Central spectral gaps of the almost Mathieu operator

TL;DR

This paper investigates the spectral properties of the almost Mathieu operator at critical coupling, demonstrating that certain spectral gaps are inherited from rational approximations of the frequency with explicit bounds.

Contribution

It establishes conditions under which the central spectral gaps of rational approximations persist in the irrational case, providing explicit lower bounds on their lengths.

Findings

Spectral gaps of rational approximations are inherited in the irrational case.

Explicit lower bounds on the size of inherited spectral gaps.

Conditions on continued fraction coefficients influence gap inheritance.

Abstract

We consider the spectrum of the almost Mathieu operator $H_\alpha$ with frequency $\alpha$ and in the case of the critical coupling. Let an irrational $\alpha$ be such that $|\alpha-p_n/q_n|<c q_n^{-\varkappa}$, where $p_n/q_n$, $n=1,2,\dots$ are the convergents to $\alpha$, and $c$, $\varkappa$ are positive absolute constants, $\varkappa<56$. Assuming certain conditions on the parity of the coefficients of the continued fraction of $\alpha$, we show that the central gaps of $H_{p_n/q_n}$, $n=1,2,\dots$, are inherited as spectral gaps of $H_\alpha$ of length at least $c'q_n^{-\varkappa/2}$, $c'>0$.