The Born–Oppenheimer Approximation: Straight-Up and with a Twist

The problem of calculating asymptotic series for low-lying eigennvalues of Schrödinger operators is solved for two classes of such operators. For both models, a version of the Born–Oppenheimer Approximation is proven. The first model considered is the family [Formula: see text] in L2(ℝ,ℋ) where H(x):ℋ→ℋ has a simple eigenvalue less than zero. The second model considered is a more specific family ℍε=-ε4Δ+H(r,ω) in [Formula: see text] where the eigenprojection P(ω) of [Formula: see text] is associated with a non-trivial, or "twisted," fibre bundle. The main tools are a pair of theorems that allow asymptotic series for eigenvalues to be corrected term by term when a family of operators is perturbed.