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

In this work, we consider the bilinear Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2(\mathcal{G},\mathbb{C})$ with $\mathcal{G}$ an infinite graph. The Laplacian $-\Delta$ 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 $D(|\Delta|^{3/2})$ 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.