Quantum dynamical semigroups
reduced description of many-body systems
nonlinear evolution equations
nonlinear frictional Schrödinger equations
quantum Boltzmann equation
Springer Online Journal Archives 1860-2000
Abstract The notion of a nonlinear quantum dynamical semigroup is introduced, and the existence and uniqueness of solutions of the corresponding nonlinear evolution equations are studied in a more abstract framework. The construction of nonlinear quantum dynamical semigroups is carried out for two different mean-field models. First a mean-field coupling between a system of noninteracting subsystems and the bath is investigated. As examples, a nonlinear frictional Schrödinger equation and a model for a quantum Boltzmann equation are discussed. Second, a many-body system with mean-field interaction coupled to a bath is considered. Here, again, the form of the generator is derived; however, it cannot be obtained rigorously, except for some particular examples. Finally, the quantum Ising-Weiss model is briefly studied.
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