Partial Data for the Neumann-to-Dirichlet Map

We show that measurements of a Neumann-to-Dirichlet map, with either inputs or outputs restricted to part of the boundary, can determine an electric potential on that domain. Given a convexity condition on the domain, either the set on which measurements are taken, or the set on which input functions are supported, can be made to be arbitrarily small. The result is analogous to the result by Kenig, Sjöstrand, and Uhlmann for the Dirichlet-to-Neumann map. The main new ingredient in the proof is an improved Carleman estimate for the Schrödinger operator with appropriate boundary conditions. This is proved by Fourier analysis of a conjugated operator along the boundary of the domain.