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.
Download
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.
Related articles
-
[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 18604-18612, 2021)
pdf, source -
[CJL20]: Analysis of a simple equation for the ground state of the Bose gas II: Monotonicity, Convexity and Condensate Fraction
Eric A. Carlen, Ian Jauslin, Elliott H. Lieb, 2020
(published in SIAM Journal on Mathematical Analysis, Volume 53, Number 5, pages 5322-5360, 2021)
pdf, source -
[Ja23b]: Evidence of a liquid phase in interacting Bosons at intermediate densities
Ian Jauslin, 2023
pdf, source -
[Ja23]: The Simplified approach to the Bose gas without translation invariance
Ian Jauslin, 2023
pdf, source -
[Ja22]: Review of a Simplified Approach to study the Bose gas at all densities
Ian Jauslin, 2022
(published in The Physics and Mathematics of Elliott Lieb, The 90th anniversary Volume 1, pages 609-635)
pdf, source -
[Ja23c]: Bose-Einstein condensation and the Simplified Approach to interacting Bosons
Ian Jauslin, 2023
pdf, source -
[CHJL20]: A simplified approach to the repulsive Bose gas from low to high densities and its numerical accuracy
Eric A. Carlen, Markus Holzmann, Ian Jauslin, Elliott H. Lieb, 2020
(published in Physical Review A, volume 103, number 053309, 2021)
pdf, source
Talks
This work has been presented at the following conferences:
-
[Wa24]: Non-perturbative behavior of interacting Bosons at intermediate densities
University of Warsaw, Poland, Jan 11 2024
pdf -
[SMM125]: Non-perturbative behavior of interacting Bosons at intermediate densities
125th Statistical Mechanics Meeting, Rutgers University, New Jersey, USA, Dec 19 2023
pdf -
[CMT23]: Non-perturbative behavior of interacting Bosons at intermediate densities
Condensed Matter Theory Fall 2023 Symposium, Rutgers University, New Jersey, USA, Sep 29 2023
pdf -
[TXST23]: Bose-Einstein condensation and the Simplified Approach to interacting Bosons
Texas State University, San Marcos, Texas, USA, Jul 24 2023
-
[Da23]: Non-perturbative behavior of interacting Bosons at intermediate densities
UC Davis, California, USA, May 30 2023
pdf -
[Pr23]: Non-perturbative behavior of interacting Bosons at intermediate densities
Princeton University, New Jersey, USA, Apr 18 2023
pdf -
[SISSA23]: Interacting Bosons at intermediate densities
SISSA, Trieste, Italy, Mar 16 2023
pdf -
[Ru23]: Interacting Bosons at intermediate densities
Rutgers University, New Brunswick, New Jersey, USA, Mar 09 2023
pdf -
[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
pdf -
[Mi22]: An effective equation to study Bose gases at all densities
University of Milan, Italy, Feb 21 2022
pdf -
[Tu22]: An effective equation to study Bose gases at all densities
University of Tubingen, Germany, Jan 03 2022
pdf -
[LMU21]: An effective equation to study Bose gases at all densities
Ludwig Maximilian University, Munich, Germany, Apr 28 2021
pdf -
[Ye21]: An effective equation to study Bose gases at all densities
Yeshiva University, New York, NY, USA, Apr 07 2021
pdf -
[Ja21]: An effective equation to study Bose gases at all densities
Jacobs University, Bremen, Germany, Apr 01 2021
pdf -
[Pe21]: An effective equation to study Bose gases at all densities
Penn State, University Park, Pennsylvania, USA, Mar 12 2021
pdf -
[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 -
[Cop21]: An effective equation to study Bose gasses at all densities
University of Copenhagen, Denmark, Jan 13 2021
pdf -
[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 -
[TAMU20b]: A new approach to the Mathematics of the Bose gas
Texas A&M, College Station, Texas, USA, Nov 30 2020
pdf -
[SIAM20]: An Effective Equation To Study Bose Gasses At All Densities
SIAM Texas-Louisiana Sectionnal Meeting, Mini-Symposium on Spectral Theory and Mathematical Physics, online, Oct 17 2020
pdf -
[IAMP20]: An effective equation to study Bose gasses at all densities
International Association of Mathematical of Mathematical Physics, One World Seminars, online, Sep 22 2020
pdf -
[TAMU20]: A simple equation to study interacting Bose gasses
Texas A&M, College Station, Texas, USA, May 15 2020
pdf -
[HaRM20]: A simplified approach to interacting Bose gases
Harvard University, Cambridge, Massachusetts, USA, Apr 01 2020
pdf -
[To20]: A simplified approach to interacting Bose gases
University of Toronto, Ontario, Canada, Mar 06 2020
pdf -
[Pr20]: A simplified approach to interacting Bose gases
Princeton University, New Jersey, USA, Mar 03 2020
video, pdf -
[VATech20]: A simplified approach to interacting Bose gases
VirginiaTech, Blacksburg, Virginia, USA, Feb 14 2020
pdf -
[Davis20]: A simplified approach to interacting Bose gases
UC Davis, California, USA, Feb 06 2020
pdf -
[GaTech20]: A simplified approach to interacting Bose gases
GeorgiaTech, Atlanta, Georgia, USA, Jan 21 2020
pdf -
[UBC20]: A simplified approach to interacting Bose gases
University of British Columbia, Vancouver, British Columbia, Canada, Jan 09 2020
pdf -
[GLaMP19]: Lieb's simplified approach to interacting Bose gases
Great Lakes Mathematical Physics Meeting 2019, Oberlin, Ohio, USA, Jun 29 2019
pdf