Controllability of a quantum particle in a moving potential well

We consider a nonrelativistic charged particle in a 1D moving potential well. This quantum system is subject to a control, which is the acceleration of the well. It is represented by a wave function solution of a Schrödinger equation, the position of the well together with its velocity. We prove the following controllability result for this bilinear control system: given ψ0 close enough to an eigenstate and ψf close enough to another eigenstate, the wave function can be moved exactly from ψ0 to ψf in finite time. Moreover, we can control the position and the velocity of the well. Our proof uses moment theory, a Nash–Moser implicit function theorem, the return method and expansion to the second order.