Discrete and Continuous Dirichlet-to-Neumann Maps in the Layered Case

Every sufficiently regular nonnegative function $\gamma$ (conductivity) on the closed unit disk $\overline{\mathbb{D}}$ induces the Dirichlet-to-Neumann map $\Lambda_{\gamma}$ on functions on $\partial\mathbb{D}$. The main inverse problems are to give a characterization of the maps $\Lambda_{\gamma}$ and to find out when $\Lambda_{\gamma}$ uniquely determines $\gamma$. In this paper we consider the case of conductivities that are constant on circles centered at the origin, and a discrete analogue of this so called i layered case. We characterize a closure of the set of the layered Dirichlet-to-Neumann maps in terms of their kernels and spectra. We give sharp conditions for the uniqueness in the discrete inverse problem, and conditions on $\gamma$ for the uniqueness in the continuous problem that we conjecture to be sharp. The characterization in terms of the spectra shows that continuous Dirichlet-to-Neumann maps can be viewed as limits of the discrete Dirichlet-to-Neumann maps. The characterization in te...