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One considers a quantum particle in a 1D moving infinite square potential well. It is a nonlinear control system in which the state is the wave function of the particle and the control is the acceleration of the potential well. One proves the local controllability around any eigenstate, and the steady state controllability (controllability between eigenstates) of this control system. In particular, the wave function can be moved from one eigenstate to another one, exactly and in finite time, by moving the potential well in a suitable way.
The proof uses moment theory, a Nash-Moser theorem, Coron’s return method and expansions to the second order.
This article summarizes two works : [4] and a joint work with Jean-Michel Coron [5].
@article{SEDP_2005-2006____A9_0,
author = {Beauchard, Karine},
title = {Controllability of {Schr\"odinger} equations},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
note = {talk:9},
pages = {1--18},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2005-2006},
mrnumber = {2276075},
language = {en},
url = {https://geodesic-test.mathdoc.fr/item/SEDP_2005-2006____A9_0/}
}
TY - JOUR AU - Beauchard, Karine TI - Controllability of Schrödinger equations JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:9 PY - 2005-2006 SP - 1 EP - 18 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://geodesic-test.mathdoc.fr/item/SEDP_2005-2006____A9_0/ LA - en ID - SEDP_2005-2006____A9_0 ER -
%0 Journal Article %A Beauchard, Karine %T Controllability of Schrödinger equations %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:9 %D 2005-2006 %P 1-18 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U https://geodesic-test.mathdoc.fr/item/SEDP_2005-2006____A9_0/ %G en %F SEDP_2005-2006____A9_0
Beauchard, Karine. Controllability of Schrödinger equations. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé no. 9, 18 p. https://geodesic-test.mathdoc.fr/item/SEDP_2005-2006____A9_0/