Anderson localization as a parametric instability of the linear kicked oscillator

TL;DR

This paper establishes a rigorous link between Anderson localization in a quantum model and parametric instability in a classical kicked oscillator with noise, providing analytical expressions for localization length under weak disorder.

Contribution

It introduces a novel analytical framework connecting Anderson localization to classical parametric instability, applicable across energy bands and edges.

Findings

Localization length is related to exponential energy growth rate.

Analytical expressions valid for weak disorder inside and at the band edge.

Rigorous correspondence between quantum localization and classical instability.

Abstract

We rigorously analyse the correspondence between the one-dimensional standard Anderson model and a related classical system, the `kicked oscillator' with noisy frequency. We show that the Anderson localization corresponds to a parametric instability of the oscillator, with the localization length determined by an increment of the exponential growth of the energy. Analytical expression for a weak disorder is obtained, which is valid both inside the energy band and at the band edge.