Title: Existence and Uniqueness of Solutions of the Koopman--von Neumann Equation on Bounded Domains Show Chinese title

Title: 有界区域上Koopman--von Neumann方程解的存在性与唯一性

Abstract: 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. Show Chinese abstract

Abstract: Koopman--von Neumann方程描述了与相关经典Liouville方程给出的概率分布对应的复值波函数的演化。通常，它定义在整个欧几里得空间上。到目前为止，对有界区域的研究，特别是在涉及基于量子的动力系统模拟的实际场景中，受到的关注很少。我们考虑与有界区域上的常微分方程相关的Koopman--von Neumann方程，其轨迹包含在该集合的闭包中。我们的主要结果是构造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，提出并分析了一个与Sobolev空间相关的泛函分析框架。此外，强调了Koopman--von Neumann框架与输运方程之间的联系。