The adiabatic limit of the connection Laplacian

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).