Controllability of the discrete-spectrum Schrodinger equation driven by an external field

Thomas Chambrion (1, 2) , Paolo Mason (1, 2, 3, 4) , Mario Sigalotti (1, 2, 4) , Ugo Boscain (5, 6)

1 IECN - Institut Élie Cartan de Nancy
2 CORIDA - Robust control of infinite dimensional systems and applications
3 IAC
4 INRIA Lorraine
5 SISSA / ISAS - Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies
6 Le2i - Laboratoire Electronique, Informatique et Image [UMR6306]

We prove approximate controllability of the bilinear Schrödinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. The results that are obtained apply both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. The method relies on finite-dimensional techniques applied to the Galerkin approximations and permits, in addition, to get some controllability properties for the density matrix. Two examples are presented: the harmonic oscillator and the 3D well of potential, both controlled by suitable potentials.

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]; Physique Quantique [quant-ph]

Thomas Chambrion, Paolo Mason, Mario Sigalotti, Ugo Boscain. Controllability of the discrete-spectrum Schrodinger equation driven by an external field. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2009, 26 (1), pp.329-349. ⟨10.1016/j.anihpc.2008.05.001⟩. ⟨hal-00202218v6⟩