Controllability of the rotation of a quantum planar molecule

We consider the simplest model for controlling the rotation of a molecule by the action of an electric field, namely a quantum planar pendulum. This problem consists in characterizing the controllability of a PDE (the Schrodinger equation) on a manifold with nontrivial topology (the circle S¹). The drift has discrete spectrum and its eigenfunctions are trigonometric functions. Some controllability results for the Schrodinger equation can be applied in this context. We tackle the problem by adapting the general method proposed by some of the authors in a recent paper. This requires, in particular, proving, by perturbation arguments, the non-resonance of the spectrum of the differential operator corresponding to a small constant control. The spectrum of this operator is given by the Mathieu characteristic values and its eigenfunctions are the Mathieu sinus and cosinus. Our main result says that we have simultaneous approximate controllability separately for the even and odd components of the wave function.