The Dry Ten Martini Problem at Criticality
The paper addresses a problem related to gaps in the spectrum of an almost Mathieu operator, specifically showing that every labeled gap is open for any non-zero coupling $\lambda$ and irrational frequency. It proves that the maximum logarithmic height of each gap increases with $\lambda$, extending initial open gaps to all positive couplings. This monotonicity is established by pairing the operator with its magnetic inverse, linking it to a non-negative operator expectation value.