On global classical solutions of the time-dependent von Neumann equation for Hartree-Fock systems

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. 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 generalized Lieb-Thierring inequalities we then prove global existence and uniqueness of mild and classical 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 energy. Technically, the main challenge stems from considering the model as an evolution problem for operators. Hence, many standard tools of PDE-analysis (density results, e.g.) are not readily available for the density matrix formalism.