Lyapunov Exponent and Density of States of a One-Dimensional Non-Hermitian Schrödinger Equation

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