  We study the Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta^{L^p\otimes
E}+V$ on high tensor powers of a positive line bundle $L$ on a symplectic
manifold of bounded geometry. First, we give a rough asymptotic description of
its spectrum in terms of the spectra of the model operators. This allows us to
prove the existence of gaps in the spectrum under some conditions on the
curvature of the line bundle. Then we consider the spectral projection of such
an operator corresponding to an interval whose extreme points are not in the
spectrum and study asymptotic behavior of its kernel. First, we establish the
off-diagonal exponential estimate. Then we state a complete asymptotic
expansion in a fixed neighborhood of the diagonal.
