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

We consider Schr"odinger operators $H=-{\mathrm{\Delta }}_{g_{\epsilon }}+V$ on a fibre bundle $MB$ with compact fibres and a metric $g_{\epsilon }$ that blows up directions perpendicular to the fibres by a factor ${\epsilon }^{-1}\gg 1$. 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}$$thereexistsasubspaceof$L^2(M)$thatisinvariantunder$H$ up to errors of order ${\epsilon }^{N+1}$. The dynamical and spectral features of $H$ on this subspace can be described by an effective operator on the fibre-wise $\lambda$-eigenspace bundle $\mathcal{E}\to B$, giving detailed asymptotics for $H$.