arXiv:2609.25314

Lee-Huang-Yang Energy for Dilute Bose Gases: an Upper Bound for General Potentials

Giulia Basti, Morris Brooks, Serena Cenatiempo, Alessandro Olgiati, Benjamin Schlein

Published Sep 21, 2026• 0 citations

Abstract

We establish an upper bound for the ground state energy density of a dilute Bose gas, interacting through a general short-range, repulsive potential. To reach this goal, we extend the approach recently introduced in [1] for hard spheres.

DOI: 10.48550/arxiv.2609.25314

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arXiv:2609.25314

Lee-Huang-Yang Energy for Dilute Bose Gases: an Upper Bound for General Potentials

Giulia Basti, Morris Brooks, Serena Cenatiempo, Alessandro Olgiati, Benjamin Schlein

Published Sep 21, 2026• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2609.25314

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

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

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 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.

For a dilute system of non-relativistic bosons interacting through a positive

L^1

potential

v

with scattering length

a

we prove that the ground state energy density satisfies the bound

e(ρ) \geq 4πa ρ^2 (1+ \frac{128}{15\sqrtπ} \sqrt{ρa^3} +o(\sqrt{ρa^3}\,))

, thereby proving the Lee-Huang-Yang formula for the energy density.