Impurity bands and quasi-Bloch waves for a one-dimensional model of modulated crystal

This paper investigates a simple one-dimensional model of incommensurate “harmonic crystal” in terms of the spectrum of the corresponding Schrödinger equation. Two angles of attack are studied: the first exploits techniques borrowed from the theory of quasi-periodic functions while the second relies on periodicity properties in a higher-dimensional space. It is shown that both approaches lead to essentially the same results; that is, the lower spectrum is split between “Cantor-like zones” and “impurity bands” to which correspond critical and extended eigenstates, respectively. These “new bands” seem to emerge inside the band gaps of the unperturbed problem when certain conditions are met and display a parabolic nature. Numerical tests are extensively performed on both steady and time-dependent problems.