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

Abstract

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.