On the Liouville-von Neumann equation for unbounded Hamiltonians
Davide Lonigro, Alexander Hahn, Daniel Burgarth
TL;DR
The paper addresses the well‑posedness of the Liouville–von Neumann equation for mixed states under unbounded Hamiltonians in infinite dimensions by developing a rigorous Liouville framework on the Hilbert space of Hilbert–Schmidt operators. It establishes a precise domain description for the Liouville superoperator $\mathbf{H}$ in terms of the Hamiltonian $H$, proves that $\mathbf{H}A=\overline{[H,A]}$, and extends these results to $\mathbf{H}^2$ with analogous domain conditions, including explicit cores via Nelson’s criterion. The authors present equivalent, practically checkable characterizations (including a basis‑dependent criterion) and construct concrete cores $\mathcal{D}$ and $\mathcal{D}_0$, clarifying the role of closures and domain issues in the dynamics of density operators. The work also discusses the vectorized representation $\mathbf{H}=\overline{H\otimes I-I\otimes H}$ and situates these results in the literature, emphasizing their relevance for Gibbs states and control of thermal quantum systems. Overall, the results provide a usable, mathematically rigorous toolkit for analyzing the Liouville dynamics of infinite‑dimensional quantum systems with unbounded energies, with direct implications for quantum information, statistical mechanics, and quantum technologies.
Abstract
The evolution of mixed states of a closed quantum system is described by a group of evolution superoperators whose infinitesimal generator (the quantum Liouville superoperator, or Liouvillian) determines the mixed-state counterpart of the Schrödinger equation: the Liouville-von Neumann equation. When the state space of the system is infinite-dimensional, the Liouville superoperator is unbounded whenever the corresponding Hamiltonian is. In this paper, we provide a rigorous, pedagogically-oriented, and self-contained introduction to the quantum Liouville formalism in the presence of unbounded operators. We present and discuss a characterization of the domain of the Liouville superoperator originally due to M. Courbage; starting from that, we develop some simpler characterizations of the domain of the Liouvillian and its square. We also provide, with explicit proofs, some domains of essential self-adjointness (cores) of the Liouvillian.