Quasiperiodic Modulated-Spring Model

TL;DR

This paper investigates a quasiperiodic spring chain model, revealing that the vibrational spectrum transitions from extended to critical modes as the modulation amplitude increases, with analytical and numerical evidence.

Contribution

It provides an analytical proof of spectrum nature for low modulation and numerical analysis at the critical point, advancing understanding of quasiperiodic vibrational systems.

Findings

Spectrum is absolutely continuous for Δ<1.

Spectrum becomes purely singular continuous at Δ=1.

All modes are extended or critical depending on modulation amplitude.

Abstract

We study the classical vibration problem of a chain with spring constants which are modulated in a quasiperiodic manner, {\it i. e.}, a model in which the elastic energy is $\sum_j k_j( u_{j-1}-{u_j})^2$, where $k_j=1+\Delta cos[2\pi\sigma(j-1/2)+\theta]$ and $\sigma$ is an irrational number. For $\Delta < 1$, it is shown analytically that the spectrum is absolutely continuous, {\it i.e.}, all the eigen modes are extended. For $\Delta=1$, numerical scaling analysis shows that the spectrum is purely singular continuous, {\it i.e.}, all the modes are critical.