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

We establish four local-to-global propagation results for Dirichlet–to–Neumann maps. Our first two results are proved in the general setting of smooth compact Riemannian manifolds with boundary. The first shows that if two smooth Riemannian metrics coincide in a collar neighborhood of a connected boundary component $\mathrm{\Gamma }$, then equality of the corresponding local Dirichlet–to–Neumann maps on a nonempty open subset of $\mathrm{\Gamma }$ propagates to equality of the associated global Dirichlet–to–Neumann maps on all of $\mathrm{\Gamma }$. The proof combines unique continuation and self-adjointness arguments. The second 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. Our third and fourth results establish local-to-global propagation principles under Ingham-type quasi–analytic spectral assumptions. Assuming that the boundary manifold is respectively a compact Riemannian symmetric space or a compact quasi–analytic Riemannian manifold, they rely on the propagation theorems of Ganguly–Thangavelu and of Bhowmik–Pradhan. As an application, we consider a class of conformally warped product metrics. In this setting, the local Borg–Marchenko theorem and Weyl–Titchmarsh theory relate the required Ingham-type spectral decay to a suitable quasi–analytic boundary closeness of the conformal factors, yielding new local-to-global uniqueness results for Dirichlet–to–Neumann maps.