Semiclassical Asymptotic Expansions for Functions of the Bochner–Schrödinger Operator

The Bochner–Schrödinger 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 \(p\to \infty\). We also prove a result on the asymptotic localization of the Schwartz kernel of the spectral projection on the diagonal in the case when the curvature is of full rank.