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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.

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