Observability and controllability for the Schrödinger equation on quotients of groups of Heisenberg type

Clotilde Fermanian Kammerer,Cyril Letrouit

doi:10.5802/jep.176

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.

Highlights

We consider a nilmanifold M, that is a manifold M = Γ\G which is the left quotient of a nilpotent Lie group G by a discrete cocompact subgroup Γ
We study the controllability and the observability of the associated Schrödinger equation on M thanks to the Harmonic analysis properties of the group G
For proving the second part of Theorem 1.4 – the necessity of the condition (H-GCC) – we construct a family of initial data for which the solution (ψε(t)) of the Schrödinger equation (1.3) concentrates on the curve Φt0(x0, λ0), for any choice of (x0, λ0) ∈ M × z {0}

Summary

Introduction

— With these geometric definitions, we are able to state conditions for observability and controllability of the subelliptic Schrödinger equation with analytic potential on H-type nilmanifolds. For proving the second part of Theorem 1.4 – the necessity of the condition (H-GCC) – we construct a family of initial data (uε0) for which the solution (ψε(t)) of the Schrödinger equation (1.3) concentrates on the curve Φt0(x0, λ0), for any choice of (x0, λ0) ∈ M × z {0} As mentioned above, this set of initial data is the non-commutative counterpart to the wave packets ( called coherent states) in the Euclidean setting [14, 29]. The authors are grateful to the referees for their remarks and suggestions

Semi-classical analysis on quotient manifolds

Semi-classical pseudodifferential operators on quotient manifolds

Proof of the sufficiency of the geometric conditions

Non-commutative wave packets and the necessity of the geometric control