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We prove that the Schr\"odinger equation is approximately controllable inSobolev spaces $H^s$, $s>0$ generically with respect to the potential. We givetwo applications of this result. First, in the case of one space dimension,combining our result with a local exact controllability property, we get theglobal exact controllability of the system in higher Sobolev spaces.  Then we prove that the Schr\"odinger equation with a potential which has arandom time-dependent amplitude admits at most one stationary measure on theunit sphere $S$ in $L^2 $.

Global approximate controllability for Schr\"odinger equation in higher Sobolev norms and applications