A Local--to--Global Propagation Principle for Dirichlet--to--Neumann Maps

We establish three local-to-global propagation results for Dirichlet--to--Neumann maps. First, in a general geometric setting, we show that if two smooth Riemannian metrics coincide in a collar neighborhood of a connected boundary component $\Gamma$, then equality of the corresponding local Dirichlet--to--Neumann maps on a nonempty open subset of $\Gamma$ propagates to equality of the associated global Dirichlet--to--Neumann maps on all of $\Gamma$. The proof combines unique continuation and self-adjointness arguments. Our second result replaces the geometric collar assumption by an exponential spectral assumption on the difference of the corresponding global Dirichlet--to--Neumann maps. The proof relies on the spectral unique continuation theory of Jerison--Lebeau, through the formulation of Le~Rousseau--Lebeau. Finally, we specialize to a particular class of conformally warped product metrics. In this setting, the local Borg--Marchenko theorem identifies the exponential spectral assumption with the coincidence of the metrics in a collar neighborhood of the boundary. Assuming in addition that the boundary is a compact Riemannian symmetric space, we show that this assumption can be substantially weakened by requiring only a suitable quasi--analytic boundary closeness of the conformal factors. The proof combines Weyl--Titchmarsh theory with the quasi--analytic propagation theorem of Ganguly and Thangavelu.