Controllablity of a quantum particle in a 1D variable domain

CMLA, ENS Cachan, 61 avenue du président Wilson, 94235 Cachan cedex, France; Karine.Beauchard@cmla.ens-cachan.fr

Received: 2 January 2006
Revised: 10 July 2006

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 ϕ of the particle and the control is the length l(t) of the potential well. We prove the following controllability result : given close enough to an eigenstate corresponding to the length l = 1 and close enough to another eigenstate corresponding to the length l=1, there exists a continuous function with T > 0, such that l(0) = 1 and l(T) = 1, and which moves the wave function from to 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.