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Phys-58 · physics · mathematics · Physics

The universal quantum-depletion constant for fixed bounded repulsions

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every fixed bounded nonzero nonnegative radial finite-range pair potential v in $R^3$ with scattering length $a_v$, do all periodic-cube ground states satisfy $(1-B_\Gamma )/\operatorname{sqrt}(\rho $ $a_v^3)\to 8/(3\operatorname{sqrt}(\pi ))$ in the thermodynamic-then-dilute limit, uniformly over the ground-state space?

Short English statement

For every fixed bounded nonzero nonnegative radial finite-range pair potential v in $R^3$ with scattering length $a_v$, do all periodic-cube ground states satisfy $(1-B_\Gamma )/\operatorname{sqrt}(\rho $ $a_v^3)\to 8/(3\operatorname{sqrt}(\pi ))$ in the thermodynamic-then-dilute limit, uniformly over the ground-state space?

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Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-58-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-58

The same leading depletion coefficient is claimed for every fixed bounded nonnegative radial finite-range potential of positive scattering length, uniformly over ground-state density matrices. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The universal quantum-depletion constant for fixed bounded repulsions

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The same leading depletion coefficient is claimed for every fixed bounded nonnegative radial finite-range potential of positive scattering length, uniformly over ground-state density matrices.

Mathematical statement of the claimed result

Result 1.
Let $v\ge0$ be bounded, measurable, radial, compactly supported, and nonzero on a positive-measure set. Its scattering length $a_v>0$ is determined by $(-\Delta+v/2)f=0$, with $f(x)=1-a_v/|x|$ outside the support. For the periodic symmetric Hamiltonian $-\sum_i\Delta_i+\sum_{i<j}v_L(x_i-x_j)$, let $\mathcal G^v_{N,L}$ contain all ground-space trace-one states and $B_\Gamma=N^{-1}\langle L^{-3/2},\gamma^{(1)}_\Gamma L^{-3/2}\rangle$. For every $\varepsilon>0$, some $\rho_0(v,\varepsilon)>0$ satisfies \[ 0<\rho<\rho_0,\ N_k,L_k\to\infty,\ N_k/L_k^3\to\rho \ \Longrightarrow\ \limsup_k\sup_{\Gamma\in\mathcal G^v_{N_k,L_k}} \left|\frac{1-B_\Gamma}{\sqrt{\rho a_v^3}}-\frac8{3\sqrt\pi}\right|\le\varepsilon. \] Volume and particle number tend to infinity while density remains fixed.

Outline of the argument

Local density tails and ground-state path bounds control occupations at the relevant scattering scale. Spatial comparison transports these local estimates to the constant orbital, identifying every thermodynamic accumulation value before density tends to zero.

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.

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