Upper Bound for the Ground State Energy of a Dilute Bose Gas of Hard Spheres.

We consider a gas of bosons interacting through a hard-sphere potential with radius in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term and shows that corrections are smaller than , for a sufficiently large constant .

Correlation Energy of a Weakly Interacting Fermi Gas with Large Interaction Potential.

Recently the leading order of the correlation energy of a Fermi gas in a coupled mean-field and semiclassical scaling regime has been derived, under the assumption of an interaction potential with a small norm and with compact support in Fourier space. We generalize this result to large interaction potentials, requiring only . Our proof is based on approximate, collective bosonization in three dimensions.

The Excitation Spectrum of Two-Dimensional Bose Gases in the Gross-Pitaevskii Regime.

We consider a system of bosons, in the two-dimensional unit torus. We assume particles to interact through a repulsive two-body potential, with a scattering length that is exponentially small in (Gross-Pitaevskii regime). In this setting, we establish the validity of the predictions of Bogoliubov theory, determining the ground state energy of the Hamilton operator and its low-energy excitation spectrum, up to errors that vanish in the limit .

Effective Dynamics of Extended Fermi Gases in the High-Density Regime.

We study the quantum evolution of many-body Fermi gases in three dimensions, in arbitrarily large domains. We consider both particles with non-relativistic and with relativistic dispersion. We focus on the high-density regime, in the semiclassical scaling, and we consider a class of initial data describing zero-temperature states.

A Large Deviation Principle in Many-Body Quantum Dynamics.

We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates that are consistent with central limit theorems that have been established in the last years.

Bose-Einstein Condensation for Two Dimensional Bosons in the Gross-Pitaevskii Regime.

We consider systems of N bosons trapped on the two-dimensional unit torus, in the Gross-Pitaevskii regime, where the scattering length of the repulsive interaction is exponentially small in the number of particles. We show that low-energy states exhibit complete Bose-Einstein condensation, with almost optimal bounds on the number of orthogonal excitations.

Bose-Einstein Condensation Beyond the Gross-Pitaevskii Regime.

We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order , for . Assuming that , we show that low-energy states exhibit Bose-Einstein condensation and we provide bounds on the expectation and on higher moments of the number of excitations.

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

While Hartree-Fock 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-Fock state given by plane waves and introduce collective particle-hole pair excitations. These pairs can be approximately described by a bosonic quadratic Hamiltonian.

Rigorous derivation of the Gross-Pitaevskii equation.

The time-dependent Gross-Pitaevskii equation describes the dynamics of initially trapped Bose-Einstein condensates. We present a rigorous proof of this fact starting from a many-body bosonic Schrödinger equation with a short-scale repulsive interaction in the dilute limit. Our proof shows the persistence of an explicit short-scale correlation structure in the condensate.