The Born-Oppenheimer Expansion: Eigenvalues, Eigenfunctions and Low-Energy Scattering
In these notes I want to review some work of A. Martinez, X.P. Wang, R. Seiler and myself on the rigorous justification of the Born-Oppenheimer approximation [10, 11, 12]. The results concern firstly complete asymptotic expansions of eigenvalues and eigenfunctions of WKB-type for the full quantum mechanical Hamiltonian of a polyatomic molecule. We denote the semiclassical expansion parameter by h; its square is essentially the quotient of electronic to nuclear mass. Already Born and Oppenheimer considered (formally) complete asymptotic expansions which, however, did not include the exponential weights of WKB-type expansions describing the exponential localisation of eigenfunctions near the bottom of the potential well formed by the first eigenvalue of the electronic Hamiltonian. In honour of their seminal paper we call our results BO-expansions, although some authors might use the word Born-Oppenheimer approximation in some much more restricted sense.
KeywordsFormal Power Series Diatomic Molecule Pseudo Differential Operator Wave Operator Spectral Projection
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