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

Derrida, Bernard; Jacobsen, Jesper Lykke; Zeitak, Reuven

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

Authors

Derrida, Bernard

Jacobsen, Jesper Lykke

Zeitak, Reuven

Description

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.
25 pages including 7 figures