Central Spectral Gaps of the Almost Mathieu Operator

Abstract

We consider the spectrum of the almost Mathieu operator Hα with frequency α and in the case of the critical coupling. Let an irrational α be such that |α − pn/qn| < cq−κ n , where pn/qn, n = 1, 2, . . . are the convergents to α, and c, κ are positive absolute constants, κ < 56. Assuming certain conditions on the parity of the coefficients of the continued fraction of α, we show that the central gaps of Hpn/qn , n = 1, 2, . . . , are inherited as spectral gaps of Hα of length at least c 0 q −κ/2 n , c 0 > 0.