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Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications. (English) Zbl 1191.35257
Summary: We prove that the Schrödinger equation is approximately controllable in Sobolev spaces $$H^s$$, $$s>0$$, generically with respect to the potential. We give two applications of this result. First, in the case of one space dimension, combining our result with a local exact controllability property, we get the global exact controllability of the system in higher Sobolev spaces. Then we prove that the Schrödinger equation with a potential which has a random time-dependent amplitude admits at most one stationary measure on the unit sphere $$S$$ in $$L^2$$.

MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 93B05 Controllability 58J05 Elliptic equations on manifolds, general theory 35J10 Schrödinger operator, Schrödinger equation
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