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

Faure, Frédéric

Annales Henri Lebesgue, Tome 6 (2023), pp. 449-492

Using semi-classical analysis in ${ℝ}^n$ we present a quite general model for which the topological index formula of Atiyah–Singer predicts a spectral flow with the transition of a finite number of eigenvalues between clusters (energy bands). This model corresponds to physical phenomena that are well observed for quantum energy levels of small molecules [FZ00, FZ01], also in geophysics for the oceanic or atmospheric equatorial waves [DMV17, Mat66] and expected to be observed in plasma physics [QF22].

En utilisant l’analyse semi-classique dans ${ℝ}^n$ nous présentons un modèle assez général pour lequel la formule de l’indice topologique d’Atiyah–Singer prédit un flot spectral avec la transition d’un nombre fini de valeurs propres entre des clusters (bandes) d’énergie. Ce modèle correspond à des phénomènes physiques qui sont bien observés pour les niveaux d’énergie quantiques de petites molécules [FZ00, FZ01], également en géophysique pour les ondes équatoriales océaniques ou atmosphériques [DMV17, Mat66] et que l’on s’attend à observer en physique des plasmas [QF22].

Classification : 35Q86, 81V55, 55R50, 47A53, 19K56, 81Q20, 81Q05
Keywords: PDEs in connection with geophysics, Molecular physics, K-theory, Fredholm operators, index theories, Semiclassical techniques

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