On the Born-Oppenheimer approximation for excited states of polyatomic Schrödinger operators

Abstract

In this Letter, excited states of polyatomic molecular Schrödinger operators are investigated with the help of the Born-Oppenheimer approximation. The ratio of electronic and nuclear mass plays the role of a semi-classical parameter h ². Asymptotic series of eigenvalues at the bottom of the spectrum are constructed up to any order in h. Mathematically, this leads to the discussion of the semi-classical limit of pseudo-differential operators with the principal symbol po(x,ξ) = ξ ² + λ, where λ has a degenerate minimum (a whole manifold).