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We consider a linear Schr\"odinger equation, on a bounded interval, withbilinear control.  Beauchard and Laurent proved that, under an appropriate non degeneracyassumption, this system is controllable, locally around the ground state, inarbitrary time. Coron proved that a positive minimal time is required for thiscontrollability, on a particular degenerate example.  In this article, we propose a general context for the local controllabilityto hold in large time, but not in small time. The existence of a positiveminimal time is closely related to the behaviour of the second order term, inthe power series expansion of the solution.

Local controllability of 1D Schr\"odinger equations with bilinear control and minimal time