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

TL;DR

This paper introduces a novel method for the approximate controllability of bilinear Schrödinger equations on manifolds, enabling control in arbitrarily small time without requiring a discrete spectrum, by leveraging diffeomorphisms and phase control.

Contribution

It develops a new approach controlling radial and angular parts separately using diffeomorphisms and phases, applicable in small time and without spectral restrictions.

Findings

Achieved small-time approximate controllability on manifolds.

Extended control techniques to flows of vector fields.

Demonstrated the method on Schrödinger equations on tori and Euclidean spaces.

Abstract

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 ${\rm Diff}_c^0(M)$.