Approximate controllability for a system of Schrödinger equations modeling a single trapped ion

In this article, we analyze the approximate controllability properties for a system of Schrödinger equations modeling a single trapped ion. The control we use has a special form, which takes its origin from practical limitations. Our approach is based on the controllability of an approximate finite dimensional system for which one can design explicitly exact controls. We then justify the approximations which link up the complete and approximate systems. This yields approximate controllability results in the natural space (L²(R))²$(L^2{(\mathbb{R}))}^2$ and also in stronger spaces corresponding to the domains of powers of the harmonic operator.