, Volume 65, Issue 3-4, pp 715-723

Power law growth for the resistance in the Fibonacci model

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Abstract

Many one-dimensional quasiperiodic systems based on the Fibonacci rule, such as the tight-binding HamiltonianHψ(n)=ψ(n+1)+ψ(n−1)+λv(n) ψ(n),nεℤ,ψεl 2(ℤ),λεℝ, wherev(n)=[(n+1)α]−[],[x] denoting the integer part ofx and α the golden mean \((\sqrt 5 --1)/2\) , give rise to the same recursion relation for the transfer matrices. It is proved that the wave functions and the norm of transfer matrices are polynomially bounded (critical regime) if and only if the energy is in the spectrum of the Hamiltonian. This solves a conjecture of Kohmoto and Sutherland on the power-law growth of the resistance in a one-dimensional quasicrystal.