Simon's Problem 1: absolutely continuous spectrum for the weakly disordered Anderson model in dimension three and higher

For each $d\ge 3$ there is $\lambda_d>0$ such that for every $0<\lambda<\lambda_d$ , almost surely a fixed open interval $I_d$ near the bottom band edge (for $d\ge 4$ , $-2d+(1/200,1/100)$ ) lies in the spectrum and the spectral restriction to $I_d$ is purely absolutely continuous and nonzero. $I_d$ does not depend on $\lambda$ . Not shown: a.c. spectrum in the bulk of the band, the location of a mobility edge, quantum diffusion, uniformity in $d$ , or other laws.

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Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Simon's Problem 1: absolutely continuous spectrum for the weakly disordered Anderson model in dimension three and higher · Anderson extended states, d >= 3 (Simon's Problem 1)

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VibeMathed record: Simon's Problem 1: absolutely continuous spectrum for the weakly disordered Anderson model in dimension three and higher

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No independent mathematician has checked this yet. Checked here: the main theorem was read against Simon's Problem 1 as the manuscript paraphrases it; Simon's text itself was not opened. The proof was not refereed. No Lean formalization exists for this manuscript. The paper itself says it does not locate the mobility edge and does not resolve Simon's diffusion question (Problem 3), and that for d = 3 the disorder threshold is only existential.

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Simon's Problem 1: absolutely continuous spectrum for the weakly disordered Anderson model in dimension three and higher

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Same family: Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder

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