有界域上库普曼-冯-诺依曼方程解的存在性和唯一性

IF 2 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL
Marian Stengl, Patrick Gelß, Stefan Klus and Sebastian Pokutta
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引用次数: 0

摘要

库普曼-冯-诺依曼方程描述了一个复值波函数的演变过程,该波函数与相关经典刘维尔方程给出的概率分布相对应。通常,它定义在整个欧几里得空间上。对有界域的研究,尤其是在涉及基于量子的动态系统模拟的实际场景中,迄今为止还很少受到关注。我们考虑的是与有界域上的常微分方程相关的库普曼-冯-诺依曼方程,其轨迹包含在集合的闭合中。我们的主要成果是构建了一个强连续半群,以及相关初值问题解的存在性和唯一性。为此,我们提出并分析了一个与索波列夫空间相连的函数分析框架。此外,还强调了库普曼-冯-诺依曼框架与传输方程的联系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Existence and uniqueness of solutions of the Koopman–von Neumann equation on bounded domains
The Koopman–von Neumann equation describes the evolution of a complex-valued wavefunction corresponding to the probability distribution given by an associated classical Liouville equation. Typically, it is defined on the whole Euclidean space. The investigation of bounded domains, particularly in practical scenarios involving quantum-based simulations of dynamical systems, has received little attention so far. We consider the Koopman–von Neumann equation associated with an ordinary differential equation on a bounded domain whose trajectories are contained in the set’s closure. Our main results are the construction of a strongly continuous semigroup together with the existence and uniqueness of solutions of the associated initial value problem. To this end, a functional-analytic framework connected to Sobolev spaces is proposed and analyzed. Moreover, the connection of the Koopman–von Neumann framework to transport equations is highlighted.
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来源期刊
CiteScore
4.10
自引率
14.30%
发文量
542
审稿时长
1.9 months
期刊介绍: Publishing 50 issues a year, Journal of Physics A: Mathematical and Theoretical is a major journal of theoretical physics reporting research on the mathematical structures that describe fundamental processes of the physical world and on the analytical, computational and numerical methods for exploring these structures.
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