Controllability of the Schrödinger equation via adiabatic methods and conical intersections of the eigenvalues

We present a constructive method to control the bilinear Schrödinger equation by means of two or three controlled external fields. The method is based on adiabatic techniques and works if the spectrum of the Hamiltonian admits eigenvalue intersections, with respect to variations of the controls, and if the latter are conical. We provide sharp estimates of the relation between the error and the controllability time.