We consider a quantum charged particle in a one-dimensional infinite square potential well moving along a line. We control the acceleration of the potential well. The local controllability in large time of this nonlinear control system along the ground state trajectory has been proved recently. We prove that this local controllability does not hold in small time, even if the Schrödinger equation has an infinite speed of propagation.
On considère une particule quantique chargée dans un puits de potentiel carré infini unidimensionnel se déplaçant le long d'une droite. On contrôle l'accélération du puits de potentiel. La contrôlabilité locale autour de l'état fondamental pour des temps grands de ce système de contrôle a été récemment démontrée. Nous montrons que l'on n'a pas contrôlabilité locale pour des temps petits, bien que l'équation de Schrödinger ait une vitesse de propagation infinie.
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Jean-Michel Coron 1
@article{CRMATH_2006__342_2_103_0,
author = {Jean-Michel Coron},
title = {On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well},
journal = {Comptes Rendus. Math\'ematique},
pages = {103--108},
publisher = {Elsevier},
volume = {342},
number = {2},
year = {2006},
doi = {10.1016/j.crma.2005.11.004},
language = {en},
}
TY - JOUR AU - Jean-Michel Coron TI - On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well JO - Comptes Rendus. Mathématique PY - 2006 SP - 103 EP - 108 VL - 342 IS - 2 PB - Elsevier DO - 10.1016/j.crma.2005.11.004 LA - en ID - CRMATH_2006__342_2_103_0 ER -
%0 Journal Article %A Jean-Michel Coron %T On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well %J Comptes Rendus. Mathématique %D 2006 %P 103-108 %V 342 %N 2 %I Elsevier %R 10.1016/j.crma.2005.11.004 %G en %F CRMATH_2006__342_2_103_0
Jean-Michel Coron. On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well. Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 103-108. doi : 10.1016/j.crma.2005.11.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.11.004/
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