Special frequencies and Lifshitz singularities in binary random harmonic chains

The authors consider a one-dimensional chain of coupled harmonic oscillators; the mass of each atom is a random variable taking only two values (M or 1). They investigate the integrated density of states H(omega2) near special frequencies: a given frequency omega/sub s/ with rational wavelength becomes special if the mass ratio M exceeds a certain critical value M/sub c/. They show that H has essential singularities of the types H/sub sg/∼ exp(-C1 absolute value of omega2-omega/sub s/2/sup 1/2/) or exp(-C2absolute value of omega2-omega/sub s/2-1), according to the value of M and the sign of (omega2-omega/sub s/2). The Lifshitz singularity at the band edge is analyzed in the same way. In each case, the constant C1 or C2 is evaluated explicitly and compared with a vast amount of numerical work. All these exponential singularities are modulated by periodic amplitudes. The properties of the eigenfunctions with frequencies close to the special values are also discussed, and are illustrated by numerical data