Journal of Statistical Physics, Vol. 42, Nos. 3/4, 1986. p. 289-310.
The propagation of phonons in one-dimensional quasicrystals is investigated. We use the projection method which has been recently proposed to generate almost periodic tilings of the line. We define a natural Laplace operator on these structures, which models phonon (and also tight-binding electron) propagation. The selfsimilarity properties of the spectrum are discussed, as well as some characteristic features of the eigenstates, which are neither extended nor localized. The long-wavelength limit is examined in more detail; it is argued that one is the lower critical dimension for this type of models.