Communications in Mathematical Sciences

Volume 18 (2020)

Number 5

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

Pages: 1305 – 1314

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n5.a6

Authors

Paul M. Kielstra (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Marius Lemm (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

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: $v_n = 2 \cos ((\frac{n}{2}) \omega + ny + x)$ with $\omega$ an irrational number and $x, y \in [0, 1]$. Recently, Han, Schlag, and the second author derived a finite-size criterion in the case when $\omega$ is the golden mean, which allows the derivation of 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.

Keywords

Schrödinger cocycles, Lyapunov exponent, skew-shift dynamics

2010 Mathematics Subject Classification

37H15, 39A06, 47B36, 81Q10

Full Text (PDF format)

The computations in this paper were run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University.

Received 22 April 2019

Accepted 13 February 2020

Published 23 September 2020