The Lee-Huang-Yang Formula in the Hard-Core Simplified Approach to the Bose gas
Abstract
We study the hard-core variant of the Simplified Approach to the ground state of the dilute Bose gas. The hard-core potential has been the subject of renewed attention, and is physically relevant to realistic atomic Bose gases. The hard-core Simple Equation is an integro-differential equation, and we construct its solution via a monotone iteration scheme and prove existence and uniqueness (in a restricted sense) of the solution. The energy per particle e and the density ρ of the solution are shown to satisfy the Lee-Huang-Yang formula e = 2πρ(1 + 128/15√π√ρ + o(√ρ)) as ρ→ 0, and the condensate fraction η predicted by the hard-core Simple Equation is shown to satisfy η= 1 − 8/3√π√ρ + o(√ρ) as ρ→ 0, in agreement with Bogolyubov's theory. We thus extend results of Carlen, Jauslin, Lieb to the hard-core case. The hard-core equation is defined on its own terms, and the proofs proceed by approximating the hard-core potential with the soft-core potentials m χ{|x| ≤ 1} of increasing height m: at fixed energy, we establish the L1 convergence of the corresponding solutions and the convergence of their densities at an O(m−1/2) rate, and we show that the Lee-Huang-Yang expansions of the energy and of the condensate fraction for the soft-core equations hold with errors uniform in m, so that they survive the hard-core limit.
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BibTeX
@misc{bahri2026leehuangyang,
title = {{The Lee-Huang-Yang Formula in the Hard-Core Simplified Approach to the Bose gas}},
author = {Mohamed El Hedi Bahri and Ian Jauslin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05293},
eprint = {2611.05293},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05293},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Lee-Huang-Yang Formula in the Hard-Core Simplified Approach to the Bose gas
AU - Mohamed El Hedi Bahri
AU - Ian Jauslin
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05293
UR - https://arxiv.org/abs/2611.05293
AB - We study the hard-core variant of the Simplified Approach to the ground state of the dilute Bose gas. The hard-core potential has been the subject of renewed attention, and is physically relevant to realistic atomic Bose gases. The hard-core Simple Equation is an integro-differential equation, and we construct its solution via a monotone iteration scheme and prove existence and uniqueness (in a restricted sense) of the solution. The energy per particle e and the density ρ of the solution are shown to satisfy the Lee-Huang-Yang formula e = 2πρ(1 + 128/15√π√ρ + o(√ρ)) as ρ→ 0, and the condensate fraction η predicted by the hard-core Simple Equation is shown to satisfy η= 1 − 8/3√π√ρ + o(√ρ) as ρ→ 0, in agreement with Bogolyubov's theory. We thus extend results of Carlen, Jauslin, Lieb to the hard-core case. The hard-core equation is defined on its own terms, and the proofs proceed by approximating the hard-core potential with the soft-core potentials m χ_{|x| ≤ 1} of increasing height m: at fixed energy, we establish the L^1 convergence of the corresponding solutions and the convergence of their densities at an O(m^−1/2) rate, and we show that the Lee-Huang-Yang expansions of the energy and of the condensate fraction for the soft-core equations hold with errors uniform in m, so that they survive the hard-core limit.
ER -