# Aisenstadt Chair

##### [ Français ]

**Robert Seiringer
** (IST Austria)

Stay: September 10-14, 2018

## Lectures

All lectures will be held at the Pavillon André-Aisenstadt of the Université de Montréal.

**September 10, room 1140 à at 4:00 pm**

## Lecture I. The quantum many-body problem and Bose-Einstein condensation (Lecture suitable for a general scientific audience )

**Abstract**: A detailed understanding of the behaviour of many-particle systems in quantum mechanics poses a formidable challenge to mathematical physics. From the mathematical point of view, it requires a precise spectral analysis of the corresponding Hamiltonian operator. The recent experimental advances in cold-atom physics have led to a renewed interest in studying the quantum many-body problem. The aim of this lecture is to explain part of the progress that was made in the last decade or so. The topics covered include the question of existence of Bose-Einstein condensation, superfluidity and quantised vortices in systems in rotating traps, as well as recent investigations on the structure of the excitation spectrum for weakly interacting fluids. We will describe the mathematics involved in understanding these phenomena, starting from the underlying many-body Schrödinger equation.

**September 12, room 6254 at 4:00 pm**

## Lecture II. Bose-Einstein condensation in dilute, trapped gases at positive temperature

**Abstract;**: We consider an interacting, dilute Bose gas trapped in a harmonic potential
at positive temperature, in a combination of a
thermodynamic and a Gross-Pitaevskii (GP) limit. We prove that the
difference between the canonical free energy of the interacting gas and the one
of the noninteracting system can be obtained by minimizing the GP energy
functional. We also prove Bose-Einstein condensation, in the sense that
the one-particle density matrix of the interacting Gibbs state is to leading order given by that of the noninteracting
gas but with the free condensate wavefunction replaced by the GP minimizer.
(Joint work with A. Deuchert and J. Yngvason)

**September 13, room 6254 at 4:00 pm **

## Lecture III. Quantum many-body systems with point interactions

**Abstract**: We investigate the stability of quantum many-body systems with point interactions. In particular, we present a proof that a system of N fermions interacting with an additional particle via point interactions is stable if the ratio of the mass of the additional particle to the one of the fermions is larger than some critical value. For this impurity problem, we also show that the ground state energy of the system at given non-zero mean density differs from the one of the ideal gas by a term depending only on the density and the scattering length of the interactions, independently of N. While the general problem with more than one impurity remains open, we can show stability for the simplest such system, the one consisting of two fermions interacting with two (fermionic) impurities. (Joint work with T. Moser)