# On the finite-size Lyapunov exponent for the Schroedinger operator with skew-shift potential

## TL;DR

This paper investigates the finite-size Lyapunov exponent for a quantum system with a skew-shift potential, providing numerical verification of conditions related to localization phenomena in such systems.

## Contribution

It offers the first numerical verification of finite-size Lyapunov exponent conditions for skew-shift potentials, advancing understanding of localization in these models.

## Key findings

- Numerical verification of finite-size Lyapunov exponent conditions.
- Support for localization conjectures in skew-shift potential models.
- Extension of finite-size criteria to non-golden mean frequencies.

## Abstract

It is known that a one-dimensional quantum particle is localized when subjected to an arbitrarily weak random potential. It is conjectured that localization also occurs for an arbitrarily weak potential generated from the nonlinear skew-shift dynamics: $v_n=2\cos\left(\binom{n}{2}\omega +ny+x\right)$ with $\omega$ an irrational number. Recently, Han, Schlag, and the second author derived a finite-size criterion in the case when $\omega$ is the golden mean, which allows to derive the positivity of the infinite-volume Lyapunov exponent from three conditions imposed at a fixed, finite scale. Here we numerically verify the two conditions among these that are amenable to computer calculations.