Summary: Global existence of the von Neumann equation for
Hartree-Fock systems with Relaxation-Time
Anton Arnold, Roberta Bosi
Abstract This paper is concerned with the well-posedness analysis of the Hartree-
Fock system modeling the time evolution of a quantum system comprised of
fermions and interacting with the external environment via a relaxation-time term.
We consider quantum states with finite mass and finite kinetic energy, and the
self-consistent potential is the unbounded Coulomb interaction. This model is first
formulated as a semi-linear evolution problem for the one-particle density matrix
operator lying in the space of Hermitian trace class operators. Using semigroup
techniques and Lieb-Thierring-type inequalities we then prove global existence and
uniqueness of mild solutions. To this end we prove that the quadratic Hartree-Fock
terms are locally Lipschitz in the space of trace class operators with finite kinetic
Technically, the main challenge stems from considering the model as an evolution
problem for operators. Hence, many standard tools of PDE-analysis (density re-
sults, e.g.) are not readily available for the density matrix formalism.
Key words: Hartree-Fock system, von Neumann equation, density matrix, evo-
lution semigroups, trace class operators, open quantum systems.
AMS 2000 subject classification: 81Q15, 82C10, 35Q40, 47J35, 47H20, 81V70