Abstract
Along the lines of previous work, we give the general framework together with a detailed and rigorous study of the spectrum and Born-von Karman eigenstates of a 1D harmonic chain with controlled disorder determined by the Thue-Morse sequence. The spectrum is a Cantor-like set; we prove numerically that its measure is zero and calculate its Bouligand-Minkowski dimension (box dimension). We prove that the value of the IDS on each of the gaps is (2k+1)/(3·2p), withk andp integers. Finally, we also prove that points in a dense subset of the spectrum give rise to extended states, an exceptional property due to the symmetry of the Thue-Morse substitution which can have important applications to multilayered structures, and we illustrate this situation.
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Axel, F., Peyrière, J. Spectrum and extended states in a harmonic chain with controlled disorder: Effects of the Thue-Morse symmetry. J Stat Phys 57, 1013–1047 (1989). https://doi.org/10.1007/BF01020046
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DOI: https://doi.org/10.1007/BF01020046