Existence and Uniqueness of Solutions of the Koopman--von Neumann Equation on Bounded Domains

TL;DR

This paper establishes the existence and uniqueness of solutions for the Koopman--von Neumann equation on bounded domains, extending the theory beyond the usual Euclidean space setting.

Contribution

It introduces a functional-analytic framework on Sobolev spaces for bounded domains and constructs a strongly continuous semigroup for the equation.

Findings

Proves existence and uniqueness of solutions on bounded domains.

Connects Koopman--von Neumann equation to transport equations.

Develops a Sobolev space-based analytical framework.

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.