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Periodic Ground States in Gross–Pitaevskii Theory

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.

For a pair potential $w$, $(w*|u|^2)(x)=\int w(x-y)|u(y)|^2dy$. The stationary Gross--Pitaevskii equation is the Euler equation for a nonlocal energy at chemical potential $\mu$. Superstability controls the interaction energy from below at high local density; $\widehat w$ is its Fourier transform. Periodicity means invariance under a full-rank lattice of spatial translations.

Catalogue wording retained for traceability.
For superstable, sufficiently decaying pair interactions $w$ whose Fourier transform takes negative values, must every infinite-volume ground state of $(-\Delta+w*|u|^2)u=\mu u$ be periodic when $\mu$ exceeds the fluid–solid transition value $\mu_c$?
Scope and source qualification.
The quantifier is every ground state, not merely the existence of a periodic minimizer. Infinite-volume ground state and $\mu_c$ use the source definitions.

Short English statement

Determine whether every infinite-volume Gross--Pitaevskii ground state is crystalline above the fluid--solid transition.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Phys-384-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-384

Periodic Ground States in Gross–Pitaevskii Theory

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.

For a pair potential $w$, $(w*|u|^2)(x)=\int w(x-y)|u(y)|^2dy$. The stationary Gross--Pitaevskii equation is the Euler equation for a nonlocal energy at chemical potential $\mu$. Superstability controls the interaction energy from below at high local density; $\widehat w$ is its Fourier transform. Periodicity means invariance under a full-rank lattice of spatial translations.

Catalogue wording retained for traceability.
For superstable, sufficiently decaying pair interactions $w$ whose Fourier transform takes negative values, must every infinite-volume ground state of $(-\Delta+w*|u|^2)u=\mu u$ be periodic when $\mu$ exceeds the fluid–solid transition value $\mu_c$?
Scope and source qualification.
The quantifier is every ground state, not merely the existence of a periodic minimizer. Infinite-volume ground state and $\mu_c$ use the source definitions.

Short English statement

Determine whether every infinite-volume Gross--Pitaevskii ground state is crystalline above the fluid--solid transition.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Computed a density-wave second variation and an explicit finite-volume instability criterion; did not prove crystallinity.
Sources and literature check.
Read the full supplied definition, catalogue wording, source qualifications and listed references for 30005661. No fresh status or novelty search was performed. The argument below proves its auxiliary statements directly and does not import an unverified external theorem.
Density-wave test on a torus.
As an auxiliary finite-volume calculation take a real even periodic integrable kernel $w$, a nonzero reciprocal lattice vector $k$, density $\rho>0$, and $|\varepsilon|<1$. Normalize convolution so its Fourier multiplier is $\widehat w(k)$. Set $u_\varepsilon(x)=\sqrt{\rho(1+\varepsilon\cos(k\cdot x))}$. Its mean density is exactly $\rho$. For the energy per volume $\langle|\nabla u|^2\rangle+\frac12\langle|u|^2(w*|u|^2)\rangle$, the interaction difference is exactly $\rho^2\varepsilon^2\widehat w(k)/4$. Also \[ \langle|\nabla u_\varepsilon|^2\rangle =\frac{\rho\varepsilon^2|k|^2}{4} \left\langle\frac{\sin^2(k\cdot x)}{1+\varepsilon\cos(k\cdot x)}\right\rangle =\frac{\rho\varepsilon^2|k|^2}{8}+O(\varepsilon^4). \] The linear term inside the average vanishes; the remaining expansion is uniform for $|\varepsilon|\leq1/2$. Thus the constant state fails this local minimum test if $|k|^2+2\rho\widehat w(k)<0$. The chemical-potential term has zero difference because mean density is fixed.
Why the route stops.
A negative Fourier component can select an unstable wavelength, but this calculation selects neither a lattice nor every minimizing state. The test is periodic by construction; using it to conclude periodicity of arbitrary infinite-volume minimizers would assume the desired property. Interfaces between competing patterns and nonperiodic candidates are not excluded by a local second variation.
Scope.
The source requires every infinite-volume ground state above its actual $\mu_c$, for its superstable decaying kernels. No identification of $\mu_c$ with this finite-volume instability threshold is proved. The first missing estimate is a global energy comparison excluding all nonperiodic competitors under the source ground-state definition.
Verification and adversarial audit.
Checked exact preservation of density, the factors in the convolution energy, the even expansion and the distinction between instability and global periodicity. Model self-audit only; no independent human mathematical verification. Applied contribution prompt A before drafting and B afterwards. Checked exact preservation of density, the factors in the convolution energy, the even expansion and the distinction between instability and global periodicity. First remaining gap: Exclude nonperiodic infinite-volume ground states by a global comparison for the source potentials and transition threshold. The inherited source status is preserved; original-target completion remains zero.

Coordinator checked the density-wave interaction term, kinetic expansion and conditional local-instability sign. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.
Python 3 and local file tools used to read definitions and serialize manuscripts. Calculations are symbolic reasoning; no numerical experiment or theorem prover. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.
Final status and first remaining gap.
UNRESOLVED. Exclude nonperiodic infinite-volume ground states by a global comparison for the source potentials and transition threshold.

Completion Estimate

COMPLETION: 0% This is an elementary auxiliary calculation or an explicit test of a restricted strategy. No new portion of the full source target is established.

Sources

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