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

TL;DR

This paper establishes the global existence and uniqueness of solutions for the time-dependent Hartree-Fock equations describing fermionic quantum systems with Coulomb interactions, using advanced operator and semigroup techniques.

Contribution

It provides the first rigorous proof of well-posedness for the Hartree-Fock system in the density matrix formalism with Coulomb potential.

Findings

Proved global existence and uniqueness of solutions.

Demonstrated local Lipschitz continuity of Hartree-Fock terms.

Applied semigroup methods and Lieb-Thirring inequalities.

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. 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.