Asymptotics of spectral gaps of quasi-periodic Schrödinger operators

Abstract

For non-critical almost Mathieu operators with Diophantine frequency, we establish exponential asymptotics on the size of spectral gaps, and show that the spectrum is homogeneous. We also prove the homogeneity of the spectrum for Schödinger operators with (measure-theoretically) typical quasi-periodic analytic potentials and fixed strong Diophantine frequency. As applications, we show the discrete version of Deift's conjecture \cite{Deift, Deift17} for subcritical analytic quasi-periodic initial data and solve a series of open problems of Damanik-Goldstein et al \cite{BDGL, DGL1, dgsv, Go} and Kotani \cite{Kot97}.

References in corpus (3)

• Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II
• The absolutely continuous spectrum of the almost Mathieu operator
• Almost reducibility and absolute continuity I

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• Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,R)$ cocycles
• A series of spectral gaps for the almost Mathieu operator with a small coupling constant
• On the spectra of separable 2D almost Mathieu operators
• Hölder regularity of the integrated density of states for quasi-periodic long-range operators on ${\ell }^2({\mathbb{Z}}^d)$
• Exponential Decay of the lengths of Spectral Gaps for Extended Harper's Model with Liouvillean Frequency
• Arithmetic version of Anderson localization via reducibility
• Polynomial decay of the gap length for C^k quasi-periodic Schrodinger operators and spectral application