Small-time approximate controllability of bilinear Schr\"odinger equations and diffeomorphisms

We consider Schr\"odinger PDEs, posed on a boundaryless Riemannian manifold $M$, with bilinear control. We propose a new method to prove the global $L^2$-approximate controllability. Contrarily to previous ones, it works in arbitrarily small time and does not require a discrete spectrum. This approach consists in controlling separately the radial part and the angular part of the wavefunction thanks to the control of the group ${\rm Diff}_c^0(M)$ of diffeomorphisms of $M$ and the control of phases, which refer to the possibility, for any initial state $\psi_0\in L^2(M,\mathbb{C})$, diffeomorphism $P\in {\rm Diff}_c^0(M)$ and phase $\varphi \in L^2(M,\mathbb{R})$ to reach approximately the states $(\det DP)^{1/2}(\psi_0\circ P)$ and $e^{i \varphi}\psi_0$. The control of the radial part uses the transitivity of the group action of ${\rm Diff}_c^0(M)$ on positive densities proved by Moser. We develop this approach on two examples of Schr\"odinger equations, posed on $\mathbb{T}^d$ or $\mathbb{R}^d$, 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 thanks to Lie bracket techniques. Combining this property with the simplicity of the group ${\rm Diff}_c^0(M)$ proved by Thurston, we obtain the control of the group ${\rm Diff}_c^0(M)$.