Lieb's simplified approach to interacting Bose gases
Great Lakes Mathematical Physics Meeting 2019, Oberlin, Ohio, USA
June 29, 2019
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. In this talk, I will discuss some recent work about this effective equation, in which we proved that the ground state energy obtained from it agrees exactly with the asymptotic prediction of Lee, Huang and Yang for the full Nbody system at low densities, and that, for potentials whose Fourier transform is positive, the effective theory produces the correct asymptotics for large densities as well. This is joint work with E. Carlen and E.H. Lieb.
Slides
PDF:
LaTeX source:
 tarball: 19glamp1.0.tar.gz
 git repository: 19glampgit (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)
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