  We study asymptotic spectral properties of the Bochner-Schr\"odinger operator
$H_{p}=\frac 1p\Delta^{L^p\otimes E}+V$ on high tensor powers of a Hermitian
line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian
manifold $X$ of bounded geometry under assumption that the curvature form of
$L$ is non-degenerate. At an arbitrary point $x_0$ of $X$ the operator $H_p$
can be approximated by a model operator $\mathcal H^{(x_0)}$, which is a
Schr\"odinger operator with constant magnetic field. For large $p$, the
spectrum of $H_p$ asymptotically coincides, up to order $p^{-1/4}$, with the
union of the spectra of the model operators $\mathcal H^{(x_0)}$ over $X$. We
show that, if the union of the spectra of $\mathcal H^{(x_0)}$ over the
complement of a compact subset of $X$ has a gap, then the spectrum of $H_{p}$
in the gap is discrete and the corresponding eigensections decay exponentially
away the compact subset.
