The adiabatic limit of Schr\'odinger operators on fibre bundles

We consider Schrodinger operators $$H=-\Delta _{g_\varepsilon } + V$$ on a fibre bundle $$M\mathop {\rightarrow }\limits ^{\pi }B$$ with compact fibres and a metric $$g_\varepsilon $$ that blows up directions perpendicular to the fibres by a factor $${\varepsilon ^{-1}\gg 1}$$ . We show that for an eigenvalue $$\lambda $$ of the fibre-wise part of H, satisfying a local gap condition, and every $$N\in \mathbb {N}$$ there exists a subspace of $$L^2(M)$$ that is invariant under H up to errors of order $$\varepsilon ^{N+1}$$ . The dynamical and spectral features of H on this subspace can be described by an effective operator on the fibre-wise $$\lambda $$ -eigenspace bundle $$\mathcal {E}\rightarrow B$$ , giving detailed asymptotics for H.