Ian Jauslin

## Analysis of a simple equation for the ground state of the Bose gas II: Monotonicity, Convexity and Condensate Fraction

2020

### Abstract

In a recent paper we studied an equation (called the "simple equation") introduced by one of us in 1963 for an approximate correlation function associated to the ground state of an interacting Bose gas. Solving the equation yields a relation between the density $\rho$ of the gas and the energy per particle. Our construction of solutions gave a well-defined function $\rho(e)$ for the density as a function of the energy $e$. We had conjectured that $\rho(e)$ is a strictly monotone increasing function, so that it can be inverted to yield the strictly monotone increasing function $e(\rho)$. We had also conjectured that $\rho e(\rho)$ is convex as a function of $\rho$. We prove both conjectures here for small densities, the context in which they have the most physical relevance, and the monotonicity also for large densities. Both conjectures are grounded in the underlying physics, and their proof provides further mathematical evidence for the validity of the assumptions underlying the derivation of the simple equation, at least for low or high densities, if not intermediate densities, although the equation gives surprisingly good predictions for all densities $\rho$. Another problem left open in our previous paper was whether the simple equation could be used to compute accurate predictions of observables other than the energy. Here, we provide a recipe for computing predictions for any one- or two-particle observables for the ground state of the Bose gas. We focus on the condensate fraction and the momentum distribution, and show that they have the same low density asymptotic behavior as that predicted for the Bose gas. Along with the computation of the low density energy of the simple equation in our previous paper, this shows that the simple equation reproduces the known and conjectured properties of the Bose gas at low densities.

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LaTeX source:
• tarball: 20cjl-0.2.tar.gz
• git repository: 20cjl-git (the git repository contains detailed information about the changes in the paper as well as the source code for all previous versions).

### Talks

This work has been presented at the following conferences:

• [LMU21]: An effective equation to study Bose gases at all densities
Ludwig Maximilian University, Munich, Germany, Apr 28 2021
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• [Ye21]: An effective equation to study Bose gases at all densities
Yeshiva University, New York, NY, USA, Apr 07 2021
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• [Ja21]: An effective equation to study Bose gases at all densities
Jacobs University, Bremen, Germany, Apr 01 2021
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• [Pe21]: An effective equation to study Bose gases at all densities
Penn State, University Park, Pennsylvania, USA, Mar 12 2021
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• [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
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• [Cop21]: An effective equation to study Bose gasses at all densities
University of Copenhagen, Denmark, Jan 13 2021
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• [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
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• [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

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