Cantor spectrum for a class of $C^2$ quasiperiodic Schrödinger operators

Abstract

We show that for a class of $C^2$ quasiperiodic potentials and for any Diophantine frequency, the spectrum of the corresponding Schrödinger operators is Cantor. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to general $\mathrm{SL}(2,\mathbb{R})$ cocycles, and obtain that uniform hyperbolic systems form a open and dense set in some one-parameter family.

arXiv comment21 pages, preliminary version, all comments are welcome

References in corpus (4)

• Positive Lyapunov Exponents for Quasiperiodic Szego cocycles
• Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrödinger Operators
• Quasi-Periodic Schrödinger Cocycles with Positive Lyapunov Exponent are not Open in the Smooth Topology
• Uniform Positivity and Continuity of Lyapunov Exponents for a Class of $C^2$ Quasiperiodic Schrödinger Cocycles 

Cited by in corpus (4)

• Schrödinger Operators with Dynamically Defined Potentials: A Survey
• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Genericity of mode-locking for quasiperiodically forced circle maps
• Abundance of mode-locking for quasiperiodically forced circle maps