Related papers: On the Liouville-von Neumann equation for unbounded Hamiltonians

On the Liouville-von Neumann equation for unbounded Hamiltonians

URL: http://arxiv.org/abs/2408.04980v1
Date: Fri, 9 Aug 2024 10:13:35 GMT
Title: On the Liouville-von Neumann equation for unbounded Hamiltonians
Authors: Davide Lonigro, Alexander Hahn, Daniel Burgarth,
Abstract summary: We present and discuss a characterization of the domain of the Liouville superoperator originally due to M. Courbage. We also provide, with explicit proofs, some domains of essential self-adjointness (cores) of the Liouvillian.
Score: 44.99833362998488
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
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 Schr\"odinger 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.

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