## Abstract

We present a semigroup analysis of the quantum Liouville equation, which models the temporal evolution of the (quasi) distribution of an electron ensemble under the action of a scalar potential. By employing the density matrix formulation of quantum physics we prove that the quantum Liouville operator generates a unitary group on L^{2} if the corresponding Hamiltonian is essentially self‐adjoint. Also, we analyse the existence and non‐negativity of the particale density and prove that the solutions of the quantum Liouville equation converge to weak solutions of the classical Liouville equation as the Planck constant tends to zero (assuming that the potential is sufficiently smooth).

Original language | English (US) |
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Pages (from-to) | 459-469 |

Number of pages | 11 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 11 |

Issue number | 4 |

DOIs | |

State | Published - 1989 |

Externally published | Yes |

## ASJC Scopus subject areas

- General Mathematics
- General Engineering