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Adiabatic Expansions for Gapless Quantum Systems

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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 slowly varied Hamiltonian, an adiabatic expansion approximates the evolved state or observables in powers of the driving rate. A spectral gap separates a ground-state sector from the rest of the spectrum. Gapless systems lack a uniform positive separation. Imaginary-time correlations are correlations under exponential rather than unitary evolution.

Catalogue wording retained for traceability.
Can the adiabatic-expansion framework, whose convergence follows from imaginary-time correlation estimates for gapped systems, be extended to gapless quantum systems even without interactions?
Scope and source qualification.
An extension must state its observables, time scaling and error estimates; the question does not assume an adiabatic theorem with the same hypotheses as in the gapped case.

Short English statement

Determine whether the source adiabatic-expansion method extends to gapless quantum systems, including noninteracting ones.

Sources

Catalogue entry

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

Contributions 1

Phys-385-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-385

Adiabatic Expansions for Gapless Quantum Systems

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 slowly varied Hamiltonian, an adiabatic expansion approximates the evolved state or observables in powers of the driving rate. A spectral gap separates a ground-state sector from the rest of the spectrum. Gapless systems lack a uniform positive separation. Imaginary-time correlations are correlations under exponential rather than unitary evolution.

Catalogue wording retained for traceability.
Can the adiabatic-expansion framework, whose convergence follows from imaginary-time correlation estimates for gapped systems, be extended to gapless quantum systems even without interactions?
Scope and source qualification.
An extension must state its observables, time scaling and error estimates; the question does not assume an adiabatic theorem with the same hypotheses as in the gapped case.

Short English statement

Determine whether the source adiabatic-expansion method extends to gapless quantum systems, including noninteracting ones.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Calculated the exact order of finite imaginary-time moments in a gapless spectral model, identifying a failure of a naive all-orders argument.
Sources and literature check.
Read the full supplied definition, catalogue wording, source qualifications and listed references for 30005662. No fresh status or novelty search was performed. The argument below proves its auxiliary statements directly and does not import an unverified external theorem.
Testing imaginary-time moment integrability.
Consider a nonnegative spectral measure $d\nu(E)=E^\alpha\mathbf1_{(0,1)}(E)\,dE$, with $\alpha>-1$, and its correlation $C(t)=\int_0^1 e^{-tE}E^\alpha dE$, $t\geq0$. This is realized on $L^2((0,1),E^\alpha dE)$ by multiplication by $E$ and the vector $1$. The spectrum touches zero. For an integer $j\geq0$, nonnegativity permits exchanging integrals, and repeated integration by parts in the inner integral gives \[ \int_0^\infty t^j C(t)\,dt =j!\int_0^1 E^{\alpha-j-1}\,dE =\frac{j!}{\alpha-j}\quad\hbox{if }\alpha>j. \] For $\alpha\leq j$ it diverges; equality is a logarithmic divergence. Thus the zeroth time moment can exist while the first fails, for example $0<\alpha\leq1$.
Attempted extension and obstruction.
Any source expansion coefficient that requires this absolute $j$-th moment cannot be justified from decay sufficient only for lower moments. The example supplies a gapless correlation with an exactly computable order cutoff. It is not a no-go theorem for adiabatic evolution: cancellations, a different observable or a resummation may avoid that moment requirement.
Full-source gap.
To obtain the requested expansion one must derive appropriate low-energy spectral weights for specified noninteracting or interacting observables, control higher connected correlations, and bound the remainder on the chosen slow time scale. The example has no driven Hamiltonian and no proved dynamical error bound. Its role is to expose the precise loss of uniform moment estimates when a gapped proof is copied to a gapless system.
Verification and adversarial audit.
Checked the spectral realization, nonnegative interchange, factorial, alpha>-1, strict integrability threshold and logarithmic endpoint. Model self-audit only; no independent human mathematical verification. Applied contribution prompt A before drafting and B afterwards. Checked the spectral realization, nonnegative interchange, factorial, alpha>-1, strict integrability threshold and logarithmic endpoint. First remaining gap: Establish source-observable correlation estimates and a driven-evolution remainder with stated time and volume uniformity. The inherited source status is preserved; original-target completion remains zero.

Coordinator checked the nonnegative spectral moment interchange and its exact threshold and logarithmic boundary divergence. 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. Establish source-observable correlation estimates and a driven-evolution remainder with stated time and volume uniformity.

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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