Reducibility of finitely differentiable quasi-periodic cocycles and its spectral applications

2017/12/25 by Cai, Ao, Ge, Lingrui
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1712.09041

Abstract

In this paper, we prove the generic version of Cantor spectrum for quasi-periodic Schrödinger operators with finitely smooth and small potentials, and we also show pure point spectrum for a class of multi-frequency C^k long-range operators on ℓ²(\Z^d). These results are based on reducibility properties of finitely differentiable quasi-periodic SL(2,\R) cocycles. More precisely, we prove that if the base frequency is Diophantine, then a C^k SL(2,\R)-valued cocycle is reducible if it is close to a constant cocycle, sufficiently smooth and the rotation number of it is Diophantine or rational with respect to the frequency.

Related

• Quantitative Reducibility of C^k Quasi-Periodic Cocycles
• Reducibility without KAM
• The absolutely continuous spectrum of finitely differentiable quasi-periodic Schrödinger operators
• Construction of finite differentiable quasi-periodic Schrödinger operators with cantor spectrum
• Cantor spectrum for a class of C² quasiperiodic Schrödinger operators