arXiv:2506.04153 · Communications in Mathematical Physics

Third Order Upper Bound for the Ground State Energy of the Dilute Bose Gas

Morris Brooks, Jakob Oldenburg, Diane Saint Aubin, Benjamin Schlein

Published Jul 9, 2026• 0 citations

Abstract

We consider a dilute Bose gas in the thermodynamic limit. We prove an upper bound for the ground state energy per unit volume, capturing the expected third order term, as predicted by Wu, Hugenholtz-Pines and Sawada.

DOI: 10.1007/s00220-026-05656-8

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arXiv:2506.04153 · Communications in Mathematical Physics

Third Order Upper Bound for the Ground State Energy of the Dilute Bose Gas

Morris Brooks, Jakob Oldenburg, Diane Saint Aubin, Benjamin Schlein

Published Jul 9, 2026• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.1007/s00220-026-05656-8

Related papers

A new second-order upper bound for the ground state energy of dilute Bose gases

Giulia Basti, Serena Cenatiempo, Benjamin Schlein

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arXiv:2101.06222Forum of Mathematics SigmaJan 1, 202144 citations

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arXiv:2203.01841Journal of Mathematical PhysicsJul 1, 20224 citations

V \in L^3 (\mathbb {R}^3)

, gives better rates and, in our opinion, is substantially simpler.

We consider a gas of bosons interacting through a hard-sphere potential with radius

\frak{a}

in the thermodynamic limit. We derive a simple upper bound for the ground state energy per particle at low density. Our bound captures the leading term

4πρ\frak{a}

and shows that corrections are smaller than

C ρ\frak{a} (ρ\frak{a}^3)^{1/2}

, for a sufficiently large constant

C > 0

. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy is, in fact, of the order

ρ\frak{a} (ρ\frak{a}^3)^{1/2}

, in agreement with the Lee-Huang-Yang prediction.

Abstract We consider a gas of bosons interacting through a hard-sphere potential with radius

\mathfrak {a}

a in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term

4\pi \rho \mathfrak {a}

4 π ρ a and shows that corrections are smaller than

C \rho \mathfrak {a} (\rho {{\mathfrak {a}}}^3)^{1/2}

C ρ a ( ρ a 3 ) 1 / 2 , for a sufficiently large constant

C > 0

C > 0 . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order

\rho \mathfrak {a}(\rho {{\mathfrak {a}}}^3)^{1/2}

ρ a ( ρ a 3 ) 1 / 2 , in agreement with the Lee–Huang–Yang prediction.

We consider a Bose gas in spatial dimension

n>3

with a repulsive, radially symmetric two-body potential

V

. In the limit of low density

ρ

, the ground state energy per particle in the thermodynamic limit is shown to be

(n-2)|\mathbb S^{n-1}|a^{n-2}ρ

, where

|\mathbb S^{n-1}|

denotes the surface measure of the unit sphere in

\mathbb{R}^n

and

a

is the scattering length of

V

. Furthermore, for smooth and compactly supported two-body potentials, we derive upper bounds to the ground state energy with a correction term

(1+Cγ)8π^4a^6ρ^2|\ln(a^4ρ)|

in dimension

n=4

, where

γ:=\int V(x)|x|^{-2}\, dx

, and a correction term which is

\mathcal{O}(ρ^2)

in higher dimensions.