  The Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta^{L^p\otimes E}+V$ on
tensor powers $L^p$ of a Hermitian line bundle $L$ twisted by a Hermitian
vector bundle $E$ on a Riemannian manifold of bounded geometry is studied. For
any function $\varphi\in \mathcal S(\mathbb R)$, we consider the bounded linear
operator $\varphi(H_p)$ in $L^2(X,L^p\otimes E)$ defined by the spectral
theorem and describe an asymptotic expansion of its smooth Schwartz kernel in a
fixed neighborhood of the diagonal in the semiclassical limit $p\to \infty$. In
particular, we prove that the trace of the operator $\varphi(H_p)$ admits a
complete asymptotic expansion in powers of $p^{-1/2}$ as $p\to \infty$.
