A Series of Spectral Gaps for the Almost Mathieu Operator with a Small Coupling Constant

For the almost Mathieu operator with a small coupling constant $$\lambda$$ , for a series of spectral gaps, we describe the asymptotic locations of the gaps and get lower bounds for their lengths. The number of the gaps we consider can be of the order of $$\ln 1/\lambda$$ , and the length of the $$k$$ th gap is roughly of the order of $$\lambda^k$$ .