Abstract :

We consider a quantum particle in an infinite square potential well of $\mathbb{R}^{n}$, $n=2,3$, subjected to a control which is a uniform (in space) electric field. Under the dipolar moment approximation, the wave function solves a PDE of Schr\"odinger type. We study the spectral controllability in finite time of the linearized system around the ground state. We characterize one necessary condition for spectral controllability in finite time: $(Kal)$ if $\Omega$ is the bottom of the well, then for every eigenvalue $\lambda $ of $-\Delta_\Omega^D$, the projections of the dipolar moment onto every (normalized) eigenvector associated to $\lambda$ are linearly independent in $\R^n$.