Inverse problems for a nonlinear dynamical Schrödinger operator with magnetic potential

Abstract

We study two inverse problems for a nonlinear dynamical Schrödinger equation with time-dependent magnetic and electric potentials. Under suitable analyticity assumptions, we show that the associated Dirichlet-to-Neumann map uniquely determines the linear magnetic potential and all coefficients of the nonlinear electric potential. We establish both full-data and partial-data uniqueness results. For the partial data problem, assuming that the coefficients are known in a neighborhood of the boundary, uniqueness is obtained using measurements made on arbitrarily small open subsets of the boundary. In addition, we establish the well-posedness of the forward problem.