A Uniqueness Theorem for a Discrete Calderón Problem

The work in this thesis provides a proof of uniqueness of the potentials for which a discrete Calderón problem can be formulated. The Calderón problem will be defined with an associated Schrödinger equation and studied in a discrete setting. Specifically, the derivatives used to define the Schrödinger equation and the Dirichlet-to-Neumann map are determined using finite difference approximations, and the functions are discrete functions defined on a uniform grid in dimensions $n \geq 3.$ In the continuous setting, the uniqueness proof of the Calderón problem utilizes a Carleman estimate and a particular form of solutions known as Complex Geometrical Optics solutions, or CGO solutions (as presented in the well-known paper by J. Sylvester and G. Uhlmann). S. Ervedoza and F. de Gournay presented a discrete version of this Carleman estimate and a construction of discrete CGO solutions to the Schrödinger equation. This paper expands on the constructions in this previous work to define a specific set of CGO solutions. These particular constructions will then be used to complete the uniqueness theorem.