Recovery of a Time-Dependent Hermitian Connection and Potential Appearing in the Dynamic Schrödinger Equation

We consider, on a trivial vector bundle over a Riemannian manifold with boundary, the inverse problem of uniquely recovering time- and space-dependent coefficients of the dynamic, vector-valued Schrödinger equation from the knowledge of the Dirichlet-to-Neumann (D-to-N) map. We show that the D-to-N map uniquely determines both the connection form and the potential appearing in the Schrödinger equation, under the assumption that the manifold is either (a) two-dimensional and simple or (b) of higher dimension with strictly convex boundary and admits a smooth, strictly convex function.