Observability and Controllability for the Schr{o}dinger Equation on Quotients of Groups of Heisenberg Type

We give necessary and sufficient conditions for the controllability of a Schrodinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups.This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schrodinger equations is subelliptic, and, contrarily to what happens for the usual elliptic Schrodinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main tools used in the proofs are (operator-valued) semi-classical measures constructed by use of representation theory and a notion of semi-classical wave packets that we introduce here in the context of groups of Heisenberg type.