Control for Schrödinger operators on 2-tori: rough potentials

Abstract

For the Schrödinger equation, $(i{\partial }_t+\Delta )u=0$ on a torus, an arbitrary non-empty open set $\Omega$ provides control and observability of the solution: $\Vert u{|}_{t=0}{\Vert }_{L^2({\mathbb{T}}^2)}\le K_T\Vert u{\Vert }_{L^2([0,T]\times \Omega )}$ . We show that the same result remains true for $(i{\partial }_t+\Delta -V)u=0$ where $V\in L^2({\mathbb{T}}^2)$ , and ${\mathbb{T}}^2$ is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for $V\in C({\mathbb{T}}^2)$ and conjectured for $V\in L^{\infty }({\mathbb{T}}^2)$ . The higher dimensional generalization remains open for $V\in L^{\infty }({\mathbb{T}}^n)$