Minimal time for the approximate bilinear control of Schrödinger equations

Karine Beauchard (1, 2) , Jean-Michel Coron (3, 4) , Holger Teismann (5)

1 ENS Rennes - École normale supérieure - Rennes
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
4 CaGE - Control And GEometry
5 Department of Mathematics & Statistics

We consider a quantum particle in a potential $V (x) (x ∈ R^N)$ in a time-dependent electric eld E(t) (the control). Boscain, Caponigro, Chambrion and Sigalotti proved in [2] that, under generic assumptions on V , this system is approximately controllable on the $L^2 (R^N$ , C)-sphere, in suciently large time T. In the present article we show that approximate controllability does not hold in arbitrarily small time, no matter what the initial state is. This generalizes our previous result for Gaussian initial conditions. Moreover, we prove that the minimal time can in fact be arbitrarily large.

Mots clés: Schrödinger equation; quantum control; minimal time

Domaines: Mathématiques [math]

Karine Beauchard, Jean-Michel Coron, Holger Teismann. Minimal time for the approximate bilinear control of Schrödinger equations. Mathematical Methods in the Applied Sciences, 2018, 41 (5), pp.1831-1844. ⟨10.1002/mma.4710⟩. ⟨hal-01333537⟩