Good Lie Brackets for classical and quantum harmonic oscillators

IF 2.5 3区计算机科学 Q3 AUTOMATION & CONTROL SYSTEMS

Systems & Control Letters Pub Date : 2025-09-11 DOI:10.1016/j.sysconle.2025.106233

Andrei Agrachev , Bettina Kazandjian , Eugenio Pozzoli

引用次数: 0

Abstract

We study the small-time controllability problem on the Lie groups

S L_{2} (R)

and

S L_{2} (R) ⋉ H_{d} (R)

with Lie bracket methods (here

H_{d} (R)

denotes the

(2 d + 1)

-dimensional real Heisenberg group). Then, using unitary representations of

S L_{2} (R) ⋉ H_{d} (R)

L^{2} (R^{d}, ℂ)

and

L^{r} (T^{*} R^{d}, R), r \in [1, \infty]

, we recover small-time reachability properties of the Schrödinger PDE for the quantum harmonic oscillator, and find new small-time reachability properties of the Liouville PDE for the classical harmonic oscillator.

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经典谐振子和量子谐振子的Good Lie托架

利用李括号方法研究了李群SL2(R)和SL2(R) × Hd(R)上的小时可控性问题(这里Hd（R)表示（2d+1）维实数Heisenberg群）。然后，利用SL2(R) Hd(R)在L2（Rd,）和Lr(T * Rd，R), R∈[1，∞]上的幺正表示，我们恢复了量子谐振子Schrödinger PDE的小时可达性，并找到了经典谐振子Liouville PDE的新的小时可达性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Systems & Control Letters 工程技术-运筹学与管理科学

CiteScore

4.60

自引率

3.80%

发文量

144

审稿时长

6 months

期刊介绍： Founded in 1981 by two of the pre-eminent control theorists, Roger Brockett and Jan Willems, Systems & Control Letters is one of the leading journals in the field of control theory. The aim of the journal is to allow dissemination of relatively concise but highly original contributions whose high initial quality enables a relatively rapid review process. All aspects of the fields of systems and control are covered, especially mathematically-oriented and theoretical papers that have a clear relevance to engineering, physical and biological sciences, and even economics. Application-oriented papers with sophisticated and rigorous mathematical elements are also welcome.