Singular behavior of the density of states and the Lyapunov coefficient in binary random harmonic chains

Abstract

We study the integrated density of statesH(ω2) of a chain of harmonic oscillators with a binary random distribution of the masses. We show in particular that there is a dense set of values of the squared frequency for which the differenceH(ω2+ɛ)-H(ω2) has a singularity of the type ¦ɛ¦2α, multiplied by a periodic function of ln ¦ɛ¦, where the exponent α and the period depend continuously onω2. In the region where α < 1/2,H is not differentiate on a dense set of points. The same type of singularities is also present in the Lyapunov coefficient.