We will present sharp quantifications of the uncertainty principle for a Schrödinger operator with a potential function $V=V(x)$, which is assumed to be an analytic symbol of negative degree, hence not necessarily a short range perturbation. Our approach relies on spectral inequalities, adapted to the unbounded case using holomorphic extension, spectral projections and suitable Carleman estimates for the D-bar operator. These results are motivated from control theory.