Small-time local control of a Schrödinger equation: a negative and a positive quadratic result

We study the small-time local controllability (STLC) of a bilinear Schrödinger equation with Neumann boundary conditions near its ground state. We focus on the degenerate case where the linearized system is not controllable, necessitating a second-order analysis. We prove two complementary results. The negative result provides a new PDE instance of Sussmann’s classical quadratic obstruction, corresponding to a non-vanishing Lie bracket. The positive result appears to be the first to establish STLC at the quadratic order for a physical PDE with a single scalar control. Both proofs rely on a Fourier-based approach, which is crucial because the integral kernel of the second-order term lacks the regularity required by standard integration-by-parts arguments. Along the way, we develop tools valid in a more general setting to analyze such quadratic forms. In particular, we prove results that allow for the multiplication of a kernel by a modulation function.