Homogeneity of the spectrum for quasi-periodic Schrödinger operators
Bibliographic record
Abstract
We consider the one-dimensional discrete Schrödinger operator \bigl[H(x,\omega)\varphi\bigr](n)\equiv -\varphi(n-1)-\varphi(n+1) + V(x + n\omega)\varphi(n)\ , n \in \mathbb Z , x,\omega \in [0, 1] with real-analytic potential V(x) . Assume L(E,\omega)>0 for all E . Let \mathcal{S}_\omega be the spectrum of H(x,\omega) . For all \omega obeying the Diophantine condition \omega \in \mathbb{T}_{c,a} , we show the following: if \mathcal{S}_\omega \cap (E',E'')\neq \emptyset , then \mathcal{S}_\omega \cap (E',E'') is homogeneous in the sense of Carleson (see [Car83]). Furthermore, we prove, that if G_i , i=1,2 are two gaps with 1 > |G_1| \ge |G_2| , then |G_2|\lesssim \exp\left(-(\log \mathrm {dist} (G_1,G_2))^A\right) , A\gg 1 . Moreover, the same estimates hold for the gaps in the spectrum on a finite interval, that is, for \mathcal S_{N,\omega}:=\cup_{x\in\mathbb T}\mathrm {spec} \; H_{[-N,N]}(x,\omega) , N \ge 1 , where H_{[-N, N]}(x, \omega) is the Schrödinger operator restricted to the interval [-N,N] with Dirichlet boundary conditions. In particular, all these results hold for the almost Mathieu operator with |\lambda| \neq 1 . For the supercritical almost Mathieu operator, we combine the methods of [GolSch08] with Jitomirskaya's approach from [Jit99] to establish most of the results from [GolSch08] with \omega obeying a strong Diophantine condition.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".