Inverse problems for a nonlinear dynamical Schr\"odinger operator with magnetic potential

We study two inverse problems for a nonlinear dynamical Schr\"odinger operator with magnetic and electric potentials. Under suitable analyticity assumptions, we show that the Dirichlet-to-Neumann map uniquely determines time-dependent magnetic and electric potentials. We establish the uniqueness of these potentials from both full data and partial data. In particular, for the partial data problem, the desired uniqueness is established by assuming that the potentials are known near the boundary, and the Neumann data is measured on arbitrarily small open subsets of the boundary. In addition, we establish the well-posedness of the forward problem, where we obtain the optimal Sobolev regularity for solutions.