  We restate the adiabatic elimination approximation as the first term in a
singular perturbation expansion. We use the invariant manifold formalism for
singular perturbations in dynamical systems to identify systematic improvements
on adiabatic elimination, connecting with well established quantum mechanical
perturbation methods. We prove convergence of the expansions when energy scales
are well separated. We state and solve the problem of hermiticity of improved
effective hamiltonians.
