Lyapunov exponents in continuum Bernoulli-Anderson models

TL;DR

This paper investigates the positivity of Lyapunov exponents in one-dimensional continuum Bernoulli-Anderson models, identifying critical energies and analyzing eigenfunction behavior at those energies.

Contribution

It provides a proof of Lyapunov exponent positivity away from critical energies and explicitly characterizes these energies using scattering coefficients.

Findings

Lyapunov exponents are positive except at a discrete set of energies.

Critical energies are explicitly described via scattering coefficients.

Eigenfunction behavior at critical energies is analyzed.

Abstract

We study one-dimensional, continuum Bernoulli-Anderson models with general single-site potentials and prove positivity of the Lyapunov exponent away from a discrete set of critical energies. The proof is based on F\"urstenberg's Theorem. The set of critical energies is described explicitly in terms of the transmission and reflection coefficients for scattering at the single-site potential. In examples we discuss the asymptotic behavior of generalized eigenfunctions at critical energies.