Small-time approximate controllability of bilinear Schrödinger equations and diffeomorphisms

We consider Schrödinger PDEs, posed on a connected closed Riemannian manifold M or \mathbb{R}^{d} , with bilinear control. We propose a new method to prove the global L^{2} -approximate controllability. Contrary to previous ones, it works in arbitrarily small times and does not require a discrete spectrum. This approach consists in controlling the radial and the angular parts of the wavefunction separately, thanks to the control of the group \mathrm{Diff}_{c}^{0}(M) of (compactly supported and isotopic to the identity) diffeomorphisms of M and the control of phases. They refer to the capability, for any initial state \psi_{0}\in L^{2}(M,\mathbb{C}) , diffeomorphism P\in \mathrm{Diff}_{c}^{0}(M) and phase \varphi \in L^{2}(M,\mathbb{R}) , to approximately reach the states (\det DP)^{1/2}(\psi_{0}\circ P) and e^{i \varphi}\psi_{0} . We develop this approach for two examples of Schrödinger equations, posed on \mathbb{T}^{d} and \R^{d} , for which the small-time control of phases was recently proved. We prove that it implies the small-time control of flows of vector fields, with Lie bracket techniques. Combining this property with the simplicity of the group \mathrm{Diff}_{c}^{0}(M) proved by Thurston, we obtain the control of \mathrm{Diff}_{c}^{0}(M) . The small-time control of the radial part then follows from the transitivity of the group action of \mathrm{Diff}_{c}^{0}(M) on suitable densities, proved by Moser.