Analysis on the derivative of the discriminant polynomial for the critical almost Mathieu operator

Abstract

This article first introduces the discriminant polynomial and the central spectral band of the critical almost Mathieu operator. The Hardy–Littlewood method and the symbolic coding of continued fractions are introduced and applied to estimate the first-order and the second-order derivatives of the discriminant polynomial. Finally, these results are applied to obtain a lower bound for the measure of the central spectral band.