An analysis of the quantum Liouville equation. (English) Zbl 0682.46047

Summary: We present an analysis of the quantum Liouville equation under the assumption of a globally bounded potential energy. By using methods of semigroup theory we prove existence and uniqueness results. We also show the existence of the particle density. The last section is concerned with the classical limit. We show that the solutions of the quantum Liouville equation converge to the solution of the classical Liouville equation as the Planck constant h tends to zero.


46N99 Miscellaneous applications of functional analysis
47D03 Groups and semigroups of linear operators
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
Full Text: DOI


[1] : Sobolev spaces. Acad. Press 1975. · Zbl 0314.46030
[2] ; : Lehrbuch der Theoretischen Physik. 3. Quantenmechanik. Akademie Verlag, Berlin 1960.
[3] : The nuclear Vlasov equation: Methods and results that can not be taken over from the classical case. In: Fluid dynamical approaches to the many body problem. Report, Fachbereich Mathematik, Universität Kaiserslautern, D-6750 Kaiserslautern, FRG.
[4] : Semigroups of linear operators and applications to partial differential equations. Springer Verlag, Berlin etc. 1983. · Zbl 0516.47023
[5] ; : Fourier analysis. Self adjointness 2. Academic Press 1975.
[6] : Pseudodifferential operators and spectral theory. Springer Verlag, Berlin etc. 1986.
[7] Tatarskii, Sov. Phys. Usp. 26 pp 311– (1983)
[8] : Shock waves and reaction diffusion equations. Springer, Berlin etc. 1982.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.