In these notes we review the material presented at the summer school on “Mathematical
Physics, Analysis and Stochastics” held at the University of Heidelberg in July 2014. We …

Starting from first‐principle many‐body quantum dynamics, we show that the dynamics of
Bose‐Einstein condensates can be approximated by the time‐dependent nonlinear Gross …

We consider the many-body quantum dynamics of systems of bosons interacting through a
two-body potential N^ 3 β-1 V (N^ β x) N 3 β-1 V (N β x), scaling with the number of particles …

In this paper we study the dynamics of fermionic mixed states in the mean‐field regime. We
consider initial states that are close to quasi‐free states and prove that, under suitable …

We consider a system of N interacting bosons in the mean-field scaling regime and construct
corrections to the Bogoliubov dynamics that approximate the true N-body dynamics in norm …

We consider a system of N bosons interacting through a singular two-body potential scaling
with N and having the form N 3 β− 1 V (N β x), for an arbitrary parameter β∈(0, 1). We …

We prove that Gibbs measures of nonlinear Schrödinger equations arise as high-
temperature limits of thermal states in many-body quantum mechanics. Our results hold for …

We consider the evolution of a gas of N bosons in the three-dimensional Gross–Pitaevskii
regime (in which particles are initially trapped in a volume of order one and interact through …

We consider the ground state and the low-energy excited states of a system of N identical
bosons with interactions in the mean-field scaling regime. For the ground state, we derive a …

The success of the Gross–Pitaevskii and Bogoliubov theories in the description of large
systems of interacting bosons led to a substantial effort into rigorously deriving these …