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Using semi-classical analysis in $\mathbb{R}^{n}$ we present a quite generalmodel for which the topological index formula of Atiyah-Singer predicts aspectral flow with the transition of a finite number of eigenvalues betweenclusters (energy bands). This model corresponds to physical phenomena that arewell observed for quantum energy levels of small molecules[faure_zhilinskii_2000,2001], also in geophysics for the oceanic or atmosphericequatorial waves [Matsuno_1966, Delplace_Marston_Venaille_2017] and expected tobe observed in plasma physics [Qin, Fu 2022].

Manifestation of the topological index formula in quantum waves and geophysical waves