Controllability of a quantum particle in a 1D variable domain
ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 1, pp. 105-147.

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 φ 0 close enough to an eigenstate corresponding to the length l=1 and φ f close enough to another eigenstate corresponding to the length l=1, there exists a continuous function l:[0,T] + * with T>0, such that l(0)=1 and l(T)=1, and which moves the wave function from φ 0 to φ 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.

DOI : 10.1051/cocv:2007047
Classification : 35B37, 35Q55, 93B05, 93C20
Mots clés : controllability, Schrödinger equation, Nash-Moser theorem, moment theory
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     title = {Controllability of a quantum particle in a {1D} variable domain},
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Beauchard, Karine. Controllability of a quantum particle in a 1D variable domain. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 1, pp. 105-147. doi : 10.1051/cocv:2007047. http://archive.numdam.org/articles/10.1051/cocv:2007047/

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