The Absolutely Continuous Spectrum of Finitely Differentiable Quasi-Periodic Schrödinger Operators

Randomness versus quasi-periodicity

“…It is not hard to see that randomness dominates quasi-periodicity in the sense of positive Lyapunov exponents. For example, for any quasi-periodic Schrödinger operator in the almost reducible regime (small coupling constant, Diophantine frequency and smooth enough), it has purely absolutely continuous spectrum with zero Lyapunov exponents [1] [2]. However, when the randomness comes in, the new mixed cocycle has positive Lyapunov exponent which is consistent with the random case.…”

“…Back then, I just finished my PhD with my dissertation "Reducibility of finitely differentiable quasi-periodic cocycles and its applications" based on Kolmogorov-Arnold-Moser (KAM) theory under the supervision of You. Obviously, I was a purely quasiperiodic person [1,2,8,9] while Duarte and Klein had already many collaborations on random cocycles as well as quasi-periodic ones, see the two excellent books [10,11] and the references therein.…”

Reducibility of Finitely Differentiable Quasi-Periodic Cocycles and Its Spectral Applications

“…Proof. We will only prove estimates for the non-resonant case because it is more delicate and the proof of the resonant case is the same compared with those in [11].…”

“…Indeed, it is well known that conditions (11) and ( 12) play a role in addressing the small denominator problem in KAM theory.…”

“…We know the homogeneity of the spectrum is related to polynomial decay of gap length and Hölder continuity of the integrated density of states. Therefore, we need to change the assumption to k > 17τ + 2 so that we can further ensure the 1 2 -Hölder continuity of IDS (see Theorem 3.3 and Theorem 3.4 in [11]). This theorem greatly reduces the initial regularity assumption in the C k case.…”