We consider a non relativistic charged particle in a 1-D box of potential. This quantum system is subject to a control, which is a uniform electric field. It is represented by a complex probability amplitude solution of a Schrödinger equation. We prove the local controllability of this nonlinear system around the ground state. Our proof uses the return method, a Nash-Moser implicit function theorem and moment theory.Résumé: On considère une particule non relativiste dans un puits de potentiel en dimension un d'espace. Ce systeme quantique est soumisà un champélectrique uniforme, qui constitue un contrôle. Il est représenté par une densité de probabilité complexe solution d'uneéquation de Schrödinger. On démontre la contrôlabilité locale de ce système non linéaire au voisinage de l'état fondamental. La preuve utilise la méthode du retour, un théorème de Nash-Moser et la théorie des moments.

We consider a linear Schrödinger equation, on a bounded interval, with bilinear control, that represents a quantum particle in an electric field (the control). We prove the controllability of this system, in any positive time, locally around the ground state. Similar results were proved for particular models (by the first author and with J.M. Coron), in non optimal spaces, in long time and the proof relied on the Nash-Moser implicit function theorem in order to deal with an a priori loss of regularity. In this article, the model is more general, the spaces are optimal, there is no restriction on the time and the proof relies on the classical inverse mapping theorem. A hidden regularizing effect is emphasized, showing there is actually no loss of regularity. Then, the same strategy is applied to nonlinear Schrödinger equations and nonlinear wave equations, showing that the method works for a wide range of bilinear control systems

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