Quantum Evolution And Sub-laplacian Operators On Groups Of Heisenberg Type

TL;DR

This paper studies quantum evolution on Heisenberg-type groups using semi-classical analysis, describing how energy densities evolve and establishing an Egorov's theorem for sub-Laplacian operators.

Contribution

It introduces a semi-classical framework for analyzing quantum dynamics on stratified Lie groups and proves an Egorov's theorem in this setting.

Findings

Characterization of semi-classical measures on Heisenberg-type groups

Proof of an Egorov's theorem for sub-Laplacian operators

Description of quantum limits in stratified Lie group context

Abstract

In this paper we analyze the evolution of the time averaged energy densities associated with a family of solutions to a Schr{\"o}dinger equation on a Lie group of Heisenberg type. We use a semi-classical approach adapted to the stratified structure of the group and describe the semi-classical measures (also called quantum limits) that are associated with this family. This allows us to prove an Egorov's type Theorem describing the quantum evolution of a pseudodifferential semi-classical operator through the semi-group generated by a sub-Laplacian.