Quantum dynamical semigroups for diffusion models with Hartree interaction

Abstract

We consider a class of evolution equations in Lindblad form, which model the dynamics of dissipative quantum mechanical systems with mean-field interaction. Particularly, this class includes the so-called Quantum Fokker-Planck-Poisson model. The existence and uniqueness of global-in-time, mass preserving solutions is proved, thus establishing the existence of a nonlinear conservative quantum dynamical semigroup. The mathematical difficulties stem from combining an unbounded Lindblad generator with the Hartree nonlinearity.Comment: 30 pages; Introduction changed, title changed, easier and shorter proofs due to new energy norm. to appear in Comm. Math. Phy