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

Markowich, P. A.; Ringhofer, C. A.

Z. Angew. Math. Mech. 69, No. 3, 121-127 (1989).

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.

MSC:

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

Keywords:

quantum Liouville equation; globally bounded potential energy; semigroup theory; existence and uniqueness; existence of the particle density; classical limit

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