Koopman-von Neumann Mechanics

The Koopman--von Neumann (KvN) equation is a vital concept in the field of dynamical systems, providing a mathematical framework to understand complex behaviors in various scientific disciplines. In this blog post, we delve into the paper titled "Existence and Uniqueness for the Koopman--von Neumann Equation" [1], which explores the existence and uniqueness of solutions to this intriguing equation.

Involving complex function spaces, such as $L^2_\mathbb{C}(\Omega)$, $H^1(\Omega)$, and $H(\textrm{div}, \Omega)$ defined as sets of complex-valued functions on a domain $\Omega$ with specific properties, the heart of the paper lies in defining and understanding the Koopman--von Neumann generator. This operator, denoted as $\mathcal{L}^*_{\textrm{KvN}}$, is constructed based on the Perron--Frobenius generator $\mathcal{L}^*$ within the real-valued function space. The KvN generator is defined as:
$$
\langle \mathcal{L}^*_{\textrm{KvN}} \psi, \phi \rangle_{H(\mathcal{L}^*,\Omega)^*, H(\mathcal{L}^*,\Omega)} = \frac{1}{2} \left((L^* \psi, \phi)_{\mathcal{L}^2(\Omega)} - (\psi, \mathcal{L}^* \phi)_{L^2(\Omega)} \right),
$$
where $H(\mathcal{L}^*,\Omega)$ denotes the so-called Perron-Frobenius-Sobolev space.

The paper explores the properties of the KvN generator, such as its skew-symmetry and its association with the real part of functions. It also discusses the KvN equation within the context of transport equations, providing a robust foundation for further analysis. Consider the following autonomous initial value problem $$\begin{aligned} \dot x &= F(x), \\ x(0) &= x_0 \end{aligned}$$ for a vector field $F \colon \bar\Omega \to \mathbb{R}^d$. The KvN equation is the key focus of our paper. This equation describes the evolution of solutions driven by the KvN operator. It is presented as follows:

$$\begin{aligned}\partial_t \psi &= \mathcal{L}^*_{\textrm{KvN}} \psi, &&\text{for all } t \geq 0 \text{ in } \Omega, \\\psi F \cdot \nu &= 0, &&\text{for all } t \geq 0 \text{ on } \partial \Omega, \\\psi(0) &= \psi_0, &&\text{in } \Omega.\end{aligned}\tag{1}$$

The KvN framework holds significant relevance for operator-based numerical simulations and the analysis of dynamical systems, particularly when executed on quantum computers. This framework transforms the classical Liouville equation, which governs the conservation of probability distributions, into a Schrödinger equation operating within a Hilbert space. The solution of this Schrödinger equation produces a complex-valued wave function denoted as $\psi$, which undergoes propagation through a semigroup of unitary operators. Employing Born's rule, this wave function facilitates the extraction of the probability density $\rho$, defined as $|\psi|^2$. Notably, this probability density adheres to the Liouville equation.

Originally, Koopman and von Neumann primarily considered the KvN framework in the context of Hamiltonian systems. However, subsequent extensions have broadened its applicability to encompass general dynamical systems. Consequently, the KvN framework serves as a pivotal bridge connecting classical mechanics and quantum mechanics, offering valuable insights for the study of complex phenomena and the simulation of dynamical systems on quantum computers. The main result of our paper can be summarized as follows:

Theorem (Existence and uniqueness of solutions of (1)). Let $\Omega \subseteq \mathbb{R}^d$ be a bounded, open domain with Lipschitz boundary and let $\psi_0 \in H_{\mathbb{C},0}(\mathcal{L}^*,\Omega)$ be given. Then the KvN generator induces a $C_0$-semigroup of contractions $(T(t))_{t \geq 0}$ and (1) has a unique solution defined by $\psi(t) = T(t)\psi_0$. Moreover, for all $t \geq 0$ it holds that $\|\psi(t)\|_{L^2_{\mathbb{C}(\Omega)}} = \|\psi_0\|_{L^2_{\mathbb{C}(\Omega)}}$.

The KvN equation plays a vital role in understanding complex behaviors and phenomena in various scientific and engineering disciplines, making it important to provide a profound exploration of the KvN equation, its generator, and their properties. Our paper establishes the existence and uniqueness of solutions for the Koopman--von Neumann equation and offers essential mathematical frameworks and definitions for researchers in the field of dynamical systems. For a more in-depth understanding and detailed mathematical derivations, we recommend reading the full paper:

[1] M. Stengl, P. Gelß, S. Klus, S. Pokutta. Existence and Uniqueness of Solutions of the Koopman--von Neumann Equation on Bounded Domains. arXiv: 2306.13504