Persistence of Unitary Almost Mathieu Operator Gaps Ten Dry Martinis, Five on the Rim

Abstract:We determine exactly which gap labels are realized by open spectral gaps for the unitary almost Mathieu operator at every irrational frequency and all coupling constants ($0<\lambda_1,\lambda_2\leq1$). When both couplings are strictly less than one, every permitted gap label is realized by an open gap, including in the critical case ($\lambda_1=\lambda_2$). When at least one coupling equals one, both lifts of every odd winding label are realized by open gaps, whereas all even winding labels are collapsed. The proof combines quantitative almost reducibility and Aubry duality with a monotonicity principle for gap heights.

Mathematical Physics (math-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Spectral Theory (math.SP); Quantum Physics (quant-ph)

arXiv:2610.42120 [math-ph]

(or arXiv:2610.42120v1 [math-ph] for this version)

https://doi.org/10.48550/arXiv.2610.42120 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.