A remark on the attainable set of the Schrödinger equation

We discuss the set of wavefunctions ψ[V] (t) that can be obtained from a given initial condition ψ_0 by applying the flow of the Schrödinger operator −∆ + V (t, x) and varying the potential V (t, x). We show that this set has empty interior, both as a subset of the sphere in L^2(R^d) and as a set of trajectories