Positivity and continuity of Lyapunov exponents for one-frequency, smooth quasi-periodic Schrodinger operators.

摘要

We investigate the dynamical behavior of one-frequency quasiperiodic Schrodinger cocycles. These cocycles arise naturally in the study of the spectral properties of one-dimensional discrete.;Schrodinger operators. Quasiperodicity refers to the potentials vn= v(x + nalpha), which sample along irrational rotations on the unit circle. In particular, we study the positivity and regularity of Lyapunov exponents.;We will classify our results by the regularity of v. For the real analytic case, inspired by a new notion of acceleration that is introduced in [Av2], we give a different proof of the uniform positivity of Lyapunov exponents for nonconstant real analytic potentials (originally proven in [SoSp]).;In the Cr case with 1 ≤ r ≤ infinity, we first show that for a certain type of C1 potential with large coupling constants, the cocyles are nonuniformly hyperbolic for a large set of energies. Then for some C2 potentials, we further show that Lyapunov exponents are uniformly positive and weak Holder continuous as function of energy. In particular, a version of the Large Deviation Theorem for potentials in Cr category with 1 ≤ r ≤ infinity will be established for the first time.