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

Marian Stengl, Patrick Gelß, Stefan Klus, Sebastian Pokutta

TL;DR

This work addresses the Koopman--von Neumann equation on bounded domains for ODEs with trajectories contained in the domain's closure by introducing a no-outflow boundary condition $F(x) \cdot \nu \le 0$ on $\partial\Omega$ and proving well-posedness. A Perron--Frobenius--Sobolev framework is developed to encode the vector field $F$ into the domain, yielding a KvN generator that is skew-symmetric and admits a closed graph, enabling semigroup analysis. The authors establish that the KvN generator on the subspace $H_0({\mathcal L}^*,\Omega)$ generates a strongly continuous contraction semigroup on $L^2_{\mathbb{C}}(\Omega)$, guaranteeing existence, uniqueness, and mass conservation of the KvN evolution, with Green's formula and trace theory underpinning boundary treatment. The results provide a rigorous mathematical foundation for KvN on bounded domains, linking KvN dynamics to transport-equation methods and supporting bounded-domain quantum simulations, while outlining future directions for nonautonomous dynamics, spectral analysis, and quantum-discretization approaches.

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.

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

TL;DR

This work addresses the Koopman--von Neumann equation on bounded domains for ODEs with trajectories contained in the domain's closure by introducing a no-outflow boundary condition on and proving well-posedness. A Perron--Frobenius--Sobolev framework is developed to encode the vector field into the domain, yielding a KvN generator that is skew-symmetric and admits a closed graph, enabling semigroup analysis. The authors establish that the KvN generator on the subspace generates a strongly continuous contraction semigroup on , guaranteeing existence, uniqueness, and mass conservation of the KvN evolution, with Green's formula and trace theory underpinning boundary treatment. The results provide a rigorous mathematical foundation for KvN on bounded domains, linking KvN dynamics to transport-equation methods and supporting bounded-domain quantum simulations, while outlining future directions for nonautonomous dynamics, spectral analysis, and quantum-discretization approaches.

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.
Paper Structure (23 sections, 17 theorems, 113 equations)

This paper contains 23 sections, 17 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Semigroup Theory
  4. Function Spaces
  5. Continuous and Differentiable Functions
  6. Lebesgue Spaces
  7. Sobolev Space and H(div) Space
  8. Derivation of the Koopman--von Neumann Generator
  9. Perron--Frobenius and Koopman Operators
  10. Koopman--von Neumann Operator
  11. Perron--Frobenius--Sobolev Space
  12. Trace Operator and Green's Formula for the Perron--Fro-be-nius--Sobolev Functions
  13. Existence and Uniqueness for the Koopman--von Neumann Equation
  14. Koopman--von Neumann Generator
  15. Koopman--von Neumann Equation
  16. ...and 8 more sections

Key Result

Theorem 1

Let $A \colon {\mathcal{D}}(A) \subseteq X \to X$ be the infinitesimal generator of a $C_0$-semigroup $(T(t))_{t \geq 0}$ with $T(t) \in {\mathcal{L}}(X)$ for all $t \geq 0$, then the mapping $u\colon t \mapsto T(t)u_0$ is the unique solution $u \in C^1([0,\infty), X) \cap C([0,\infty), {\mathcal{D}

Theorems & Definitions (33)

  • Theorem 1: Solution of the Cauchy problem
  • Corollary 2: Corollary of Lumer--Phillips Theorem, see bib:Pazy_Semigroup
  • Theorem 3: Product rule, cf. bib:AltNuernberg
  • Theorem 4: Traces of Sobolev functions, cf. bib:EvansGariepy
  • Theorem 5: Traces of $H(\operatorname{div})$ functions, see cachereva.bib:GiraultRaviart_FEMNavierStokes
  • Theorem 6: Green's formula, see cachereva.bib:GiraultRaviart_FEMNavierStokes
  • Remark 1
  • Definition 7: Perron--Frobenius--Sobolev Spaces
  • Theorem 8: Trace for Perron--Frobenius--Sobolev Functions
  • proof
  • ...and 23 more