Mean field evolution of fermions with Coulomb interaction

Abstract

We study the many body Schrödinger evolution of weakly coupled fermions interacting through a Coulomb potential. We are interested in a joint mean field and semiclassical scaling, that emerges naturally for initially confined particles. For initial data describing approximate Slater determinants, we prove convergence of the many-body evolution towards Hartree-Fock dynamics. Our result holds under a condition on the solution of the Hartree-Fock equation, that we can only show in a very special situation (translation invariant data, whose Hartree-Fock evolution is trivial), but that we expect to hold more generally.

arXiv comment21 pages. Minor changes, typos corrected

References in corpus (6)

• Mean-field Evolution of Fermionic Systems
• From the Hartree dynamics to the Vlasov equation
• Mean-Field Dynamics of Fermions with Relativistic Dispersion
• A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics
• Kinetic Energy Estimates for the Accuracy of the Time-Dependent Hartree-Fock Approximation with Coulomb Interaction
• The classical limit of the time dependent Hartree-Fock equation. I. The Weyl symbol of the solution

Cited by in corpus (27)

• The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations
• Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime
• Empirical Measures and Quantum Mechanics: Application to the Mean-Field Limit
• The dilute Fermi gas via Bogoliubov theory
• Propagation of Moments and Semiclassical Limit from Hartree to Vlasov Equation
• Hartree Corrections in a Mean-field Limit for Fermions with Coulomb Interaction
• Semiclassical limit to the Vlasov equation with inverse power law potentials
• From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials
• Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data
• Combined mean-field and semiclassical limits of large fermionic systems
• Classical Dynamics Generated by Long-Range Interactions for Lattice Fermions and Quantum Spins
• Mean-field Dynamics for the Nelson Model with Fermions
• On Quantum Sobolev Inequalities
• Strong semiclassical limit from Hartree and Hartree-Fock to Vlasov-Poisson equation
• Effective Dynamics of Extended Fermi Gases in the High-Density Regime
• Global-in-time Semiclassical Regularity for the Hartree-Fock Equation
• Bosonization of Fermionic Many-Body Dynamics
• Optimal Semiclassical Regularity of Projection Operators and Strong Weyl Law
• The Maximal Negative Ion of the Time-Dependent Thomas-Fermi and the Vlasov Atom
• A mixed-norm estimate of the two-particle reduced density matrix of many-body Schrödinger dynamics for deriving the Vlasov equation
• Convergence towards the Vlasov-Poisson Equation from the $N$-Fermionic Schrödinger Equation
• Dynamics of mean-field Fermi systems with nonzero pairing
• On the semiclassical regularity of thermal equilibria
• On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate
• Effective Dynamics of Interacting Fermions from Semiclassical Theory to the Random Phase Approximation
• Stability of homogeneous equilibria of the Hartree-Fock equation, for its equivalent formulation for random fields
• Derivation of Vlasov equations for multi-species fermions