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Controllability of localized quantum states on infinite graphs through bilinear control fields

In this work, we consider the bilinear Schrödinger equation $i\partial_tψ=-Δψ+u(t)Bψ$ in the Hilbert space $L^2(\mathcal{G},\mathbb{C})$ with $\mathcal{G}$ an infinite graph. The Laplacian $-Δ$ is equipped with self-adjoint boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We study the well-posedness in suitable subspaces of $D(|Δ|^{3/2})$ preserved by the dynamics despite the dispersive behaviour of the equation. In such spaces, we study the global exact controllability and the {\virgolette{energetic controllability}}. We provide examples involving for instance infinite tadpole graphs.