Abstract
The Born–Oppenheimer approximation can generally be applied when a quantum system is coupled with another comparatively slower system which is treated classically: for a fixed classical state, one considers a stationary quantum vector of the quantum system. Geometrically, this gives a vector bundle over the classical phase space of the slow motion. The topology of this bundle is characterized by integral Chern classes. In the case where the whole system is isolated with a discrete energy spectrum, we show that these integers have a direct manifestation in the qualitative structure of this spectrum: the spectrum is formed by groups of levels and these integers determine the precise number of levels in each group.
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Faure, F., Zhilinskii, B. Topological Properties of the Born–Oppenheimer Approximation and Implications for the Exact Spectrum. Letters in Mathematical Physics 55, 219–238 (2001). https://doi.org/10.1023/A:1010912815438
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DOI: https://doi.org/10.1023/A:1010912815438