Global exact controllability in infinite time of Schr\"odinger equation: multidimensional case

Abstract

We prove that the multidimensional Schr\"odinger equation is exactly controllable in infinite time near any point which is a finite linear combination of eigenfunctions of the Schr\"odinger operator. We prove that, generically with respect to the potential, the linearized system is controllable in infinite time. Applying the inverse mapping theorem, we prove the controllability of the nonlinear system.