Transport and spectral properties of strongly disordered chains

As a function of a parameter α, measuring the amount of disorder, we derive asymptotic results (including higher corrections) for the density of eigenstates scrN(E) and the inverse localization length γ(E) at frequencies ${\omega }^2$=E→0 in a harmonic chain with random masses. The same is done for the frequency-dependent diffusion coefficient $D^{\mathrm{}\left(0\right)\mathrm{}}$(z) and Burnett coefficient $D^{\mathrm{}\left(2\right)\mathrm{}}$(z) as z→0 in the hopping random barrier (RBM) and random jump rate models (RJM) with static disorder (bond and site problem, respectively). The disorder is described by random jump rates $w_n$ or random masses $m_n$=1/$w_n$ with a probability distribution ρ(w)=(1-α)$w^{\mathrm{-}\alpha }$CTHETA(1-w) with α<1. With increasing disorder, i.e., increasing α, the above quantities pass from universal behavior in weakly disordered systems through several regions of nonuniversal behavior. A typical result is the long-time tail of the velocity autocorrelation function in the RBM: & with ${\delta }_0$=2/(2-α) for 0<α<1, ${\delta }_0$=1-α/2 for -1<α<0, and ${\delta }_0$=(3/2) (universal) for α<-1, with crossover behavior ${\phi }^{\mathrm{}\left(0\right)\mathrm{}}$(t)∼$t^{\mathrm{-}1}$ (ln t${)}^{\mathrm{-}2}$ at α=0 and ${\phi }_0$(t)∼$t^{\mathrm{-}3/2}$ ln t at α=-1.