Abstract
We 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). We investigate the integrated density of statesH(ω2) near special frequencies: a given frequencyω s with rational wavelength becomes “special” if the mass ratioM exceeds a certain critical valueM c . We show thatH has essential singularities of the typesH sg ∼ exp(−C 1 ¦ω2−ω 2 s ¦−1/2) or exp(−C 2¦ω2−ω 2 s ¦−1), according to the value ofM and the sign of (ω2−ω 2 s ). The Lifshitz singularity at the band edge is analyzed in the same way. In each case, the constantC 1 orC 2 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.
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Nieuwenhuizen, T.M., Luck, J.M., Canisius, J. et al. Special frequencies and Lifshitz singularities in binary random harmonic chains. J Stat Phys 45, 395–417 (1986). https://doi.org/10.1007/BF01021078
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DOI: https://doi.org/10.1007/BF01021078