Controllability of Localized Quantum States on Infinite Graphs through Bilinear Control Fields

International audienceIn this work, we consider the bilinear Schr\"odinger equation (BSE) $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2(\mathscr{G},\mathbb{C})$ with $\mathscr{G}$ 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 of the (BSE) 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 "energetic controllability". We provide examples involving for instance infinite tadpole graphs