[Contrôle exact global d'équations de Schrödinger 1d avec un terme de polarisabilité]
On considère une particule quantique dans un intervalle 1d, soumise à un potentiel. L'évolution de cette particule est contrôlée par un champ électrique extérieur. En prenant en compte dans le modèle le terme dit de polarisabilité (quadratique par rapport au contrôle), on prouve la contrôlabilité exacte globale dans un espace approprié pour des potentiels et des moments dipolaires arbitraires. Ce terme est intéressant à la fois d'un point de vue mathématique et physique. La preuve utilise des outils issus du cadre bilinéaire et un argument de perturbation.
We consider a quantum particle in a 1d interval submitted to a potential. The evolution of this particle is controlled using an external electric field. Taking into account the so-called polarizability term in the model (quadratic with respect to the control), we prove global exact controllability in a suitable space for arbitrary potential and arbitrary dipole moment. This term is relevant both from the mathematical and physical points of view. The proof uses tools from the bilinear setting and a perturbation argument.
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@article{CRMATH_2014__352_5_425_0,
author = {Morancey, Morgan and Nersesyan, Vahagn},
title = {Global exact controllability of 1d {Schr\"odinger} equations with a polarizability term},
journal = {Comptes Rendus. Math\'ematique},
pages = {425--429},
publisher = {Elsevier},
volume = {352},
number = {5},
year = {2014},
doi = {10.1016/j.crma.2014.03.013},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2014.03.013/}
}
TY - JOUR AU - Morancey, Morgan AU - Nersesyan, Vahagn TI - Global exact controllability of 1d Schrödinger equations with a polarizability term JO - Comptes Rendus. Mathématique PY - 2014 SP - 425 EP - 429 VL - 352 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2014.03.013/ DO - 10.1016/j.crma.2014.03.013 LA - en ID - CRMATH_2014__352_5_425_0 ER -
%0 Journal Article %A Morancey, Morgan %A Nersesyan, Vahagn %T Global exact controllability of 1d Schrödinger equations with a polarizability term %J Comptes Rendus. Mathématique %D 2014 %P 425-429 %V 352 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2014.03.013/ %R 10.1016/j.crma.2014.03.013 %G en %F CRMATH_2014__352_5_425_0
Morancey, Morgan; Nersesyan, Vahagn. Global exact controllability of 1d Schrödinger equations with a polarizability term. Comptes Rendus. Mathématique, Tome 352 (2014) no. 5, pp. 425-429. doi : 10.1016/j.crma.2014.03.013. http://www.numdam.org/articles/10.1016/j.crma.2014.03.013/
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