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Results · Transport in quasi-periodic systems

Band edges of the almost Mathieu operator at Liouville scales

Seth Douglas and Nidhal Mghirbi · version 2.0.0 · Zenodo, · 60 pages · doi:10.5281/zenodo.23877075 · PDF

In plain wordsWhen the frequency is close enough to fractions, some gaps in the spectrum of the Aubry–André model shrink only like a power of their label, not exponentially; below that threshold, some gaps shrink more slowly than at typical frequencies, by an amount set by how well the frequency is approximated by fractions; and the paper maps which regimes are proved and which are conjectures.

The result

As the paper states it, in briefFor the supercritical almost Mathieu operator, write L=log⁡λ, β=β(α) and x=2β/L, and let ρ be the exponential gap-decay rate, the liminf of −log⁡|Gm|/(L|m|). Above the Liouville threshold, for finite β>L/2, infinitely many gaps are only polynomially small in their integer label, so ρ=0: at a convergent pn/qn, with δn=qnα−pn and bn=−qn−1log⁡|δn|, the gaps with labels m=−Dqnsgnδn at the top of the spectrum have width at least exp⁡(−(bn+L/2+o(1))qn), uniformly over an explicit growing range of D, and a separate surplus argument includes β=∞. Below the threshold, for 0<x<1, ρ≤1−x: at β-attaining convergents, the gap with label −Dqnsgnδn has an explicit lower bound on its width for every fixed D. With the theorem of Liu and Shi this gives 1/C≤ρ≤1−x<1 for 0<x≤2/C, which contradicts their stated expectation of a sharper upper gap bound in that range but not their theorem. Equality ρ=1−x, and the line x=1, remain open.

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Novelty and credit

Novelty
Extends named prior work: all noncritical gaps are open by Avila, You and Zhou; Ge, You and Zhou prove the Diophantine rate; polynomial gap phenomena at critical coupling are due to Krasovsky; the orbital displacement is the classical Rammal–Wilkinson correction. The contribution is their uniform combination at one Liouville scale with growing labels, and the resulting theorem. Below the threshold, the bound ρ ≤ 1 − x contradicts the sharper upper gap bound that Liu and Shi expected in an overlapping range; it does not contradict their theorem.

Versionstwo, each with its own DOI

Every archived version, newest first. Dates are Zenodo publication dates. All versions: 10.5281/zenodo.23873796, which always resolves to the newest.
VersionDateDOINotes
2.0.010.5281/zenodo.23877075Adds the other side of the Liouville threshold: for 0 < x < 1 the gap-decay rate satisfies ρ ≤ 1 − x, and with the theorem of Liu and Shi, 1/C ≤ ρ ≤ 1 − x for 0 < x ≤ 2/C. New sections and appendices give the proof; the Numerical Record gains three independent checks, archived with their code and certified outputs in the release's verification folder. PDF hosted here.
1.0.010.5281/zenodo.23873797

Paper, code and links

PDF

papers/band-edges-v2.0.0.pdf, version 2.0.0, 980,638 bytes, 60 pages.

SHA-256 c54e27925352e1914dabd6bf2fd22b7c616f21631e2239dd8114cb72fd03f11c

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Cite

@misc{douglas2026bandedges,
  author  = {Douglas, Seth and Mghirbi, Nidhal},
  title   = {Band edges of the almost {M}athieu operator at {L}iouville scales},
  year    = {2026},
  version = {2.0.0},
  doi     = {10.5281/zenodo.23877075},
  url     = {https://doi.org/10.5281/zenodo.23877075}
}

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