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Growth of Sobolev norms and controllability of the Schrödinger equation. (English) Zbl 1180.93017
Summary: We obtain a stabilization result for both linear and nonlinear Schrödinger equations under generic assumptions on the potential. Then we consider the Schrödinger equations 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 the system is almost surely non-bounded in Sobolev spaces.

MSC:
93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
35Q40 PDEs in connection with quantum mechanics
93D21 Adaptive or robust stabilization
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