A simplified approach to interacting Bose gases
University of Toronto, Ontario, Canada
March 6, 2020
In 1963, Lieb introduced an effective theory to approximate the ground state energy of a system of Bosons interacting with each other via a repulsive pair potential, in the thermodynamic limit. Lieb showed that in one dimension, this effective theory predicts a ground state energy that differs at most by 20% from its exact value, for any density. The main idea is that instead of considering marginals of the square of the wave function, as in Hartree theory, we consider marginals of the wave function itself, which is positive in the ground state. The effective theory Lieb obtained is a nonlinear integrodifferential equation, whose nonlinearity is an autoconvolution. In this talk, I will discuss some recent work about this effective equation. In particular, we proved the existence of a solution. We also proved that the ground state energy obtained from this simplified equation agrees exactly with that of the full Nbody system at asymptotically low and at high densities. In fact, preliminary numerical work has shown that, for some potentials, the ground state energy can be computed in this way with an error of at most 5% over the entire range of densities. This is joint work with E. Carlen and E.H. Lieb.
Slides
PDF:
LaTeX source:
 tarball: 20toronto1.0.tar.gz
 git repository: 20torontogit (the git repository contains detailed information about the changes in the slides as well as the source code for all previous versions).
References
This presentation is based on

[CJL19]: Analysis of a simple equation for the ground state energy of the Bose gas
Eric Carlen, Ian Jauslin, Elliott H. Lieb, 2019
(published in Pure and Applied Analysis, volume 2, issue 3, pages 659684)
pdf, source 
[CJLL20]: On the convolution inequality f>f*f
Eric A. Carlen, Ian Jauslin, Elliott H. Lieb, Michael Loss, 2020
(published in International Mathematics Research Notices, Volume 2021, Issue 24, pages 1860418612, 2021)
pdf, source