Good Lie Brackets for classical and quantum harmonic oscillators

TL;DR

The paper develops a geometric control framework for small-time controllability on $SL_2$ and its semidirect extension with the Heisenberg group, via Good Lie Brackets and compatibility. It proves a small-time controllability result on $SL_2\ltimes H_d$ and translates this into explicit small-time approximate reachability statements for both the quantum harmonic oscillator Schrödinger equation and the classical harmonic oscillator Liouville equation using unitary (metaplectic) representations. The core method builds unitary representations of covering groups and leverages compatibility and Lie-bracket generating properties to connect group-theoretic controllability with PDE reachability. This yields concrete families of physically meaningful states and densities that can be steered in arbitrarily small time, illuminating controllability aspects of fundamental oscillator dynamics and suggesting broader applicability to bilinear PDE control problems.

Abstract

We study the small-time controllability problem on the Lie groups $SL_2(\mathbb{R})$ and $SL_2(\mathbb{R})\ltimes H_{d}(\mathbb{R})$ with Lie bracket methods (here $H_{d}(\mathbb{R})$ denotes the $(2d+1)$-dimensional real Heisenberg group). Then, using unitary representations of $SL_2(\mathbb{R})\ltimes H_{d}(\mathbb{R})$ on $L^2(\mathbb{R}^d,\mathbb{C})$ and $L^p(T^*\mathbb{R}^d,\mathbb{R}), p\in[1,\infty)$, we recover small-time approximate reachability properties of the Schrödinger PDE for the quantum harmonic oscillator, and find new small-time approximate reachability properties of the Liouville PDE for the classical harmonic oscillator.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Definition 4
• Theorem 2
• Definition 5
• Theorem 3
• Theorem 4
• proof