Mathematics of Bose-Einstein Condensation

Introduction

We propose a project in mathematics with a focus on many-body theory in mathematical physics. We are especially interested in the mathematical tools involved in the description and analysis of the recent experimental realizations of Bose-Einstein Condensation. It remains one of the most important challenges of mathematical physics to rigorously understand the formation of condensates in interacting systems. This project aims to address that challenge.

Objectives

Progress on the problem of condensation has been made on certain length scales, and we aim to push the boundaries of these lengths with a view towards the end goal of actually having a mathematical proof of condensation in a continuum system of interacting quantum particles in the thermodynamic limit.

Approach

To approach this objective, we will study various related systems and problems with the expectation of getting improved understanding by seeing the methods in a new light.

Development of Tools

To fully solve these simpler problems, it will require the development of new mathematical tools and the gain of critical insight. Some of these simplified problems are concerned with:

1. The energy of the Bose gas in the dilute limit.
2. Systems in dimensions different from 3.
3. LHY-physics specially prepared systems where the normally lower order correction terms become dominant.