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We consider a 1D linear Schr{\"o}dinger equation, on a bounded interval, withDirichlet boundary conditions and bilinear control. We study itscontrollability around the ground state when the linearized system is notcontrollable. More precisely, we study to what extent the nonlinear terms ofthe expansion can recover the directions lost at the first order.In previousworks, for any positive integer $n$, assumptions have been formulated underwhich the quadratic term induces a drift in the nonlinear dynamics, quantifiedby the $H^{-n}$ norm of the control. This drift is an obstruction to thesmall-time local controllability (STLC) under a smallness assumption on thecontrols in regular spaces. In this paper, we prove that for controls small inless regular spaces, the cubic term can recover the controllability lost at thelinear level, despite the quadratic drift. The proof is inspired by Sussman'smethod to prove the sufficiency of the $\mathcal{S}(\theta)$ condition for STLCof ODEs. However, it uses a different global strategy relying on a new conceptof tangent vector, better adapted to the infinite-dimensional setting of PDEs.Given a target, we first realize the expected motion along the lost directionby using control variations for which the cubic term dominates the quadraticone. Then, we correct the other components exactly, by using a STLC inprojection result, with simultaneous estimates of weak norms of the control.These estimates ensure that the new error along the lost direction isnegligible, and we conclude with the Brouwer fixed point theorem.

Small-time local controllability of the bilinear Schr\"odinger equation, despite a quadratic obstruction, thanks to a cubic term