  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 $-\Delta + V(t,x)$ 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.
