Controllability of a quantum particle in a 1D variable domain
ESAIM: Control, Optimisation and Calculus of Variations, Volume 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 $\phi$ of the particle and the control is the length $l\left(t\right)$ 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 $l:\left[0,T\right]\to {ℝ}_{+}^{*}$ with $T>0$, such that $l\left(0\right)=1$ and $l\left(T\right)=1$, 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.

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

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