A remark on the attainable set of the Schr\"odinger equation

We discuss the set of wavefunctions ${\psi }_V(t)$ that can be obtained from a given initial condition ${\psi }_0$ by applying the flow of the Schr"odinger operator $-\mathrm{\Delta }+V(t,x)$ and varying the potential $V(t,x)$. We show that this set has empty interior, both as a subset of the sphere in $L^2({\mathbb{R}}^d)$ and as a set of trajectories.