THE BORN-OPPENHEIMER APPROXIMATION - STRAIGHT-UP AND WITH A TWIST

The problem of calculating asymptotic series for low-lying eigennvalue s of Schrodinger 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 H-epsilon = -epsilon( 4) d2/dx2 + H(x) in L-2(R,H) where H(x) : H --> H has a simple eigenva lue less than zero. The second model considered is a more specific fam ily H-epsilon = -epsilon(4) Delta + H(r,w) in L-2(R-3,C-2) where the e igenprojection P(w) of H(r,w) : C-2 --> C-2 is associated with a non-t rivial, or '''cwisted,'' fibre bundle. The main tools are a pair of th eorems that allow asymptotic series for eigenvalues to be corrected te rm by term when a family of operators is perturbed.