Quantum Depletion for Fixed Bounded Repulsive Potentials

OpenAI

Published Oct 6, 2026• 0 citations

Abstract

We prove the Bogoliubov quantum-depletion asymptotic for the ground state of a three-dimensional Bose gas with a fixed bounded, nonnegative, radial interaction of finite range and positive scattering length a. The thermodynamic limit is taken at each fixed density ρ before the dilute limit. Every thermodynamic accumulation value of the fraction outside the constant mode is \frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3}) as \rho\downarrow0. The assertion is uniform over ground-state density matrices and does not require the occupation to have a unique thermodynamic limit.

DOI: 10.83295/6ppw-k9zr

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Quantum Depletion for Fixed Bounded Repulsive Potentials

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Published Oct 6, 2026• 0 citations

Publisher

Abstract

DOI: 10.83295/6ppw-k9zr

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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.