Quasiperiodic Modulated-Spring Model

Abstract

We study the classical vibration problem of a chain with spring constants which are modulated in a quasiperiodic manner, i.e., a model in which the elastic energy is ∑ j k j ( u j -1 - u j ) 2 , where k j =1+Δcos [2πσ( j -1/2)+θ] and σ is an irrational number. For Δ < 1, it is shown analytically that the spectrum is absolutely continuous, i.e., all the eigen modes are extended. For Δ=1, numerical scaling analysis shows that the spectrum is purely singular continuous, i.e., all the modes are critical.