Document ESAIM: COCV, Vol. 14, N°1, pp. 105-147
DOI: 10.1051/cocv:2007047
Controllablity of a quantum particle in a 1D variable domain
Karine BeauchardCMLA, ENS Cachan, 61 avenue du président Wilson, 94235 Cachan cedex, France; Karine.Beauchard@cmla.ens-cachan.fr
(Received January 2, 2006. Revised July 10, 2006. Published online September 21, 2007.)
Abstract
We consider a quantum particle in a 1D infinite square potential well with variable length. It is a nonlinear control system in which the state is the wave function 
of the particle and the control is the length l(t) of the potential well. We prove the following controllability result :
given 
close enough to an eigenstate corresponding to the length l = 1 and 
close enough to another eigenstate corresponding to the length l=1, there exists a continuous function
![$l:[0,T] \rightarrow \mathbb_{+}$](/articles/cocv/abs/2008/01/cocv0601/img4.gif){kind=small}
with T > 0, such that l(0) = 1 and l(T) = 1, and which moves the wave function from to in time T.
In particular, we can move the wave function from one eigenstate to another one by acting on the length of the potential well in a suitable way.
Our proof relies on local controllability results proved with moment theory,
a Nash-Moser implicit function theorem and expansions to the second order.
Mathematics Subject Classification. 35B37, 35Q55, 93B05, 93C20
Key words: Controllability, Schrödinger equation, Nash-Moser theorem, moment theory
© EDP Sciences, SMAI 2007