  We study the behaviour of Laplace-type operators H on a complex vector bundle
E $\rightarrow$ M in the adiabatic limit of the base space. This space is a
fibre bundle M $\rightarrow$ B with compact fibres and the limit corresponds to
blowing up directions perpendicular to the fibres by a factor 1/$\epsilon$.
Under a gap condition on the fibre-wise eigenvalues we prove existence of
effective operators that provide asymptotics to any order in $\epsilon$ for H
(with Dirichlet boundary conditions), on an appropriate almost-invariant
subspace of L${}^2$(E).
