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 elimin
ation, 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.
We present a general method, based on a multiple-scale approach, for deriving the perturbative solutions of the scaling equations governing the expansion of superfluid ultracold quantum gases released from elongated harmonic traps. We discuss how to
treat the secular terms appearing in the usual naive expansion in the trap asymmetry parameter epsilon, and calculate the next-to-leading correction for the asymptotic aspect ratio, with significant improvement over the previous proposals.
