Mean-field evolution of fermions with singular interaction

We consider a system of N fermions in the mean-field regime interacting though an inverse power law potential $V(x)=1/|x|^{\alpha}$, for $\alpha\in(0,1]$. We prove the convergence of a solution of the many-body Schr\"{o}dinger equation to a solution of the time-dependent Hartree-Fock equation in the sense of reduced density matrices. We stress the dependence on the singularity of the potential in the regularity of the initial data. The proof is an adaptation of [22], where the case $\alpha=1$ is treated.Comment: 16 page

Remarks on the derivation of Gross-Pitaevskii equation with magnetic Laplacian

The effective dynamics for a Bose-Einstein condensate in the regime of high dilution and subject to an external magnetic field is governed by a magnetic Gross-Pitaevskii equation. We elucidate the steps needed to adapt to the magnetic case the proof of the derivation of the Gross-Pitaevskii equation within the "projection counting" scheme

Mean-field dynamics of fermions with relativistic dispersion

We extend the derivation of the time-dependent Hartree-Fock equation recently obtained by Benedikter et al. ["Mean-field evolution of fermionic systems," Commun. Math. Phys. (to be published)] to fermions with a relativistic dispersion law. The main new ingredient is the propagation of semiclassical commutator bounds along the pseudo-relativistic Hartree-Fock evolution. (C) 2014 AIP Publishing LLC

Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime

While Hartree\u2013Fock theory is well established as a fundamental approximation for interacting fermions, it has been unclear how to describe corrections to it due to many-body correlations. In this paper we start from the Hartree\u2013Fock state given by plane waves and introduce collective particle\u2013hole pair excitations. These pairs can be approximately described by a bosonic quadratic Hamiltonian. We use Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann\u2013Brueckner\u2013type upper bound to the ground state energy. Our result justifies the random-phase approximation in the mean-field scaling regime, for repulsive, regular interaction potentials

Effective non-linear dynamics of binary condensates and open problems

We report on a recent result concerning the effective dynamics for a mixture of Bose-Einstein condensates, a class of systems much studied in physics and receiving a large amount of attention in the recent literature in mathematical physics; for such models, the effective dynamics is described by a coupled system of non-linear Sch\"odinger equations. After reviewing and commenting our proof in the mean field regime from a previous paper, we collect the main details needed to obtain the rigorous derivation of the effective dynamics in the Gross-Pitaevskii scaling limit.Comment: Corrected typos, updated reference

Derivation of renormalized Gibbs measures from equilibrium many-body quantum Bose gases

We review our recent result on the rigorous derivation of the renormalized Gibbs measure from the many-body Gibbs state in 1D and 2D. The many-body renormalization is accomplished by simply tuning the chemical potential in the grand-canonical ensemble, which is analogous to the Wick ordering in the classical field theory.Comment: Contribution to Proceedings of the International Congress of Mathematical Physics, Montreal, Canada, July 23-28, 201