A new work determines exactly which gap labels are realized as open spectral gaps in the unitary almost Mathieu operator, for any irrational frequency and for all coupling constants between 0 and 1. The result covers the subcritical and supercritical regimes as well as the critical case, in which the two couplings coincide, a point that had resisted a unified treatment.

The almost Mathieu operator is one of the paradigmatic models in the spectral theory of almost-periodic operators, and its spectral gaps are labelled by topological invariants which, in the unitary case, are interpreted as winding numbers. The question of which gap labels are effectively realized as open gaps, and which collapse, is classical and has for decades conditioned the understanding of the Aubry-André transition and of localization in quasiperiodic systems.

The proof combines three ingredients: quantitative reducibility, Aubry duality, and a monotonicity principle for gap heights. The result is clean-cut: when both couplings are strictly less than one, every permitted gap label is realized as an open gap, including the critical case. When at least one coupling reaches the value one, both lifts of every odd winding label are realized as open gaps, whereas all even winding labels collapse.

The classification is exact and leaves no gap undecided in any coupling regime, thus closing an open question on the realization of gap labels in this model. The monotonicity principle for gap heights, introduced in the proof, may prove useful beyond this particular operator, in other spectral gap problems for almost-periodic operators.