Small-time approximate controllability of bilinear Schrödinger equations and diffeomorphisms

Abstract

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 $\phi \in L^2(M,\mathbb{R})$, to approximately reach the states $(\det DP{)}^{1/2}({\psi }_0\circ P)$ and $e^{i\phi }{\psi }_0$. We develop this approach for two examples of Schrödinger equations, posed on ${\mathbb{T}}^d$ and ${\mathbb{R}}^d$, 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, 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.