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The Bogoliubov momentum law for the dilute hard-sphere ground state

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For fixed hard-sphere radius $a>0$, do all pure or mixed bosonic ground states in periodic cubes have a common dilute-limit nonzero-momentum occupation law: after the thermodynamic limit at fixed rho and then rho $a^3\to 0$, the measure (N sqrt(rho $a^3))^{-1}sum_{k\neq 0}n_\Gamma (k)\delta _(k/\operatorname{sqrt}(8\pi $ rho a)) converges against every bounded continuous real test to the source Bogoliubov measure? Require uniformity over the whole ground-state space.

Short English statement

For fixed hard-sphere radius $a>0$, do all pure or mixed bosonic ground states in periodic cubes have a common dilute-limit nonzero-momentum occupation law: after the thermodynamic limit at fixed rho and then rho $a^3\to 0$, the measure (N sqrt(rho $a^3))^{-1}sum_{k\neq 0}n_\Gamma (k)\delta _(k/\operatorname{sqrt}(8\pi $ rho a)) converges against every bounded continuous real test to the source Bogoliubov measure? Require uniformity over the whole ground-state space.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-61-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-61

The hard-sphere ground state is claimed to have the full scaled Bogoliubov momentum law, with depletion $\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})$. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The Bogoliubov momentum law for the dilute hard-sphere ground state

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The hard-sphere ground state is claimed to have the full scaled Bogoliubov momentum law, with depletion $\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})$.

Mathematical statement of the claimed result

Result 1.
For the three-dimensional periodic hard-sphere Bose gas of exclusion distance $a>0$, let $\mathcal G_{N,L,a}$ be all trace-one states supported on the lowest eigenspace, and $n_{\Gamma,L}(k)$ the occupation of $L^{-3/2}e^{ik\cdot x}$. Define \[ g(t)=\frac{|t|^2+1}{2\sqrt{|t|^4+2|t|^2}}-\frac12,\qquad d\nu_{\mathrm{Bog}}(t)=\frac{(8\pi)^{3/2}}{(2\pi)^3}g(t)\,dt. \] For each bounded continuous real $f$ and $\varepsilon>0$, the source gives $\rho_0(a,f,\varepsilon)>0$. If $0<\rho<\rho_0$, $\eta=\rho a^3$, and $N_j,L_j\to\infty$, $N_j/L_j^3\to\rho$, then \[ \limsup_j\sup_{\Gamma\in\mathcal G_{N_j,L_j,a}} \left|\frac{\sum_{k\in(2\pi/L_j)\mathbb Z^3,\ k\ne0} f(k/\sqrt{8\pi\rho a})n_{\Gamma,L_j}(k)}{N_j\sqrt\eta} -\int f\,d\nu_{\mathrm{Bog}}\right|\le\varepsilon. \] The measure has total mass $8/(3\sqrt\pi)$.

Outline of the argument

Local energy estimates keep excitations instead of discarding them. One-body covariance and short-trajectory certificates are compared through smoothed deletion measures, while random encounter routes connect local momentum data to the thermodynamic occupation measure before the dilute limit.

Formal verification

For family 267, the selected statement is a density-uniform hard-sphere ground-state condensate lower bound for pure and mixed ground states. Positive-temperature condensation and quantum-depletion asymptotics are beyond it. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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