The adiabatic limit of the connection Laplacian

TL;DR

This paper investigates the asymptotic behavior of Laplace-type operators on vector bundles over fibered manifolds in the adiabatic limit, deriving effective operators under spectral gap conditions.

Contribution

It introduces a method to obtain asymptotics of Laplace-type operators in the adiabatic limit with a gap condition, providing effective operators for complex vector bundles.

Findings

Effective operators approximate the Laplace-type operators in the adiabatic limit.

Asymptotic expansions are valid to all orders in epsilon.

Results apply to operators with Dirichlet boundary conditions.

Abstract

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).