A lower bound for the Lyapounov exponents of the random Schrodinger operator on a strip

TL;DR

This paper establishes a lower bound for the positive Lyapounov exponents of the random Schrödinger operator on a strip, showing they decay roughly exponentially with the strip width, using Green's function decay without Furstenberg's theory.

Contribution

It provides a new lower bound for Lyapounov exponents on a strip, avoiding Furstenberg's theory and employing Green's function decay and barrier construction techniques.

Findings

Lyapounov exponents are bounded below exponentially in strip width W

Method avoids reliance on Furstenberg's random matrix theory

Constructs barriers using RSO theory on b Z

Abstract

We consider the random Schrodinger operator on a strip of width $W$, assuming the site distribution of bounded density. It is shown that the positive Lyapounov exponents satisfy a lower bound roughly exponential in $-W$ or $W\to \infty$. The argument proceeds directly by establishing Green's function decay, but does not appeal to Furstenberg's random matrix theory on the strip. One ingredient involved is the construction of `barriers' using the RSO theory on $\mathbb Z$.