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Quantum Depletion for Fixed Bounded Repulsive Potentials

OpenAI · October 5, 2026

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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.

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Cite

@misc{OAI:Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026,
  author = {{OpenAI}},
  title = {{Quantum Depletion for Fixed Bounded Repulsive Potentials}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026/fixed-repulsion-quantum-depletion.pdf}{OAI:Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026}},
  year = {2026}
}

Lean formalization

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Family 267: Positive-temperature Bose–Einstein condensation and exact quantum depletion

Proves Bose–Einstein condensation for the exact canonical Gibbs state of the three-dimensional hard-sphere gas: each fixed exclusion distance and sufficiently small fixed density admit a strictly positive temperature, independent of volume, with positive condensate fraction in the thermodynamic limit. At zero temperature, proves the Bogoliubov leading quantum-depletion law for hard spheres and fixed bounded nonnegative radial finite-range potentials of positive scattering length, taking the thermodynamic limit before the dilute limit.

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