Lyapounov exponent and density of states of a one-dimensional non-Hermitian Schroedinger equation

TL;DR

This paper numerically investigates the Lyapounov exponent and density of states in a one-dimensional non-Hermitian Schrödinger equation with off-diagonal disorder, revealing a maximum in the density at a specific energy and deriving an asymptotic expansion.

Contribution

It provides the first detailed numerical analysis of gamma(E) and rho(E) for this model, including an asymptotic expansion of the density of states.

Findings

Density of states rho(|E|) has a maximum at E=2/sqrt(3).

Density of states can be expanded asymptotically in powers of E.

Both gamma(E) and rho(E) depend only on the modulus of E.

Abstract

We calculate, using numerical methods, the Lyapounov exponent gamma(E) and the density of states rho(E) at energy E of a one-dimensional non-Hermitian Schroedinger equation with off-diagonal disorder. For the particular case we consider, both gamma(E) and rho(E) depend only on the modulus of E. We find a pronounced maximum of rho(|E|) at energy E=2/sqrt(3), which seems to be linked to the fixed point structure of an associated random map. We show how the density of states rho(E) can be expanded in powers of E. We find rho(|E|) = 1/pi^2 + 4/(3 pi^3) |E|^2 + ... This expansion, which seems to be asymptotic, can be carried out to an arbitrarily high order.