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Extended States with Extensive Disorder

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Establish, in some energy range, the existence of extended eigenstates or continuous spectrum for linear operators with extensive disorder, such as a discrete Schrödinger operator with an i.i.d. random potential acting on $\ell^2(\mathbb{Z}^d)$. Also clarify the borderline case $d=2$.

Formulation references: [S1], [S2].

Short English statement

Establish, in some energy range, the existence of extended eigenstates or continuous spectrum for linear operators with extensive disorder, such as a discrete Schrödinger operator with an i.i.d. random potential acting on $\ell^2(\mathbb{Z}^d)$. Also clarify the borderline case $d=2$.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-27-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-27

The theorem supplies a purely absolutely continuous window for iid uniform disorder in every d>=3. The problem statement bundles a broader extended-state/continuous-spectrum existence discussion with the borderline d=2 question, and the latter is not answered by this manuscript. This entry records partial support; complete coverage of the question is not established by this catalogue.

Extended States with Extensive Disorder

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. For uniform iid Anderson site disorder in every fixed dimension at least three, sufficiently small fixed disorder is claimed to give purely absolutely continuous spectrum on a nonempty open interval.

Mathematical statement of the claimed result

Result 1.
For independent $V_x$ uniformly distributed on $[-1,1]$, consider \[ (H_{\lambda,\omega}\psi)(x)=-\sum_{j=1}^d[\psi(x+e_j)+\psi(x-e_j)]+\lambda V_x\psi(x) \] on $\ell^2(\mathbb Z^d)$. The claimed fixed constant $e_2\in(0,1/100)$ and thresholds $\lambda_d>0$ give, for each $d\ge3$ and $0<\lambda<\lambda_d$, almost surely \[ I_d\subset\sigma(H_{\lambda,\omega}),\qquad \chi_{I_d}(H_{\lambda,\omega})\ell^2(\mathbb Z^d)\subset\mathcal H_{\mathrm{ac}}(H_{\lambda,\omega}), \] where \[ I_3=(-6+e_2/2,-6+e_2),\qquad I_d=-2d+(1/200,1/100)\quad(d\ge4). \] The projection is nonzero; the assertion is purely absolutely continuous spectrum on these windows, for this uniform-distribution model.

Outline of the argument

A finite-volume matrix-state certificate is propagated through controlled ordinary sweeps. Local trials and clearing of pending components maintain the comparison forms needed for resolvent bounds, which identify an interval with nonzero absolutely continuous weight.

Formal verification

For family 261, the selected planar Anderson statement constructs the self-adjoint operator and gives its almost-sure spectral interval. Spectral type, including the claimed pure-point conclusion, is not contained in 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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