Analysis of a simple equation for the ground state energy of the Bose gas

Eric Carlen, Ian Jauslin, Elliott H. Lieb

2019

Abstract

In 1963 a partial differential equation with a convolution non-linearity was introduced in connection with a quantum mechanical many-body problem, namely the gas of bosonic particles. This equation is mathematically interesting for several reasons. (1) Although the equation was expected to be valid only for small values of the parameters, further investigation showed that predictions based on the equation agree well over the entire range of parameters with what is expected to be true for the solution of the true many-body problem. (2) The novel nonlinearity is easy to state but seems to have almost no literature up to now. (3) The earlier work did not prove existence and uniqueness of a solution, which we provide here along with properties of the solution such as decay at infinity.

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LaTeX source:

• tarball: 19cjl-1.0.1.tar.gz
• git repository: 19cjl-git (the git repository contains detailed information about the changes in the paper as well as the source code for all previous versions).

Other releases

• arXiv preprint: arXiv:1912.04987.
• This article was peer reviewed for: Pure and Applied Analysis, Volume 2, Issue 3, pages 659-684, 2020.

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Talks

This work has been presented at the following conferences:

• [Rut20]: Analysis of a non-linear, non-local PDE to study Bose gases at all densities
  Rutgers University, New Brunswick, New Jersey, USA, Dec 16 2020
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• [TAMU20b]: A new approach to the Mathematics of the Bose gas
  Texas A&M, College Station, Texas, USA, Nov 30 2020
  pdf, source


• [SIAM20]: An Effective Equation To Study Bose Gasses At All Densities
  SIAM Texas-Louisiana Sectionnal Meeting, Mini-Symposium on Spectral Theory and Mathematical Physics, Oct 17 2020
  pdf, source


• [IAMP20]: An effective equation to study Bose gasses at all densities
  International Association of Mathematical of Mathematical Physics, One World Seminars, Sep 22 2020
  pdf, source


• [TAMU20]: A simple equation to study interacting Bose gasses
  Texas A&M, College Station, Texas, USA, May 15 2020
  pdf, source


• [HaRM20]: A simplified approach to interacting Bose gases
  Harvard University, Cambridge, Massachusetts, USA, Apr 01 2020
  pdf, source


• [To20]: A simplified approach to interacting Bose gases
  University of Toronto, Ontario, Canada, Mar 06 2020
  pdf, source


• [Pr20]: A simplified approach to interacting Bose gases
  Princeton University, New Jersey, USA, Mar 03 2020
  video, pdf, source


• [VATech20]: A simplified approach to interacting Bose gases
  VirginiaTech, Blacksburg, Virginia, USA, Feb 14 2020
  pdf, source


• [Davis20]: A simplified approach to interacting Bose gases
  UC Davis, California, USA, Feb 06 2020
  pdf, source


• [GaTech20]: A simplified approach to interacting Bose gases
  GeorgiaTech, Atlanta, Georgia, USA, Jan 21 2020
  pdf, source


• [UBC20]: A simplified approach to interacting Bose gases
  University of British Columbia, Vancouver, British Columbia, Canada, Jan 09 2020
  pdf, source


• [GLaMP19]: Lieb's simplified approach to interacting Bose gases
  Great Lakes Mathematical Physics Meeting 2019, Oberlin, Ohio, USA, Jun 29 2019
  pdf, source