{\partial \Omega}$$ (not necessarily zero)? I am just learning these things, and it seems to me that this is too stringent a condition to be satisfied, but I cannot come up with a rigorous justification.