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 $-Δ+ V(t,x)$ 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.

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