Positive Lyapunov exponents and localization bounds for strongly mixing potentials

TL;DR

This paper derives a perturbative formula for the Lyapunov exponent of a one-dimensional Schrödinger operator with a strongly mixing potential, showing positivity across the spectrum and implications for spectral measure and quantum dynamics.

Contribution

It provides a novel perturbative approach to establish positivity of Lyapunov exponents for strongly mixing potentials with power law decay.

Findings

Lyapunov exponent is positive for all energies including band edges and center

Spectral measure has Hausdorff dimension zero

Quantum dynamics grow at most logarithmically in time

Abstract

For a one-dimensional discrete Schr\"odinger operator with a weakly coupled potential given by a strongly mixing dynamical system with power law decay of correlations, we derive for all energies including the band edges and the band center a perturbative formula for the Lyapunov exponent. Under adequate hypothesis, this shows that the Lyapunov exponent is positive on the whole spectrum. This in turn implies that the Hausdorff dimension of the spectral measure is zero and that the associated quantum dynamics grows at most logarithmically in time.