Controllablity of a quantum particle in a 1D variable domain

Issue		ESAIM: COCV Volume 14, Number 1, January-March 2008


Page(s)		105 - 147
DOI		10.1051/cocv:2007047

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 Beauchard

CMLA, 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 $\phi$ 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.

Mathematics Subject Classification. 35B37, 35Q55, 93B05, 93C20

Key words: Controllability, Schrödinger equation, Nash-Moser theorem, moment theory

© EDP Sciences, SMAI 2007

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