Semiclassical trace formula for the Bochner-Schrödinger operator
Yuri A. Kordyukov
TL;DR
This work develops a semiclassical trace framework for the Bochner-Schrödinger operator $H_p=\frac{1}{p^2}\Delta^{L^p\otimes E}+V$ on manifolds of bounded geometry. By localizing to normal coordinates, rescaling, and applying Dynkin-Helffer-Sjöstrand functional calculus, the authors obtain a complete near-diagonal asymptotic expansion of the Schwartz kernel $K_{\varphi(H_p)}$ as $p\to\infty$, with diagonal coefficients $f_r(x_0)$ expressed as universal polynomials in derivatives of $V$ and the curvatures. In the compact case this yields a full trace expansion $\operatorname{tr}\varphi(H_p)\sim p^{d}\sum_{r=0}^{\infty}f_r(\varphi) p^{-r}$, where the coefficients are local and gauge-invariant. The approach avoids magnetic pseudodifferential calculus by adapting Bergman-kernel localization techniques and provides explicit formulas for $f_1$ and $f_2$ in the Euclidean gauge model, illustrating the structure of the coefficients. These results extend prior magnetic trace formulas to a broad geometric setting and offer a robust tool for semiclassical spectral analysis on manifolds with bounded geometry.
Abstract
We study the semiclassical Bochner-Schrödinger operator $H_{p}=\frac{1}{p^2}Δ^{L^p\otimes E}+V$ 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 $\varphi\in C^\infty_c(\mathbb R)$, we consider the bounded linear operator $\varphi(H_p)$ 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 $p\to \infty$. In particular, when the manifold is compact, we get a complete asymptotic expansion for the trace of $\varphi(H_p)$.