Simultaneous global exact controllability in projection of infinite 1D bilinear Schr\"odinger equations

The aim of this work is to study the controllability of infinite bilinear Schr\"odinger equations on a segment. We consider the equations (BSE) $i\partial_t\psi^{j}=-\Delta\psi^j+u(t)B\psi^j$ in the Hilbert space $L^2((0,1),\mathbb{C})$ for every $j\in\mathbb{N}^*$. The Laplacian $-\Delta$ is equipped with Dirichlet homogeneous boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We prove the simultaneous local and global exact controllability of infinite (BSE) in projection. The local controllability is guaranteed for any positive time and we provide explicit examples of $B$ for which our theory is valid. In addition, we show that the controllability of infinite (BSE) in projection onto suitable finite dimensional spaces is equivalent to the controllability of a finite number of (BSE) (without projecting). In conclusion, we rephrase our controllability results in terms of density matrices.