Central spectral gaps of the almost Mathieu operator

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 $c^{\prime }{q}_n^{-\varkappa /2}$, $c^{\prime }>0$.

22 pages

Cited by in corpus (6)

• Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies
• Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum
• Positive Hausdorff dimensional spectrum for the critical almost Mathieu operator
• Almost Mathieu operators with completely resonant phases
• A series of spectral gaps for the almost Mathieu operator with a small coupling constant
• Exponential Decay of the lengths of Spectral Gaps for Extended Harper's Model with Liouvillean Frequency