Lyapunov exponent of the random Schr\"{o}dinger operator with short-range correlated noise potential

TL;DR

This paper derives explicit formulas for the Lyapunov exponent of a one-dimensional Schrödinger operator with short-range correlated noise, revealing how disorder affects wave propagation and comparing it to white noise cases.

Contribution

It provides the first explicit expressions for the Lyapunov exponent with short-range correlated noise, extending previous white noise results.

Findings

Lyapunov exponent explicitly expressed as a function of noise intensity and frequency.

Uniform asymptotic formulas for the Lyapunov exponent across different parameter regimes.

Lyapunov exponent with short-range correlated noise is smaller than in the white noise case.

Abstract

We study the influence of disorder on propagation of waves in one-dimensional structures. Transmission properties of the process governed by the Schr\"{o}dinger equation with the white noise potential can be expressed through the Lyapunov exponent $\gamma$ which we determine explicitly as a function of the noise intensity \sigma and the frequency \omega. We find uniform two-parameter asymptotic expressions for $\gamma$ which allow us to evaluate $\gamma$ for different relations between \sigma and \omega. The value of the Lyapunov exponent is also obtained in the case of a short-range correlated noise, which is shown to be less than its white noise counterpart.