Generalized Aubry-Andr\'e formula and continuity of the intersection spectrum of the Almost Mathieu operator

TL;DR

This paper generalizes the Aubry-Andre9 formula for the Almost Mathieu operator's spectrum, showing polynomial moments of the Lebesgue measure restriction depend analytically on parameters, with implications for spectral measure continuity.

Contribution

It introduces a generalized formula for the spectrum measure of the AMO, establishing its polynomial dependence on coupling and continuity and smoothness properties of the spectral measure.

Findings

Moments of the Lebesgue measure restriction are polynomials in  with trigonometric polynomial coefficients.

The measure restriction _{,} depends continuously on  and  in weak-* topology.

The measure's dependence on  and  is analytic in  and , respectively.

Abstract

We consider the spectrum of the Almost Mathieu operator (AMO) and show that the moments of the restriction of the Lebesgue measure to the intersection spectrum $\text{Leb}|_{\Sigma_{\alpha,\lambda}}$ are polynomials in coupling $\lambda$ with coefficients that are trigonometric polynomials in frequency $\alpha$. The statement can be considered as a generalization of the Aubry-Andr\'e formula for the measure of the spectrum of AMO. As a corollary, we obtain that the restriction of the Lebesgue measure to the intersection spectrum that we denote by $\mu^{-}_{\alpha, \lambda}$ depends continuously on the parameters (frequency $\alpha$ and coupling $\lambda$) in weak-* topology. Moreover, we prove that the dependence is not just continuous but analytic in $\lambda$ and $C^{\infty}$ in $\alpha$ in a sense that an integral of an analytic test function $\varphi(x)$ with respect to $\mu^{-}_{\alpha, \lambda}$ has the same kind of dependence. In particular, this implies that the Lebesgue measure of the part of the spectrum $\Sigma_{\alpha,\lambda}$ that lies between two gaps depends analytically on the coupling constant $\lambda$ and $C^{\infty}$ on the frequency $\alpha$ in an open domain (away from the critical coupling $\lambda=1$) where these gaps do not bifurcate.