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Periodic and quasiperiodic wavefunctions in a class of one-dimensional quasicrystals: an analytical treatment

The one-dimensional, discrete Schrodinger equation is studied analytically for a class of quasiperiodic Hamiltonians known as the copper mean. The lattice is described by a recursion formula SL+1=SL-1SL-1SL, L=2,3, . . ., given the two initial sequences S0 and S1. Extended states are shown to exist for energies satisfying TrTL=0 and Tr((T0)-2T1)=2 where TL is the transfer matrix for the Lth generation of the quasicrystal. Also, periodic states are shown to exist quite generally in a subclass of the copper mean. A specific one-dimensional quasicrystal is given as an example of this, and is shown to have exclusively periodic states.