Generic controllability of the bilinear Schroedinger equation on 1-D domains: the case of measurable potentials

Several sufficient conditions for the controllability of the Schroedinger equation have been proposed in the last few years. In this article we discuss the genericity of these conditions with respect to the variation of the controlled or the uncontrolled potential. In the case where the Schr¨odinger equation is set on a domain of dimension one, we improve the results in the literature, removing from the previously known genericity results some unnecessary technical assumptions on the regularity of the potentials