Controllability of periodic bilinear quantum systems on infinite graphs

Abstract: In this work, we study the controllability of the bilinear Schr\"odinger equation on infinite graphs for periodic quantum states. We consider the bilinear Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2_p$ composed by functions defined on an infinite graph $\mathscr{G}$ verifying periodic boundary conditions on the infinite edges. The Laplacian $-\Delta$ is equipped with specific boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We present the well-posedness of the system in suitable subspaces of $D(|\Delta|^{3/2})$. In such spaces, we study the global exact controllability and we provide examples involving for instance tadpole graphs and star graphs with infinite spokes.