We prove approximate controllability of the bilinear Schrödinger equation in the case in which the uncontrolled Hamiltonian has discrete nonresonant spectrum. The results that are obtained apply both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. The method relies on finite-dimensional techniques applied to the Galerkin approximations and permits, in addition, to get some controllability properties for the density matrix. Two examples are presented: the harmonic oscillator and the 3D well of potential, both controlled by suitable potentials.Résumé Nous montrons la contrôlabilité approchée de l'équation de Schrödinger bilinéaire dans le cas où l'hamiltonien non contrôlé a un spectre discret et nonrésonnant. Les résultats obtenus sont valables que le domaine soit borné ou non, et que le potentiel de contrôle soit borné ou non. La preuve repose sur des méthodes de dimension finie appliquées aux approximations de Galerkyn du système. Ces méthodes permettent en plus d'obtenir des résultats de contrôlabilité des matrices de densité. Deux exemples sont présentés, l'oscillateur harmonique et le puit de potentiel en dimension trois, munis de potentiels de contrôle adéquats.

In this paper we prove an approximate controllability result for the bilinear Schrödinger equation. This result requires less restrictive non-resonance hypotheses on the spectrum of the uncontrolled Schrödinger operator than those present in the literature. The control operator is not required to be bounded and we are able to extend the controllability result to the density matrices. The proof is based on fine controllability properties of the finite dimensional Galerkin approximations and allows to get estimates for the L 1 norm of the control. The general controllability result is applied to the problem of controlling the rotation of a bipolar rigid molecule confined on a plane by means of two orthogonal external fields.