Uniqueness for the Schr\"odinger equation with an inverse square potential and application to controllability and inverse problems

In this paper, we prove a sharp uniqueness result for the singular Schr\"odinger equation with an inverse square potential. This will be done without assuming geometrical restrictions on the observation region. The proof relies on a recent technique transforming the Schr\"odinger equation into an elliptic equation. We show that this technique is still applicable for singular equations. In our case, substantial difficulties arise when dealing with singular potentials of cylindrical type. Using the uniqueness result, we show the approximate controllability of the equation using a distributed control. The uniqueness result is also applied to prove the uniqueness for an inverse source problem.