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In this paper we prove an approximate controllability result for the bilinearSchr\"odinger equation. This result requires less restrictive non-resonancehypotheses on the spectrum of the uncontrolled Schr\"odinger operator thanthose present in the literature. The control operator is not required to bebounded and we are able to extend the controllability result to the densitymatrices. The proof is based on fine controllability properties of the finitedimensional Galerkin approximations and allows to get estimates for the $L^{1}$norm of the control. The general controllability result is applied to theproblem of controlling the rotation of a bipolar rigid molecule confined on aplane by means of two orthogonal external fields.

communications in mathematical physics/communications in mathematical physics

A weak spectral condition for the controllability of the bilinear  Schr\"odinger equation with application to the control of a rotating planar  molecule