Controllability of a quantum particle in a 1D variable domain

Beauchard, Karine

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 $\phi$ 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 $l:[0,T]\to {ℝ}_{+}^{*}$ 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.

Classification : 35B37, 35Q55, 93B05, 93C20
Mots clés : controllability, Schrödinger equation, Nash-Moser theorem, moment theory

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