Discrete inverse problems for the Schrödinger operator on the multi-dimensional square lattice with partial Cauchy data

Motivated by the known results for an ongoing problem, in this paper we define the partial Cauchy data set for the discrete Schrödinger operator on a finite subset of the grid, and we prove a reconstruction theorem for the potential. Our methods are applicable in some general situations; we give a potential reconstruction theorem in the case of the hexagonal lattice as an example.