Topological properties of the Born-Oppenheimer approximation and implications for the exact spectrum

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 quant um vector of the quantum system. Geometrically, this gives a vector bundle over the classical phase space of the slow motion. The topology of this bun dle is characterized by integral Chern classes. In the case where the whole system is isolated with a discrete energy spectrum, we show that these int egers have a direct manifestation in the qualitative structure of this spec trum: the spectrum is formed by groups of levels and these integers determi ne the precise number of levels in each group.