Control for Schr\"odinger equations on 2-tori: rough potentials

For the Schr\"odinger equation, $(i \partial_t + \Delta) u = 0$ on a torus, an arbitrary non-empty open set $\Omega$ provides control and observability of the solution: $\| u |_{t = 0} \|_{L^2 (\T^2)} \leq K_T \| u \|_{L^2 ([0,T] \times \Omega)}$. We show that the same result remains true for $(i \partial_t + \Delta - V) u = 0$ where $V \in L^2 (\T^2)$, and $\T^2$ is a (rational or irrational) torus. That extends the results of \cite{AM}, and \cite{BZ4} where the observability was proved for $V \in C (\T^2)$ and conjectured for $V \in L^\infty (\T^2)$. The higher dimensional generalization remains open for $V \in L^\infty (\T^n)$.