Simultaneous global exact controllability in projection of infinite 1D bilinear Schrödinger equations

The aim of this work is to study the controllability of infinite bilinear Schr\"odinger equations on a segment. In particular, 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 $T>0$. First, we show that simultaneously controlling infinite (BSE) by projecting onto suitable $N$ dimensional spaces is equivalent to the simultaneous controllability of $N$ equations (without projecting). Second, we prove the simultaneous local and global exact controllability of infinite bilinear Schrödinger equations in projection. The local controllability is guaranteed for any positive time and both the outcomes can be ensured for explicit $B$. In conclusion, we rephrase the results in terms of density matrices