1 results
Controllablity of a quantum particle in a 1D variable domain
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- Journal:
- ESAIM: Control, Optimisation and Calculus of Variations / Volume 14 / Issue 1 / January 2008
- Published online by Cambridge University Press:
- 21 September 2007, pp. 105-147
- Print publication:
- January 2008
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- Article
- Export citation
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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 ϕ of the particle and the control is the length l(t) of the potential well. We prove the following controllability result : given $\phi_{0}$
close enough to an eigenstate corresponding to the length l = 1 and $\phi_{f}$ 
close enough to another eigenstate corresponding to the length l=1, there exists a continuous function $l:[0,T] \rightarrow \mathbb{R}^{*}_{+}$ 
with T > 0, such that l(0) = 1 and l(T) = 1, and which moves the wave function from $\phi_{0}$ to $\phi_{f}$ 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.
