Local controllability of the bilinear 1D Schrödinger equation with simultaneous estimates

We consider the linear Schrödinger equation, in 1D, on a bounded interval, with Dirichlet boundary conditions and bilinear scalar control. The small-time local exact controllability around ground state was proved [5], under an appropriate nondegeneracy assumption. Here, we work weaker assumption prove projection, state, estimates control (depending linearly target) simultaneously several spaces. These are obtained at level of linearized system, thanks to new result about trigonometric moment problems. Then, they transported nonlinear system by inverse mapping theorem, error between dynamics.