标题： Bochner-Schrödinger算子函数的半经典渐近展开 显示英文标题

标题： Semiclassical asymptotic expansions for functions of the Bochner-Schrödinger operator

摘要： 博赫纳-薛定谔算子$H_{p}=\frac 1p\Delta^{L^p\otimes E}+V$在由一个埃尔米特线丛$L$扭转的埃尔米特向量丛$E$的张量幂$L^p$上在有界几何的黎曼流形上被研究。 对于任何函数$\varphi\in \mathcal S(\mathbb R)$，我们考虑在$L^2(X,L^p\otimes E)$中由谱定理定义的有界线性算子$\varphi(H_p)$，并描述其在半经典极限$p\to \infty$下在对角线固定邻域中的光滑 Schwartz 核的渐近展开式。 特别地，我们证明算子$\varphi(H_p)$的迹在$p\to \infty$时在$p^{-1/2}$的幂次中 admits 完全渐近展开式。 显示英文摘要

摘要： 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 $p\to \infty$.