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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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PDF:

• Carlen_Jauslin_Lieb_2019.pdf

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.

Numerical computations

The numerical computations in this paper were carried out using an early version of simplesolv. Note that this early version is not compatible with the version of simplesolv published on this website. The code is included with the LaTeX source, in the 'figs' directory.

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Talks

This work has been presented at the following conferences:

• [Ga22]: An effective equation to study Bose gases at all densities
  Advances in Classical, Quantum and Statistical Mechanics - A celebration of the work and contributions of Giovanni Gallavotti - On the occasion of his 80th birthday, Rome, Italy, May 13 2022
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• [Mi22]: An effective equation to study Bose gases at all densities
  University of Milan, Italy, Feb 21 2022
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• [Tu22]: An effective equation to study Bose gases at all densities
  University of Tubingen, Germany, Jan 03 2022
  pdf, source


• [LMU21]: An effective equation to study Bose gases at all densities
  Ludwig Maximilian University, Munich, Germany, Apr 28 2021
  pdf, source


• [Ye21]: An effective equation to study Bose gases at all densities
  Yeshiva University, New York, NY, USA, Apr 07 2021
  pdf, source


• [Ja21]: An effective equation to study Bose gases at all densities
  Jacobs University, Bremen, Germany, Apr 01 2021
  pdf, source


• [Pe21]: An effective equation to study Bose gases at all densities
  Penn State, University Park, Pennsylvania, USA, Mar 12 2021
  pdf, source


• [MC21]: Many interacting quantum particles: open problems, and a new point of view on an old problem
  Mathematical Conversations, Institute for Advanced Study, Princeton, NJ, USA, Mar 10 2021
  pdf, source


• [Cop21]: An effective equation to study Bose gasses at all densities
  University of Copenhagen, Denmark, Jan 13 2021
  pdf, source


• [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
  pdf, source


• [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