 
Summary: Global existence of the von Neumann equation for
HartreeFock systems with RelaxationTime
Anton Arnold, Roberta Bosi
Abstract This paper is concerned with the wellposedness 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 relaxationtime term.
We consider quantum states with finite mass and finite kinetic energy, and the
selfconsistent potential is the unbounded Coulomb interaction. This model is first
formulated as a semilinear evolution problem for the oneparticle density matrix
operator lying in the space of Hermitian trace class operators. Using semigroup
techniques and LiebThierringtype inequalities we then prove global existence and
uniqueness of mild solutions. To this end we prove that the quadratic HartreeFock
terms are locally Lipschitz in the space of trace class operators with finite kinetic
energy.
Technically, the main challenge stems from considering the model as an evolution
problem for operators. Hence, many standard tools of PDEanalysis (density re
sults, e.g.) are not readily available for the density matrix formalism.
Key words: HartreeFock 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
