Self-consistent relaxation-time models in quantum mechanics

This paper is concerned with the relaxation-time von Neumann-Poisson (or quantum Liouville-Poisson) equation in three spatial dimensions which describes the self-consistent time evolution of an open quantum mechanical system that includes some relaxation mechanism. This model and the equivalent relaxation-time Wigner-Poisson system play an important role in the simulation of quantum semiconductor devices. For initial density matrices with finite kinetic energy, we prove that this problem, formulated in the space of Hermitian trace class operators, admits a unique global strong solution. A key ingredient for our analysis is a new generalization of the Lieb-Thirring inequality for density matrix operators.