  We consider Schr\"odinger operators $H=-\Delta_{g_\varepsilon} + V$ on a
fibre bundle $M\stackrel{\pi}{\to}B$ 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 $\mathcal{E}\to B$, giving detailed
asymptotics for $H$.
