Jon, I've looked at p.360 and the following calculations but I did not see a theorem, if you meant that it should be possible to prove a theorem using the calculations in those pages, I am actually looking for a source where this is done. As for the two-scale approach being "well-supported by the adiabatic approximation", I'm not sure whether you are making a plausibility argument or claiming it should be possible to give proofs along those lines. If it is the latter case, once again, I would love to see a book/paper where this is done.

Thanks, Jon. I did a quick scan of Kevorkian and Cole and did not immediately see theorems of the sort I've mentioned (i.e. guarantees of accuracy). Do they have such theorems in the book?

Various nice properties (e.g., orthogonality) of the eigenvectors follow easily from the Hermiticity of $A$ and $B$, and I was hoping that the perturbation theory would be better-behaved in this form. The general theory of perturbations of eigenvalues (of a possibly non-normal matrix, which $B^{-1}A$ may be) seemed a little scary, but maybe it will turn out to be ok, and I may end up trying that. The case of positive-definite $B$ was an almost trivial generalization of the usual perturbation theory for a Hermitian matrix with all the usual niceties, so I was hoping to have something similar.

@Steve Huntsman, I had read parts of Barenblatt's book years ago (I remember it as a fun and insightful book), does it have a discussion of the method of multiple scales, or did you cite it as a reference of general relevance along these lines?