Description: 

In this course, I am going to discuss some recent progress in the mathematical analysis of many-body quantum systems. We will start with the Bogoliubov functional, a recently understood, rigorous version of Bogoliubov theory and we will describe how it can be used to approximate the ground state energy of an interacting, dilute Bose gas. We will also describe Bogoliubov’s original approach and how that also includes statements about the excitation spectrum.

The energy of the dilute Bose gas stands as one of the fundamental problems of many-body mathematical quantum mechanics. The paper of Dyson from 1957 is a defining achievement of mathematical physics. In recent decades, we have been able to rigorously establish the first terms in the energy expansion of the energy of the interacting system based on work by Lieb, Yngvason, Seiringer, Schlein, and many others. The result of these works is the proof of the celebrated Lee-Huang-Yang formula. In the course we will give an overview of the ideas and techniques required for these proofs giving precise estimates on the ground state energy of dilute Bose gases in the thermodynamic limit, at low density.

Litterature:

F. J. Dyson, Ground-state energy of a hard-sphere gas, Phys. Rev. 106.1 (1957), pp. 20–26.

S. Fournais and J. P. Solovej, The energy of dilute Bose gases, Annals of Mathematics 192.3 (2020), pp. 893–976.

S. Fournais and J. P. Solovej, The energy of dilute Bose gases II: the general case, Inventiones mathematicae 232.2 (2023), pp. 863–994.

Prerequisite: This is a graduate level course. We will mainly use standard results on functional analysis, PDE and spectral theory.

Target Audience: graduate students with an interest in mathematical physics.

Teaching language: English