The local exact controllability of the one-dimensional bilinear
Schr{\”o}dinger equation with Dirichlet boundary conditions has been
extensively studied in subspaces of H 3 since the seminal work of K. Beauchard.
Our first objective is to revisit this result and establish the controllability
in H 1 0 for suitable discontinuous control potentials. In the second part, we
consider the equation in the presence of periodic boundary conditions and a
constant magnetic field. We prove the local exact controllability of periodic H
1 -states, thanks to a Zeeman-type effect induced by the magnetic field which
decouples the resonant spectrum. Finally, we discuss open problems and partial
results for the Neumann case and the harmonic oscillator.

Cet article explore les excursions dans le temps et leurs implications.

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