Growth of Sobolev norms and controllability of Schr\"odinger equation

In this paper we obtain a stabilization result for the Schr"odinger equation under generic assumptions on the potential. Then we consider the Schr"odinger equation with a potential which has a random time-dependent amplitude. We show that if the distribution of the amplitude is sufficiently non-degenerate, then any trajectory of system is almost surely non-bounded in Sobolev spaces.