Semiclassical trace formula for the Bochner-Schrödinger operator

We study the semiclassical Bochner-Schrödinger operator $H_p=\frac{1}{p^2}{\mathrm{\Delta }}^{L^p\otimes E}+V$ 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. For any function $\phi \in C_c^{\mathrm{\infty }}(\mathbb{R})$, we consider the bounded linear operator $\phi (H_p)$ in $L^2(X,L^p\otimes E)$ defined by the spectral theorem. We prove that its smooth Schwartz kernel on the diagonal admits a complete asymptotic expansion in powers of $p^{-1}$ in the semiclassical limit $p\to \mathrm{\infty }$. In particular, when the manifold is compact, we get a complete asymptotic expansion for the trace of $\phi (H_p)$.