$L^2$ Analysis of the Multi-Configuration Time-Dependent Hartree-Fock Equations

TL;DR

This paper develops an $L^2$ mathematical framework for the multi-configuration time-dependent Hartree-Fock equations, broadening the understanding of their well-posedness for general interactions including Coulomb, which supports numerical methods in quantum physics.

Contribution

It extends the analysis of MCTDHF equations by establishing an $L^2$ theory applicable to general interactions, including Coulomb, enhancing the mathematical foundation for numerical approximations.

Findings

Established $L^2$ well-posedness for MCTDHF equations.

Extended theoretical results to include Coulomb interactions.

Provided a basis for regularized numerical methods that preserve mass.

Abstract

The multiconfiguration methods are widely used by quantum physicists and chemists for numerical approximation of the many electron Schr\"odinger equation. Recently, first mathematically rigorous results were obtained on the time-dependent models, e.g. short-in-time well-posedness in the Sobolev space $H^2$ for bounded interactions (C. Lubichand O. Koch} with initial data in $H^2$, in the energy space for Coulomb interactions with initial data in the same space (Trabelsi, Bardos et al.}, as well as global well-posedness under a sufficient condition on the energy of the initial data (Bardos et al.). The present contribution extends the analysis by setting an $L^2$ theory for the MCTDHF for general interactions including the Coulomb case. This kind of results is also the theoretical foundation of ad-hoc methods used in numerical calculation when modification ("regularization") of the density matrix destroys the conservation of energy property, but keeps invariant the mass.