Derivation of the time-dependent Hartree equations for strongly interacting dense fermionic systems

TL;DR

This paper rigorously derives the time-dependent Hartree equations from the microscopic Schrödinger dynamics for large, strongly interacting fermionic systems, using gauge transformations to handle dominant interaction contributions.

Contribution

It provides a rigorous derivation of the time-dependent Hartree equations for dense fermionic systems with strong interactions, advancing the mathematical understanding of mean-field limits.

Findings

Derivation of the Hartree equations as the large-N limit of fermionic Schrödinger dynamics.

Implementation of gauge transformations to manage strong interaction effects.

Mathematical framework applicable to dense, strongly interacting fermionic systems.

Abstract

The time-dependent Hartree and Hartree-Fock equations provide effective mean-field descriptions for the dynamics of large fermionic systems and play a fundamental role in many areas of physics. In this work, we rigorously derive the time-dependent Hartree equations as the large-$N$ limit of the microscopic Schr\"odinger dynamics of $N$ fermions confined to a volume of order one and interacting via strong pair potentials. A central step in our analysis is the implementation of time-dependent gauge transformations, which eliminate the dominant contribution from the interaction potential in both the Schr\"odinger and Hartree evolutions.