KAM, Lyapunov exponents, and the Spectral Dichotomy for typical one-frequency Schrodinger operators

TL;DR

This paper establishes a spectral dichotomy for typical one-frequency Schrödinger operators, linking Lyapunov exponents to reducibility and spectral properties, and showing that such operators lack singular continuous spectrum.

Contribution

It proves a new connection between Lyapunov exponents, reducibility, and spectral types for one-frequency Schrödinger operators without arithmetic restrictions.

Findings

Analytic linearizability occurs if and only if the Lyapunov exponent is zero.

Regularity implies almost reducibility regardless of arithmetic assumptions.

Typical operators decompose into large-like and small-like components with no singular continuous spectrum.

Abstract

We show that a one-frequency analytic SL(2,R) cocycle with Diophantine rotation vector is analytically linearizable if and only if the Lyapunov exponent is zero through a complex neighborhood of the circle. More generally, we show (without any arithmetic assumptions) that regularity implies almost reducibility, i.e., the range of validity of the perturbative analysis near constants is specified by a condition on the Lyapunov exponents. Together with our previous work, this establishes a Spectral Dichotomy for typical one-frequency Schrodinger operators: they can be written as a direct sum of large-like and small-like operators. In particular, the typical operator has no singular continuous spectrum.