On the exact observability of Schrödinger equations associated with Laguerre operators

In this paper, we consider three systems of Laguerre functions and their corresponding Schrödinger equations. We establish both necessary and sufficient conditions for the exact observability of the Schrödinger equations associated with these Laguerre operators in one dimension, and provide a necessary condition in higher dimensions. These results are deduced from an infinite dimensional version of the Hautus test and the Plancherel-Rotach formula for Laguerre polynomials.