The Adiabatic Limit of the Connection Laplacian

We study the behaviour of Laplace-type operators H on a complex vector bundle EM in the adiabatic limit of the base space. This space is a fibre bundle MB with compact fibres and the limit corresponds to blowing up directions perpendicular to the fibres by a factor epsilon-1 >> 1. Under a gap condition on the fibrewise eigenvalues, we prove the 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 L2(E).