Good Lie Brackets for classical and quantum harmonic oscillators

2025/03/14 by Agrachev, Andrei, Kazandjian, Bettina, Pozzoli, Eugenio · 1 citation
#FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2503.11307

Abstract

We study the small-time controllability problem on the Lie groups SL₂(ℝ) and SL₂(ℝ)\ltimes H_d(ℝ) with Lie bracket methods (here H_d(ℝ) denotes the (2d+1)-dimensional real Heisenberg group). Then, using unitary representations of SL₂(ℝ)\ltimes H_d(ℝ) on L²(ℝ^d,ℂ) and L^p(T^*ℝ^d,ℝ), p∈[1,∞), 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.

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• Orbits and attainable Hamiltonian diffeomorphisms of mechanical Liouville equations

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