Generic controllability properties for the bilinear Schr\"odinger equation

In [15] we proposed a set of sufficient conditions for the approximate controllability of a discrete-spectrum bilinear Schr\"odinger equation. These conditions are expressed in terms of the controlled potential and of the eigenpairs of the uncontrolled Schr\"odinger operator. The aim of this paper is to show that these conditions are generic with respect to the uncontrolled and the controlled potential, denoted respectively by $V$ and $W$. More precisely, we prove that the Schr\"odinger equation is approximately controllable generically with respect to $W$ when $V$ is fixed and also generically with respect to $V$ when $W$ is fixed and non-constant. The results are obtained by analytic perturbation arguments and through the study of asymptotic properties of eigenfunctions.