The logarithmic correction to dilute Bose-gas energy
Let $V \in C_c^\infty(\mathbb{R}^3)$ be nonzero, nonnegative and radial, and let $a > 0$ be its scattering length: the solution of $(-\Delta + V/2) f = 0$ with $f(x) \to 1$ as $|x| \to \infty$ satisfies $f(x) = 1 - a/|x|$ outside the support of $V$. Let $E(N, L)$ be the ground-state energy of $N$ bosons in a periodic cube of side $L$, with kinetic energy $\sum_j -\Delta_j$ and pair potential the periodic extension of $V$, and let $$e(\rho) = \lim_{N, L \to \infty,\ N/L^3 \to \rho} \frac{E(N, L)}{L^3}.$$ The first two terms of the expansion below were predicted by Lee, Huang and Yang, and the logarithmic term by Wu, and then also by Hugenholtz and Pines and by Sawada. The leading term $4\pi a \rho^2$ was proved by Dyson (upper bound, for hard spheres) and Lieb and Yngvason (lower bound); the second-order term by Yau and Yin and by Basti, Cenatiempo and Schlein (upper bounds) and by Fournais and Solovej (lower bounds). Brooks, Oldenburg, Saint Aubin and Schlein proved the upper bound $$e(\rho) \le 4\pi a \rho^2 \Bigl[1 + \frac{128}{15\sqrt{\pi}} (\rho a^3)^{1/2} + 8\Bigl(\frac{4\pi}{3} - \sqrt{3}\Bigr) \rho a^3 \log(\rho a^3) + C \rho a^3\Bigr]$$ for some constant $C$ and all sufficiently small $\rho a^3$, with $e(\rho)$ defined there in boxes with Dirichlet boundary conditions. In the Gross–Pitaevskii regime the corresponding logarithmic term was derived by Caraci, Olgiati, Saint Aubin and Schlein.
Problem. Is it true that, as $\rho \downarrow 0$ with $V$ fixed, $$e(\rho) = 4\pi a \rho^2 \Bigl[1 + \frac{128}{15\sqrt{\pi}} (\rho a^3)^{1/2} + 8\Bigl(\frac{4\pi}{3} - \sqrt{3}\Bigr) \rho a^3 \log(\rho a^3) + o\bigl(\rho a^3 \lvert \log(\rho a^3) \rvert\bigr)\Bigr]?$$
Source: Wu (1959), and independently Hugenholtz and Pines (1959) and Sawada (1959), as a prediction; posed as the matching lower bound by Brooks, Oldenburg, Saint Aubin and Schlein (2026), after their Theorem 1. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MP — Mathematical Physics
- Secondary
- SP
- Keywords
- AIM Open Applied Problems, dilute Bose gases, ground-state energies
References
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}
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}
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}
@article{brooks2026,
author = {Brooks, Morris and Oldenburg, Jakob and Saint Aubin, Diane and Schlein, Benjamin},
title = {Third order upper bound for the ground state energy of the dilute {B}ose gas},
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pages = {163},
doi = {10.1007/s00220-026-05656-8},
eprint = {2506.04153},
archivePrefix = {arXiv},
}
@article{lee1957,
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@article{basti2021,
author = {Basti, Giulia and Cenatiempo, Serena and Schlein, Benjamin},
title = {A new second-order upper bound for the ground state energy of dilute {B}ose gases},
journal = {Forum Math. Sigma},
volume = {9},
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pages = {e74},
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}
@article{fournais2020,
author = {Fournais, S{\o}ren and Solovej, Jan Philip},
title = {The energy of dilute {B}ose gases},
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}
@article{fournais2023,
author = {Fournais, S{\o}ren and Solovej, Jan Philip},
title = {The energy of dilute {B}ose gases {II}: the general case},
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volume = {232},
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}
@article{caraci2025,
author = {Caraci, Cristina and Olgiati, Alessandro and Saint Aubin, Diane and Schlein, Benjamin},
title = {Third order corrections to the ground state energy of a {B}ose gas in the {G}ross--{P}itaevskii regime},
journal = {Comm. Math. Phys.},
volume = {406},
number = {7},
year = {2025},
pages = {153},
doi = {10.1007/s00220-025-05320-7},
eprint = {2311.07433},
archivePrefix = {arXiv},
}
@misc{aim-wu-bose-energy,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The logarithmic correction to dilute Bose-gas energy}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/197-wu-bose-energy.md},
note = {Entry 197 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00489, "The logarithmic correction to dilute Bose-gas energy", clopen.solutions.