Controllability of a quantum particle in a moving potential well

Abstract

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 ψ 0 close enough to an eigenstate and ψ f close enough to another eigenstate, the wave function can be moved exactly from ψ 0 to ψ f 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.