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

TL;DR

This paper establishes necessary and sufficient conditions for controlling Schrödinger equations on nilmanifolds derived from Heisenberg-type groups, highlighting the role of subelliptic operators and minimal controllability time.

Contribution

It introduces a framework using semi-classical measures and wave packets to analyze controllability of Schrödinger equations on these complex geometric structures.

Findings

Controllability depends on specific geometric and algebraic conditions.

A minimal time for controllability exists due to subelliptic nature.

New semi-classical tools are developed for analysis on Heisenberg-type groups.

Abstract

We give necessary and sufficient conditions for the controllability of a Schr\''odinger 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 Schr\''odinger equations is subelliptic, and, contrarily to what happens for the usual elliptic Schr\''odinger 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.