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

Using semi-classical analysis in $\mathbb{R}^{n}$ we present a quite general model for which the topological index formula of Atiyah-Singer predicts a spectral flow with a transition of a finite number of eigenvalues transitions between clusters (energy bands). This model corresponds to physical phenomena that are well observed for quantum energy levels of small molecules [faure_zhilinskii_2000,2001] but also in geophysics for the oceanic or atmospheric equatorial waves [Matsuno_1966, Delplace_Marston_Venaille_2017].