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Register a new userAsymptotics of spectral gaps of quasi-periodic Schrodinger operators
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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 Schodinger operators with (measure-theoretically) typical quasi-periodic analytic potentials and fixed strong Diophantine frequency. As applications, we show the discrete version of Deifts 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}.
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We show that a generic quasi-periodic Schrodinger operator in $L^2(mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schrodinger operator with the resulting potential has empty absolutely continuous spectrum.
We prove the Holder continuity of the integrated density of states for a class of quasi-periodic long-range operators on $ell^2(Z^d)$ with large trigonometric polynomial potentials and Diophantine frequencies. Moreover, we give the Holder exponent in terms of the cardinality of the level sets of the potentials, which improves, in the perturbative regime, the result obtained by Goldstein and Schlag cite{gs2}. Our approach is a combination of Aubry duality, generalized Thouless formula and the regularity of the Lyapunov exponents of analytic quasi-periodic $GL(m,C)$ cocycles which is proved by quantitative almost reducibility method.
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