On the equivalence of the Schrödinger and the quantum Liouville equations

Peter A. Markowich*, H. Neunzert

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

69 Scopus citations


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 L2 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 languageEnglish (US)
Pages (from-to)459-469
Number of pages11
JournalMathematical Methods in the Applied Sciences
Issue number4
StatePublished - 1989
Externally publishedYes

ASJC Scopus subject areas

  • General Mathematics
  • General Engineering


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