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Controllablity of a quantum particle in a 1D variable domain. (English) Zbl 1132.35446
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 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 \(l: [0, T ] \to \mathbb{R}_+^*\) 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.

35Q40 PDEs in connection with quantum mechanics
35B37 PDE in connection with control problems (MSC2000)
35Q55 NLS equations (nonlinear Schrödinger equations)
93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
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