Seminar

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In this talk we consider the Hartree energy functional for an ensemble of identical fermions. We prove that the minimizers of this functional satisfy an important set of semi-classical commutator estimates (SCEs), which encode the uniform regularity of the states in the semi-classical parameter. In the recent years, the SCEs have been shown to play a key role in the quantitative derivation of Hartree-Fock and Vlasov dynamics from large systems of fermions, and are typically implemented as assumptions in the initial data. Proving the validity of such estimates is, however, not an easy task; up to recently only the linear smooth case was understood. In our work, we provide the first set of examples of states satisfying the SCEs which arise from a nonlinear minimization problem. In particular,  two-body interactions up to the repulsive Coulomb potential are included.  I will present — as an application — the quantitative convergence of the fermonic N-body ground state, towards the minimizer of the Vlasov functional.  Based on joint work with L. Lafleche (ENS Lyon).

Semi-classical commutator estimates in Hartree theory