Spectrum of 1D quasicrystals near the periodic chain

A tight binding model on the general 1D quasiperiodic chain is studied in the framework of perturbation theory, near the corresponding periodic chain, using a new set of coordinates. The main gaps are well described, whereas the very small ones are correctly given, only for a very small perturbation. For a given irrational number, the energies where the gaps appear in the periodic chain spectrum, are exactly derived. Moreover, a labelling for these gaps which orders them according to their decreasing width is naturally introduced, and an approached integrated density of states is explicitely written. As an application of this perturbative derivation, we give the first order expansion of δ the exponant which describes the vanishing of the total band width B of an approximant, when its size increases : B ∼ 1/n δ. The first order expression for δ does not depend on the considered quasiperiodic chain.