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 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 ] --> 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.