On the Liouville-von Neumann equation for unbounded Hamiltonians
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915086170324992 |
|---|---|
| author | Lonigro, Davide Hahn, Alexander Burgarth, Daniel |
| author_facet | Lonigro, Davide Hahn, Alexander Burgarth, Daniel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04980 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Liouville-von Neumann equation for unbounded Hamiltonians Lonigro, Davide Hahn, Alexander Burgarth, Daniel Quantum Physics Mathematical Physics 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. |
| title | On the Liouville-von Neumann equation for unbounded Hamiltonians |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2408.04980 |