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Localization in Two Dimensions

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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.
Prove that the spectrum of the Anderson model in dimension $\nu=2$ is dense pure point.

Formulation references: [S1], [S2].

Short English statement

Prove that the spectrum of the Anderson model in dimension $\nu=2$ is dense pure point.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-28-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-28

For every positive disorder width in the nearest-neighbor two-dimensional Anderson model with iid uniform potentials, the main theorem proves pure-point spectral type, an orthonormal eigenbasis, and dense eigenvalues throughout the full spectrum. Centering a uniform interval changes only the energy origin, exactly covering Simon Problem 2 as defined in the manuscript model and problem statement. Within these assumptions, this is a claimed affirmative answer to the stated question.

Localization in Two Dimensions

OpenAI's claim

OpenAI presents a proof in the selected manuscript. At every positive uniform iid disorder strength, the two-dimensional Anderson operator is claimed to have pure-point spectral type throughout its spectrum.

Mathematical statement of the claimed result

Result 1.
For each fixed $h>0$, take independent $v_x\sim\operatorname{Unif}[-h,h]$, $x\in\mathbb Z^2$, and \[ (H_vu)(x)=\sum_{|y-x|_1=1}u(y)+v_xu(x). \] The source claims that, on an event of probability one depending on $h$, $H_v$ has a complete orthonormal eigenbasis and \[ \sigma(H_v)=[-4-h,4+h], \] with eigenvalues dense in this interval. Pure-point spectral type is asserted; this statement alone does not specify an eigenfunction-decay rate.

Outline of the argument

Scattering matrices are compared through absorption, port closing, and Schur complements. Dimension-independent analytic and determinant-phase estimates control repeated finite-volume propagation, yielding localization of the spectral measure in the infinite square-lattice limit.

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. The listed Comparator files are PlanarAndersonSpectrum.lean. The inventory does not supply a named declaration for these Comparator entries. 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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