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

This work was supported by the Viennese Science Foundation (WWTF) via the project "TDDFT" (MA-45), the Austrian Science Foundation (FWF) via the Wissenschaftkolleg "Differential equations" (W17) and the START Project (Y-137-TEC) and the EU funded Marie Curie Early Stage Training Site DEASE (MEST-CT-2005-021122).The multiconfiguration methods are widely used by quantum physicists and chemists for numerical approximation of the many electron Schrödinger 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