Approximate controllability by adiabatic methods of the Schrödinger equation with nonlinear Hamiltonian

We consider quantum systems described by a controlled Schrödinger equation, where the Hamiltonian depends smoothly on three external inputs. We use adiabatic techniques to provide a constructive controllability result in some portion of the discrete spectrum, provided that it is conically connected. The case of Schrödinger Hamiltonians with electromagnetic fields fits the nonlinear setting studied in this paper.