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We consider a quantum particle in a 1D interval submitted to a potential. Theevolution of this particle is controlled using an external electric field.Taking into account the so-called polarizability term in the model (quadraticwith respect to the control), we prove global exact controllability in asuitable space for arbitrary potential and arbitrary dipole moment. This termis relevant both from the mathematical and physical points of view. The proofuses tools from the bilinear setting and a perturbation argument.

Global exact controllability of a 1D Schr\"odinger equations with a polarizability term